Metamath Proof Explorer |
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Ref | Description |
a1ii 1 | (_Note_: This inference r... |
idi 2 | This inference, which requ... |
mp2 9 | A double modus ponens infe... |
mp2b 10 | A double modus ponens infe... |
a1i 11 | Inference introducing an a... |
2a1i 12 | Inference introducing two ... |
mp1i 13 | Inference detaching an ant... |
a2i 14 | Inference distributing an ... |
mpd 15 | A modus ponens deduction. ... |
imim2i 16 | Inference adding common an... |
syl 17 | An inference version of th... |
3syl 18 | Inference chaining two syl... |
4syl 19 | Inference chaining three s... |
mpi 20 | A nested modus ponens infe... |
mpisyl 21 | A syllogism combined with ... |
id 22 | Principle of identity. Th... |
idALT 23 | Alternate proof of ~ id . ... |
idd 24 | Principle of identity ~ id... |
a1d 25 | Deduction introducing an e... |
2a1d 26 | Deduction introducing two ... |
a1i13 27 | Add two antecedents to a w... |
2a1 28 | A double form of ~ ax-1 . ... |
a2d 29 | Deduction distributing an ... |
sylcom 30 | Syllogism inference with c... |
syl5com 31 | Syllogism inference with c... |
com12 32 | Inference that swaps (comm... |
syl5 33 | A syllogism rule of infere... |
syl6 34 | A syllogism rule of infere... |
syl56 35 | Combine ~ syl5 and ~ syl6 ... |
syl6com 36 | Syllogism inference with c... |
mpcom 37 | Modus ponens inference wit... |
syli 38 | Syllogism inference with c... |
syl2im 39 | Replace two antecedents. ... |
syl2imc 40 | A commuted version of ~ sy... |
pm2.27 41 | This theorem, called "Asse... |
mpdd 42 | A nested modus ponens dedu... |
mpid 43 | A nested modus ponens dedu... |
mpdi 44 | A nested modus ponens dedu... |
mpii 45 | A doubly nested modus pone... |
syld 46 | Syllogism deduction. Dedu... |
mp2d 47 | A double modus ponens dedu... |
a1dd 48 | Double deduction introduci... |
2a1dd 49 | Double deduction introduci... |
pm2.43i 50 | Inference absorbing redund... |
pm2.43d 51 | Deduction absorbing redund... |
pm2.43a 52 | Inference absorbing redund... |
pm2.43b 53 | Inference absorbing redund... |
pm2.43 54 | Absorption of redundant an... |
imim2d 55 | Deduction adding nested an... |
imim2 56 | A closed form of syllogism... |
embantd 57 | Deduction embedding an ant... |
3syld 58 | Triple syllogism deduction... |
sylsyld 59 | A double syllogism inferen... |
imim12i 60 | Inference joining two impl... |
imim1i 61 | Inference adding common co... |
imim3i 62 | Inference adding three nes... |
sylc 63 | A syllogism inference comb... |
syl3c 64 | A syllogism inference comb... |
syl6mpi 65 | A syllogism inference. (C... |
mpsyl 66 | Modus ponens combined with... |
mpsylsyld 67 | Modus ponens combined with... |
syl6c 68 | Inference combining ~ syl6... |
syl6ci 69 | A syllogism inference comb... |
syldd 70 | Nested syllogism deduction... |
syl5d 71 | A nested syllogism deducti... |
syl7 72 | A syllogism rule of infere... |
syl6d 73 | A nested syllogism deducti... |
syl8 74 | A syllogism rule of infere... |
syl9 75 | A nested syllogism inferen... |
syl9r 76 | A nested syllogism inferen... |
syl10 77 | A nested syllogism inferen... |
a1ddd 78 | Triple deduction introduci... |
imim12d 79 | Deduction combining antece... |
imim1d 80 | Deduction adding nested co... |
imim1 81 | A closed form of syllogism... |
pm2.83 82 | Theorem *2.83 of [Whitehea... |
peirceroll 83 | Over minimal implicational... |
com23 84 | Commutation of antecedents... |
com3r 85 | Commutation of antecedents... |
com13 86 | Commutation of antecedents... |
com3l 87 | Commutation of antecedents... |
pm2.04 88 | Swap antecedents. Theorem... |
com34 89 | Commutation of antecedents... |
com4l 90 | Commutation of antecedents... |
com4t 91 | Commutation of antecedents... |
com4r 92 | Commutation of antecedents... |
com24 93 | Commutation of antecedents... |
com14 94 | Commutation of antecedents... |
com45 95 | Commutation of antecedents... |
com35 96 | Commutation of antecedents... |
com25 97 | Commutation of antecedents... |
com5l 98 | Commutation of antecedents... |
com15 99 | Commutation of antecedents... |
com52l 100 | Commutation of antecedents... |
com52r 101 | Commutation of antecedents... |
com5r 102 | Commutation of antecedents... |
imim12 103 | Closed form of ~ imim12i a... |
jarr 104 | Elimination of a nested an... |
pm2.86d 105 | Deduction associated with ... |
pm2.86 106 | Converse of axiom ~ ax-2 .... |
pm2.86i 107 | Inference associated with ... |
pm2.86iALT 108 | Alternate proof of ~ pm2.8... |
loolin 109 | The Linearity Axiom of the... |
loowoz 110 | An alternate for the Linea... |
con4 111 | Alias for ~ ax-3 to be use... |
con4i 112 | Inference associated with ... |
con4d 113 | Deduction associated with ... |
mt4 114 | The rule of modus tollens.... |
pm2.21i 115 | A contradiction implies an... |
pm2.24ii 116 | A contradiction implies an... |
pm2.21d 117 | A contradiction implies an... |
pm2.21ddALT 118 | Alternate proof of ~ pm2.2... |
pm2.21 119 | From a wff and its negatio... |
pm2.24 120 | Theorem *2.24 of [Whitehea... |
pm2.18 121 | Proof by contradiction. T... |
pm2.18i 122 | Inference associated with ... |
pm2.18d 123 | Deduction based on reducti... |
notnotr 124 | Double negation eliminatio... |
notnotri 125 | Inference associated with ... |
notnotriOLD 126 | Obsolete proof of ~ notnot... |
notnotrd 127 | Deduction associated with ... |
con2d 128 | A contraposition deduction... |
con2 129 | Contraposition. Theorem *... |
mt2d 130 | Modus tollens deduction. ... |
mt2i 131 | Modus tollens inference. ... |
nsyl3 132 | A negated syllogism infere... |
con2i 133 | A contraposition inference... |
nsyl 134 | A negated syllogism infere... |
notnot 135 | Double negation introducti... |
notnoti 136 | Inference associated with ... |
notnotd 137 | Deduction associated with ... |
con1d 138 | A contraposition deduction... |
mt3d 139 | Modus tollens deduction. ... |
mt3i 140 | Modus tollens inference. ... |
nsyl2 141 | A negated syllogism infere... |
con1 142 | Contraposition. Theorem *... |
con1i 143 | A contraposition inference... |
con4iOLD 144 | Obsolete proof of ~ con4i ... |
pm2.24i 145 | Inference associated with ... |
pm2.24d 146 | Deduction form of ~ pm2.24... |
con3d 147 | A contraposition deduction... |
con3 148 | Contraposition. Theorem *... |
con3i 149 | A contraposition inference... |
con3rr3 150 | Rotate through consequent ... |
mt4d 151 | Modus tollens deduction. ... |
mt4i 152 | Modus tollens inference. ... |
nsyld 153 | A negated syllogism deduct... |
nsyli 154 | A negated syllogism infere... |
nsyl4 155 | A negated syllogism infere... |
pm3.2im 156 | Theorem *3.2 of [Whitehead... |
mth8 157 | Theorem 8 of [Margaris] p.... |
jc 158 | Deduction joining the cons... |
impi 159 | An importation inference. ... |
expi 160 | An exportation inference. ... |
simprim 161 | Simplification. Similar t... |
simplim 162 | Simplification. Similar t... |
pm2.5 163 | Theorem *2.5 of [Whitehead... |
pm2.51 164 | Theorem *2.51 of [Whitehea... |
pm2.521 165 | Theorem *2.521 of [Whitehe... |
pm2.52 166 | Theorem *2.52 of [Whitehea... |
expt 167 | Exportation theorem expres... |
impt 168 | Importation theorem expres... |
pm2.61d 169 | Deduction eliminating an a... |
pm2.61d1 170 | Inference eliminating an a... |
pm2.61d2 171 | Inference eliminating an a... |
ja 172 | Inference joining the ante... |
jad 173 | Deduction form of ~ ja . ... |
jarl 174 | Elimination of a nested an... |
pm2.61i 175 | Inference eliminating an a... |
pm2.61ii 176 | Inference eliminating two ... |
pm2.61nii 177 | Inference eliminating two ... |
pm2.61iii 178 | Inference eliminating thre... |
pm2.01 179 | Reductio ad absurdum. The... |
pm2.01d 180 | Deduction based on reducti... |
pm2.6 181 | Theorem *2.6 of [Whitehead... |
pm2.61 182 | Theorem *2.61 of [Whitehea... |
pm2.65 183 | Theorem *2.65 of [Whitehea... |
pm2.65i 184 | Inference rule for proof b... |
pm2.21dd 185 | A contradiction implies an... |
pm2.65d 186 | Deduction rule for proof b... |
mto 187 | The rule of modus tollens.... |
mtod 188 | Modus tollens deduction. ... |
mtoi 189 | Modus tollens inference. ... |
mt2 190 | A rule similar to modus to... |
mt3 191 | A rule similar to modus to... |
peirce 192 | Peirce's axiom. This odd-... |
looinv 193 | The Inversion Axiom of the... |
bijust 194 | Theorem used to justify de... |
impbi 197 | Property of the biconditio... |
impbii 198 | Infer an equivalence from ... |
impbidd 199 | Deduce an equivalence from... |
impbid21d 200 | Deduce an equivalence from... |
impbid 201 | Deduce an equivalence from... |
dfbi1 202 | Relate the biconditional c... |
dfbi1ALT 203 | Alternate proof of ~ dfbi1... |
biimp 204 | Property of the biconditio... |
biimpi 205 | Infer an implication from ... |
sylbi 206 | A mixed syllogism inferenc... |
sylib 207 | A mixed syllogism inferenc... |
sylbb 208 | A mixed syllogism inferenc... |
biimpr 209 | Property of the biconditio... |
bicom1 210 | Commutative law for the bi... |
bicom 211 | Commutative law for the bi... |
bicomd 212 | Commute two sides of a bic... |
bicomi 213 | Inference from commutative... |
impbid1 214 | Infer an equivalence from ... |
impbid2 215 | Infer an equivalence from ... |
impcon4bid 216 | A variation on ~ impbid wi... |
biimpri 217 | Infer a converse implicati... |
biimpd 218 | Deduce an implication from... |
mpbi 219 | An inference from a bicond... |
mpbir 220 | An inference from a bicond... |
mpbid 221 | A deduction from a bicondi... |
mpbii 222 | An inference from a nested... |
sylibr 223 | A mixed syllogism inferenc... |
sylbir 224 | A mixed syllogism inferenc... |
sylbbr 225 | A mixed syllogism inferenc... |
sylbb1 226 | A mixed syllogism inferenc... |
sylbb2 227 | A mixed syllogism inferenc... |
sylibd 228 | A syllogism deduction. (C... |
sylbid 229 | A syllogism deduction. (C... |
mpbidi 230 | A deduction from a bicondi... |
syl5bi 231 | A mixed syllogism inferenc... |
syl5bir 232 | A mixed syllogism inferenc... |
syl5ib 233 | A mixed syllogism inferenc... |
syl5ibcom 234 | A mixed syllogism inferenc... |
syl5ibr 235 | A mixed syllogism inferenc... |
syl5ibrcom 236 | A mixed syllogism inferenc... |
biimprd 237 | Deduce a converse implicat... |
biimpcd 238 | Deduce a commuted implicat... |
biimprcd 239 | Deduce a converse commuted... |
syl6ib 240 | A mixed syllogism inferenc... |
syl6ibr 241 | A mixed syllogism inferenc... |
syl6bi 242 | A mixed syllogism inferenc... |
syl6bir 243 | A mixed syllogism inferenc... |
syl7bi 244 | A mixed syllogism inferenc... |
syl8ib 245 | A syllogism rule of infere... |
mpbird 246 | A deduction from a bicondi... |
mpbiri 247 | An inference from a nested... |
sylibrd 248 | A syllogism deduction. (C... |
sylbird 249 | A syllogism deduction. (C... |
biid 250 | Principle of identity for ... |
biidd 251 | Principle of identity with... |
pm5.1im 252 | Two propositions are equiv... |
2th 253 | Two truths are equivalent.... |
2thd 254 | Two truths are equivalent ... |
ibi 255 | Inference that converts a ... |
ibir 256 | Inference that converts a ... |
ibd 257 | Deduction that converts a ... |
pm5.74 258 | Distribution of implicatio... |
pm5.74i 259 | Distribution of implicatio... |
pm5.74ri 260 | Distribution of implicatio... |
pm5.74d 261 | Distribution of implicatio... |
pm5.74rd 262 | Distribution of implicatio... |
bitri 263 | An inference from transiti... |
bitr2i 264 | An inference from transiti... |
bitr3i 265 | An inference from transiti... |
bitr4i 266 | An inference from transiti... |
bitrd 267 | Deduction form of ~ bitri ... |
bitr2d 268 | Deduction form of ~ bitr2i... |
bitr3d 269 | Deduction form of ~ bitr3i... |
bitr4d 270 | Deduction form of ~ bitr4i... |
syl5bb 271 | A syllogism inference from... |
syl5rbb 272 | A syllogism inference from... |
syl5bbr 273 | A syllogism inference from... |
syl5rbbr 274 | A syllogism inference from... |
syl6bb 275 | A syllogism inference from... |
syl6rbb 276 | A syllogism inference from... |
syl6bbr 277 | A syllogism inference from... |
syl6rbbr 278 | A syllogism inference from... |
3imtr3i 279 | A mixed syllogism inferenc... |
3imtr4i 280 | A mixed syllogism inferenc... |
3imtr3d 281 | More general version of ~ ... |
3imtr4d 282 | More general version of ~ ... |
3imtr3g 283 | More general version of ~ ... |
3imtr4g 284 | More general version of ~ ... |
3bitri 285 | A chained inference from t... |
3bitrri 286 | A chained inference from t... |
3bitr2i 287 | A chained inference from t... |
3bitr2ri 288 | A chained inference from t... |
3bitr3i 289 | A chained inference from t... |
3bitr3ri 290 | A chained inference from t... |
3bitr4i 291 | A chained inference from t... |
3bitr4ri 292 | A chained inference from t... |
3bitrd 293 | Deduction from transitivit... |
3bitrrd 294 | Deduction from transitivit... |
3bitr2d 295 | Deduction from transitivit... |
3bitr2rd 296 | Deduction from transitivit... |
3bitr3d 297 | Deduction from transitivit... |
3bitr3rd 298 | Deduction from transitivit... |
3bitr4d 299 | Deduction from transitivit... |
3bitr4rd 300 | Deduction from transitivit... |
3bitr3g 301 | More general version of ~ ... |
3bitr4g 302 | More general version of ~ ... |
notnotb 303 | Double negation. Theorem ... |
notnotdOLD 304 | Obsolete proof of ~ notnot... |
con34b 305 | A biconditional form of co... |
con4bid 306 | A contraposition deduction... |
notbid 307 | Deduction negating both si... |
notbi 308 | Contraposition. Theorem *... |
notbii 309 | Negate both sides of a log... |
con4bii 310 | A contraposition inference... |
mtbi 311 | An inference from a bicond... |
mtbir 312 | An inference from a bicond... |
mtbid 313 | A deduction from a bicondi... |
mtbird 314 | A deduction from a bicondi... |
mtbii 315 | An inference from a bicond... |
mtbiri 316 | An inference from a bicond... |
sylnib 317 | A mixed syllogism inferenc... |
sylnibr 318 | A mixed syllogism inferenc... |
sylnbi 319 | A mixed syllogism inferenc... |
sylnbir 320 | A mixed syllogism inferenc... |
xchnxbi 321 | Replacement of a subexpres... |
xchnxbir 322 | Replacement of a subexpres... |
xchbinx 323 | Replacement of a subexpres... |
xchbinxr 324 | Replacement of a subexpres... |
imbi2i 325 | Introduce an antecedent to... |
bibi2i 326 | Inference adding a bicondi... |
bibi1i 327 | Inference adding a bicondi... |
bibi12i 328 | The equivalence of two equ... |
imbi2d 329 | Deduction adding an antece... |
imbi1d 330 | Deduction adding a consequ... |
bibi2d 331 | Deduction adding a bicondi... |
bibi1d 332 | Deduction adding a bicondi... |
imbi12d 333 | Deduction joining two equi... |
bibi12d 334 | Deduction joining two equi... |
imbi12 335 | Closed form of ~ imbi12i .... |
imbi1 336 | Theorem *4.84 of [Whitehea... |
imbi2 337 | Theorem *4.85 of [Whitehea... |
imbi1i 338 | Introduce a consequent to ... |
imbi12i 339 | Join two logical equivalen... |
bibi1 340 | Theorem *4.86 of [Whitehea... |
bitr3 341 | Closed nested implication ... |
con2bi 342 | Contraposition. Theorem *... |
con2bid 343 | A contraposition deduction... |
con1bid 344 | A contraposition deduction... |
con1bii 345 | A contraposition inference... |
con2bii 346 | A contraposition inference... |
con1b 347 | Contraposition. Bidirecti... |
con2b 348 | Contraposition. Bidirecti... |
biimt 349 | A wff is equivalent to its... |
pm5.5 350 | Theorem *5.5 of [Whitehead... |
a1bi 351 | Inference rule introducing... |
mt2bi 352 | A false consequent falsifi... |
mtt 353 | Modus-tollens-like theorem... |
imnot 354 | If a proposition is false,... |
pm5.501 355 | Theorem *5.501 of [Whitehe... |
ibib 356 | Implication in terms of im... |
ibibr 357 | Implication in terms of im... |
tbt 358 | A wff is equivalent to its... |
nbn2 359 | The negation of a wff is e... |
bibif 360 | Transfer negation via an e... |
nbn 361 | The negation of a wff is e... |
nbn3 362 | Transfer falsehood via equ... |
pm5.21im 363 | Two propositions are equiv... |
2false 364 | Two falsehoods are equival... |
2falsed 365 | Two falsehoods are equival... |
pm5.21ni 366 | Two propositions implying ... |
pm5.21nii 367 | Eliminate an antecedent im... |
pm5.21ndd 368 | Eliminate an antecedent im... |
bija 369 | Combine antecedents into a... |
pm5.18 370 | Theorem *5.18 of [Whitehea... |
xor3 371 | Two ways to express "exclu... |
nbbn 372 | Move negation outside of b... |
biass 373 | Associative law for the bi... |
pm5.19 374 | Theorem *5.19 of [Whitehea... |
bi2.04 375 | Logical equivalence of com... |
pm5.4 376 | Antecedent absorption impl... |
imdi 377 | Distributive law for impli... |
pm5.41 378 | Theorem *5.41 of [Whitehea... |
pm4.8 379 | Theorem *4.8 of [Whitehead... |
pm4.81 380 | Theorem *4.81 of [Whitehea... |
imim21b 381 | Simplify an implication be... |
pm4.64 386 | Theorem *4.64 of [Whitehea... |
pm2.53 387 | Theorem *2.53 of [Whitehea... |
pm2.54 388 | Theorem *2.54 of [Whitehea... |
ori 389 | Infer implication from dis... |
orri 390 | Infer disjunction from imp... |
ord 391 | Deduce implication from di... |
orrd 392 | Deduce disjunction from im... |
jaoi 393 | Inference disjoining the a... |
jaod 394 | Deduction disjoining the a... |
mpjaod 395 | Eliminate a disjunction in... |
orel1 396 | Elimination of disjunction... |
orel2 397 | Elimination of disjunction... |
olc 398 | Introduction of a disjunct... |
orc 399 | Introduction of a disjunct... |
pm1.4 400 | Axiom *1.4 of [WhiteheadRu... |
orcom 401 | Commutative law for disjun... |
orcomd 402 | Commutation of disjuncts i... |
orcoms 403 | Commutation of disjuncts i... |
orci 404 | Deduction introducing a di... |
olci 405 | Deduction introducing a di... |
orcd 406 | Deduction introducing a di... |
olcd 407 | Deduction introducing a di... |
orcs 408 | Deduction eliminating disj... |
olcs 409 | Deduction eliminating disj... |
pm2.07 410 | Theorem *2.07 of [Whitehea... |
pm2.45 411 | Theorem *2.45 of [Whitehea... |
pm2.46 412 | Theorem *2.46 of [Whitehea... |
pm2.47 413 | Theorem *2.47 of [Whitehea... |
pm2.48 414 | Theorem *2.48 of [Whitehea... |
pm2.49 415 | Theorem *2.49 of [Whitehea... |
pm2.67-2 416 | Slight generalization of T... |
pm2.67 417 | Theorem *2.67 of [Whitehea... |
pm2.25 418 | Theorem *2.25 of [Whitehea... |
biorf 419 | A wff is equivalent to its... |
biortn 420 | A wff is equivalent to its... |
biorfi 421 | A wff is equivalent to its... |
biorfiOLD 422 | Obsolete proof of ~ biorfi... |
pm2.621 423 | Theorem *2.621 of [Whitehe... |
pm2.62 424 | Theorem *2.62 of [Whitehea... |
pm2.68 425 | Theorem *2.68 of [Whitehea... |
dfor2 426 | Logical 'or' expressed in ... |
imor 427 | Implication in terms of di... |
imori 428 | Infer disjunction from imp... |
imorri 429 | Infer implication from dis... |
exmid 430 | Law of excluded middle, al... |
exmidd 431 | Law of excluded middle in ... |
pm2.1 432 | Theorem *2.1 of [Whitehead... |
pm2.13 433 | Theorem *2.13 of [Whitehea... |
pm4.62 434 | Theorem *4.62 of [Whitehea... |
pm4.66 435 | Theorem *4.66 of [Whitehea... |
pm4.63 436 | Theorem *4.63 of [Whitehea... |
imnan 437 | Express implication in ter... |
imnani 438 | Infer implication from neg... |
iman 439 | Express implication in ter... |
annim 440 | Express conjunction in ter... |
pm4.61 441 | Theorem *4.61 of [Whitehea... |
pm4.65 442 | Theorem *4.65 of [Whitehea... |
pm4.67 443 | Theorem *4.67 of [Whitehea... |
imp 444 | Importation inference. (C... |
impcom 445 | Importation inference with... |
impd 446 | Importation deduction. (C... |
imp31 447 | An importation inference. ... |
imp32 448 | An importation inference. ... |
ex 449 | Exportation inference. (T... |
expcom 450 | Exportation inference with... |
expd 451 | Exportation deduction. (C... |
expdimp 452 | A deduction version of exp... |
expcomd 453 | Deduction form of ~ expcom... |
expdcom 454 | Commuted form of ~ expd . ... |
impancom 455 | Mixed importation/commutat... |
con3dimp 456 | Variant of ~ con3d with im... |
pm2.01da 457 | Deduction based on reducti... |
pm2.18da 458 | Deduction based on reducti... |
pm3.3 459 | Theorem *3.3 (Exp) of [Whi... |
pm3.31 460 | Theorem *3.31 (Imp) of [Wh... |
impexp 461 | Import-export theorem. Pa... |
pm3.2 462 | Join antecedents with conj... |
pm3.21 463 | Join antecedents with conj... |
pm3.22 464 | Theorem *3.22 of [Whitehea... |
ancom 465 | Commutative law for conjun... |
ancomd 466 | Commutation of conjuncts i... |
ancomst 467 | Closed form of ~ ancoms . ... |
ancoms 468 | Inference commuting conjun... |
ancomsd 469 | Deduction commuting conjun... |
pm3.2i 470 | Infer conjunction of premi... |
pm3.43i 471 | Nested conjunction of ante... |
simpl 472 | Elimination of a conjunct.... |
simpli 473 | Inference eliminating a co... |
simpld 474 | Deduction eliminating a co... |
simplbi 475 | Deduction eliminating a co... |
simpr 476 | Elimination of a conjunct.... |
simpri 477 | Inference eliminating a co... |
simprd 478 | Deduction eliminating a co... |
simprbi 479 | Deduction eliminating a co... |
adantr 480 | Inference adding a conjunc... |
adantl 481 | Inference adding a conjunc... |
adantld 482 | Deduction adding a conjunc... |
adantrd 483 | Deduction adding a conjunc... |
impel 484 | An inference for implicati... |
mpan9 485 | Modus ponens conjoining di... |
syldan 486 | A syllogism deduction with... |
sylan 487 | A syllogism inference. (C... |
sylanb 488 | A syllogism inference. (C... |
sylanbr 489 | A syllogism inference. (C... |
sylan2 490 | A syllogism inference. (C... |
sylan2b 491 | A syllogism inference. (C... |
sylan2br 492 | A syllogism inference. (C... |
syl2an 493 | A double syllogism inferen... |
syl2anr 494 | A double syllogism inferen... |
syl2anb 495 | A double syllogism inferen... |
syl2anbr 496 | A double syllogism inferen... |
syland 497 | A syllogism deduction. (C... |
sylan2d 498 | A syllogism deduction. (C... |
syl2and 499 | A syllogism deduction. (C... |
biimpa 500 | Importation inference from... |
biimpar 501 | Importation inference from... |
biimpac 502 | Importation inference from... |
biimparc 503 | Importation inference from... |
animorl 504 | Conjunction implies disjun... |
animorr 505 | Conjunction implies disjun... |
animorlr 506 | Conjunction implies disjun... |
animorrl 507 | Conjunction implies disjun... |
ianor 508 | Negated conjunction in ter... |
anor 509 | Conjunction in terms of di... |
ioran 510 | Negated disjunction in ter... |
pm4.52 511 | Theorem *4.52 of [Whitehea... |
pm4.53 512 | Theorem *4.53 of [Whitehea... |
pm4.54 513 | Theorem *4.54 of [Whitehea... |
pm4.55 514 | Theorem *4.55 of [Whitehea... |
pm4.56 515 | Theorem *4.56 of [Whitehea... |
oran 516 | Disjunction in terms of co... |
pm4.57 517 | Theorem *4.57 of [Whitehea... |
pm3.1 518 | Theorem *3.1 of [Whitehead... |
pm3.11 519 | Theorem *3.11 of [Whitehea... |
pm3.12 520 | Theorem *3.12 of [Whitehea... |
pm3.13 521 | Theorem *3.13 of [Whitehea... |
pm3.14 522 | Theorem *3.14 of [Whitehea... |
iba 523 | Introduction of antecedent... |
ibar 524 | Introduction of antecedent... |
biantru 525 | A wff is equivalent to its... |
biantrur 526 | A wff is equivalent to its... |
biantrud 527 | A wff is equivalent to its... |
biantrurd 528 | A wff is equivalent to its... |
mpbirand 529 | Detach truth from conjunct... |
jaao 530 | Inference conjoining and d... |
jaoa 531 | Inference disjoining and c... |
pm3.44 532 | Theorem *3.44 of [Whitehea... |
jao 533 | Disjunction of antecedents... |
pm1.2 534 | Axiom *1.2 of [WhiteheadRu... |
oridm 535 | Idempotent law for disjunc... |
pm4.25 536 | Theorem *4.25 of [Whitehea... |
orim12i 537 | Disjoin antecedents and co... |
orim1i 538 | Introduce disjunct to both... |
orim2i 539 | Introduce disjunct to both... |
orbi2i 540 | Inference adding a left di... |
orbi1i 541 | Inference adding a right d... |
orbi12i 542 | Infer the disjunction of t... |
pm1.5 543 | Axiom *1.5 (Assoc) of [Whi... |
or12 544 | Swap two disjuncts. (Cont... |
orass 545 | Associative law for disjun... |
pm2.31 546 | Theorem *2.31 of [Whitehea... |
pm2.32 547 | Theorem *2.32 of [Whitehea... |
or32 548 | A rearrangement of disjunc... |
or4 549 | Rearrangement of 4 disjunc... |
or42 550 | Rearrangement of 4 disjunc... |
orordi 551 | Distribution of disjunctio... |
orordir 552 | Distribution of disjunctio... |
jca 553 | Deduce conjunction of the ... |
jcad 554 | Deduction conjoining the c... |
jca31 555 | Join three consequents. (... |
jca32 556 | Join three consequents. (... |
jcai 557 | Deduction replacing implic... |
jctil 558 | Inference conjoining a the... |
jctir 559 | Inference conjoining a the... |
jccir 560 | Inference conjoining a con... |
jccil 561 | Inference conjoining a con... |
jctl 562 | Inference conjoining a the... |
jctr 563 | Inference conjoining a the... |
jctild 564 | Deduction conjoining a the... |
jctird 565 | Deduction conjoining a the... |
syl6an 566 | A syllogism deduction comb... |
ancl 567 | Conjoin antecedent to left... |
anclb 568 | Conjoin antecedent to left... |
pm5.42 569 | Theorem *5.42 of [Whitehea... |
ancr 570 | Conjoin antecedent to righ... |
ancrb 571 | Conjoin antecedent to righ... |
ancli 572 | Deduction conjoining antec... |
ancri 573 | Deduction conjoining antec... |
ancld 574 | Deduction conjoining antec... |
ancrd 575 | Deduction conjoining antec... |
anc2l 576 | Conjoin antecedent to left... |
anc2r 577 | Conjoin antecedent to righ... |
anc2li 578 | Deduction conjoining antec... |
anc2ri 579 | Deduction conjoining antec... |
pm3.41 580 | Theorem *3.41 of [Whitehea... |
pm3.42 581 | Theorem *3.42 of [Whitehea... |
pm3.4 582 | Conjunction implies implic... |
pm4.45im 583 | Conjunction with implicati... |
anim12d 584 | Conjoin antecedents and co... |
anim12d1 585 | Variant of ~ anim12d where... |
anim1d 586 | Add a conjunct to right of... |
anim2d 587 | Add a conjunct to left of ... |
anim12i 588 | Conjoin antecedents and co... |
anim12ci 589 | Variant of ~ anim12i with ... |
anim1i 590 | Introduce conjunct to both... |
anim2i 591 | Introduce conjunct to both... |
anim12ii 592 | Conjoin antecedents and co... |
prth 593 | Conjoin antecedents and co... |
pm2.3 594 | Theorem *2.3 of [Whitehead... |
pm2.41 595 | Theorem *2.41 of [Whitehea... |
pm2.42 596 | Theorem *2.42 of [Whitehea... |
pm2.4 597 | Theorem *2.4 of [Whitehead... |
pm2.65da 598 | Deduction rule for proof b... |
pm4.44 599 | Theorem *4.44 of [Whitehea... |
pm4.14 600 | Theorem *4.14 of [Whitehea... |
pm3.37 601 | Theorem *3.37 (Transp) of ... |
nan 602 | Theorem to move a conjunct... |
pm4.15 603 | Theorem *4.15 of [Whitehea... |
pm4.78 604 | Implication distributes ov... |
pm4.79 605 | Theorem *4.79 of [Whitehea... |
pm4.87 606 | Theorem *4.87 of [Whitehea... |
pm3.33 607 | Theorem *3.33 (Syll) of [W... |
pm3.34 608 | Theorem *3.34 (Syll) of [W... |
pm3.35 609 | Conjunctive detachment. T... |
pm5.31 610 | Theorem *5.31 of [Whitehea... |
imp4b 611 | An importation inference. ... |
imp4a 612 | An importation inference. ... |
imp4aOLD 613 | Obsolete proof of ~ imp4a ... |
imp4bOLD 614 | Obsolete proof of ~ imp4b ... |
imp4c 615 | An importation inference. ... |
imp4d 616 | An importation inference. ... |
imp41 617 | An importation inference. ... |
imp42 618 | An importation inference. ... |
imp43 619 | An importation inference. ... |
imp44 620 | An importation inference. ... |
imp45 621 | An importation inference. ... |
imp5a 622 | An importation inference. ... |
imp5d 623 | An importation inference. ... |
imp5g 624 | An importation inference. ... |
imp55 625 | An importation inference. ... |
imp511 626 | An importation inference. ... |
expimpd 627 | Exportation followed by a ... |
exp31 628 | An exportation inference. ... |
exp32 629 | An exportation inference. ... |
exp4b 630 | An exportation inference. ... |
exp4a 631 | An exportation inference. ... |
exp4aOLD 632 | Obsolete proof of ~ exp4a ... |
exp4bOLD 633 | Obsolete proof of ~ exp4b ... |
exp4c 634 | An exportation inference. ... |
exp4d 635 | An exportation inference. ... |
exp41 636 | An exportation inference. ... |
exp42 637 | An exportation inference. ... |
exp43 638 | An exportation inference. ... |
exp44 639 | An exportation inference. ... |
exp45 640 | An exportation inference. ... |
expr 641 | Export a wff from a right ... |
exp5c 642 | An exportation inference. ... |
exp5j 643 | An exportation inference. ... |
exp5l 644 | An exportation inference. ... |
exp53 645 | An exportation inference. ... |
expl 646 | Export a wff from a left c... |
impr 647 | Import a wff into a right ... |
impl 648 | Export a wff from a left c... |
impac 649 | Importation with conjuncti... |
exbiri 650 | Inference form of ~ exbir ... |
simprbda 651 | Deduction eliminating a co... |
simplbda 652 | Deduction eliminating a co... |
simplbi2 653 | Deduction eliminating a co... |
simplbi2comt 654 | Closed form of ~ simplbi2c... |
simplbi2com 655 | A deduction eliminating a ... |
simpl2im 656 | Implication from an elimin... |
simplbiim 657 | Implication from an elimin... |
dfbi2 658 | A theorem similar to the s... |
dfbi 659 | Definition ~ df-bi rewritt... |
pm4.71 660 | Implication in terms of bi... |
pm4.71r 661 | Implication in terms of bi... |
pm4.71i 662 | Inference converting an im... |
pm4.71ri 663 | Inference converting an im... |
pm4.71d 664 | Deduction converting an im... |
pm4.71rd 665 | Deduction converting an im... |
pm5.32 666 | Distribution of implicatio... |
pm5.32i 667 | Distribution of implicatio... |
pm5.32ri 668 | Distribution of implicatio... |
pm5.32d 669 | Distribution of implicatio... |
pm5.32rd 670 | Distribution of implicatio... |
pm5.32da 671 | Distribution of implicatio... |
biadan2 672 | Add a conjunction to an eq... |
pm4.24 673 | Theorem *4.24 of [Whitehea... |
anidm 674 | Idempotent law for conjunc... |
anidms 675 | Inference from idempotent ... |
anidmdbi 676 | Conjunction idempotence wi... |
anasss 677 | Associative law for conjun... |
anassrs 678 | Associative law for conjun... |
anass 679 | Associative law for conjun... |
sylanl1 680 | A syllogism inference. (C... |
sylanl2 681 | A syllogism inference. (C... |
sylanr1 682 | A syllogism inference. (C... |
sylanr2 683 | A syllogism inference. (C... |
sylani 684 | A syllogism inference. (C... |
sylan2i 685 | A syllogism inference. (C... |
syl2ani 686 | A syllogism inference. (C... |
sylan9 687 | Nested syllogism inference... |
sylan9r 688 | Nested syllogism inference... |
mtand 689 | A modus tollens deduction.... |
mtord 690 | A modus tollens deduction ... |
syl2anc 691 | Syllogism inference combin... |
hypstkdOLD 692 | Obsolete proof of ~ mpidan... |
sylancl 693 | Syllogism inference combin... |
sylancr 694 | Syllogism inference combin... |
sylanbrc 695 | Syllogism inference. (Con... |
sylancb 696 | A syllogism inference comb... |
sylancbr 697 | A syllogism inference comb... |
sylancom 698 | Syllogism inference with c... |
mpdan 699 | An inference based on modu... |
mpancom 700 | An inference based on modu... |
mpidan 701 | A deduction which "stacks"... |
mpan 702 | An inference based on modu... |
mpan2 703 | An inference based on modu... |
mp2an 704 | An inference based on modu... |
mp4an 705 | An inference based on modu... |
mpan2d 706 | A deduction based on modus... |
mpand 707 | A deduction based on modus... |
mpani 708 | An inference based on modu... |
mpan2i 709 | An inference based on modu... |
mp2ani 710 | An inference based on modu... |
mp2and 711 | A deduction based on modus... |
mpanl1 712 | An inference based on modu... |
mpanl2 713 | An inference based on modu... |
mpanl12 714 | An inference based on modu... |
mpanr1 715 | An inference based on modu... |
mpanr2 716 | An inference based on modu... |
mpanr12 717 | An inference based on modu... |
mpanlr1 718 | An inference based on modu... |
pm5.74da 719 | Distribution of implicatio... |
pm4.45 720 | Theorem *4.45 of [Whitehea... |
imdistan 721 | Distribution of implicatio... |
imdistani 722 | Distribution of implicatio... |
imdistanri 723 | Distribution of implicatio... |
imdistand 724 | Distribution of implicatio... |
imdistanda 725 | Distribution of implicatio... |
anbi2i 726 | Introduce a left conjunct ... |
anbi1i 727 | Introduce a right conjunct... |
anbi2ci 728 | Variant of ~ anbi2i with c... |
anbi12i 729 | Conjoin both sides of two ... |
anbi12ci 730 | Variant of ~ anbi12i with ... |
syldanl 731 | A syllogism deduction with... |
sylan9bb 732 | Nested syllogism inference... |
sylan9bbr 733 | Nested syllogism inference... |
orbi2d 734 | Deduction adding a left di... |
orbi1d 735 | Deduction adding a right d... |
anbi2d 736 | Deduction adding a left co... |
anbi1d 737 | Deduction adding a right c... |
orbi1 738 | Theorem *4.37 of [Whitehea... |
anbi1 739 | Introduce a right conjunct... |
anbi2 740 | Introduce a left conjunct ... |
bitr 741 | Theorem *4.22 of [Whitehea... |
orbi12d 742 | Deduction joining two equi... |
anbi12d 743 | Deduction joining two equi... |
pm5.3 744 | Theorem *5.3 of [Whitehead... |
pm5.61 745 | Theorem *5.61 of [Whitehea... |
adantll 746 | Deduction adding a conjunc... |
adantlr 747 | Deduction adding a conjunc... |
adantrl 748 | Deduction adding a conjunc... |
adantrr 749 | Deduction adding a conjunc... |
adantlll 750 | Deduction adding a conjunc... |
adantllr 751 | Deduction adding a conjunc... |
adantlrl 752 | Deduction adding a conjunc... |
adantlrr 753 | Deduction adding a conjunc... |
adantrll 754 | Deduction adding a conjunc... |
adantrlr 755 | Deduction adding a conjunc... |
adantrrl 756 | Deduction adding a conjunc... |
adantrrr 757 | Deduction adding a conjunc... |
ad2antrr 758 | Deduction adding two conju... |
ad2antlr 759 | Deduction adding two conju... |
ad2antrl 760 | Deduction adding two conju... |
ad2antll 761 | Deduction adding conjuncts... |
ad3antrrr 762 | Deduction adding three con... |
ad3antlr 763 | Deduction adding three con... |
ad4antr 764 | Deduction adding 4 conjunc... |
ad4antlr 765 | Deduction adding 4 conjunc... |
ad5antr 766 | Deduction adding 5 conjunc... |
ad5antlr 767 | Deduction adding 5 conjunc... |
ad6antr 768 | Deduction adding 6 conjunc... |
ad6antlr 769 | Deduction adding 6 conjunc... |
ad7antr 770 | Deduction adding 7 conjunc... |
ad7antlr 771 | Deduction adding 7 conjunc... |
ad8antr 772 | Deduction adding 8 conjunc... |
ad8antlr 773 | Deduction adding 8 conjunc... |
ad9antr 774 | Deduction adding 9 conjunc... |
ad9antlr 775 | Deduction adding 9 conjunc... |
ad10antr 776 | Deduction adding 10 conjun... |
ad10antlr 777 | Deduction adding 10 conjun... |
ad2ant2l 778 | Deduction adding two conju... |
ad2ant2r 779 | Deduction adding two conju... |
ad2ant2lr 780 | Deduction adding two conju... |
ad2ant2rl 781 | Deduction adding two conju... |
adantl3r 782 | Deduction adding 1 conjunc... |
adantl4r 783 | Deduction adding 1 conjunc... |
adantl5r 784 | Deduction adding 1 conjunc... |
adantl6r 785 | Deduction adding 1 conjunc... |
simpll 786 | Simplification of a conjun... |
simplld 787 | Deduction form of ~ simpll... |
simplr 788 | Simplification of a conjun... |
simplrd 789 | Deduction eliminating a do... |
simprl 790 | Simplification of a conjun... |
simprld 791 | Deduction eliminating a do... |
simprr 792 | Simplification of a conjun... |
simprrd 793 | Deduction form of ~ simprr... |
simplll 794 | Simplification of a conjun... |
simpllr 795 | Simplification of a conjun... |
simplrl 796 | Simplification of a conjun... |
simplrr 797 | Simplification of a conjun... |
simprll 798 | Simplification of a conjun... |
simprlr 799 | Simplification of a conjun... |
simprrl 800 | Simplification of a conjun... |
simprrr 801 | Simplification of a conjun... |
simp-4l 802 | Simplification of a conjun... |
simp-4r 803 | Simplification of a conjun... |
simp-5l 804 | Simplification of a conjun... |
simp-5r 805 | Simplification of a conjun... |
simp-6l 806 | Simplification of a conjun... |
simp-6r 807 | Simplification of a conjun... |
simp-7l 808 | Simplification of a conjun... |
simp-7r 809 | Simplification of a conjun... |
simp-8l 810 | Simplification of a conjun... |
simp-8r 811 | Simplification of a conjun... |
simp-9l 812 | Simplification of a conjun... |
simp-9r 813 | Simplification of a conjun... |
simp-10l 814 | Simplification of a conjun... |
simp-10r 815 | Simplification of a conjun... |
simp-11l 816 | Simplification of a conjun... |
simp-11r 817 | Simplification of a conjun... |
jaob 818 | Disjunction of antecedents... |
adant423OLD 819 | Obsolete as of 2-Oct-2021.... |
jaoian 820 | Inference disjoining the a... |
jao1i 821 | Add a disjunct in the ante... |
jaodan 822 | Deduction disjoining the a... |
mpjaodan 823 | Eliminate a disjunction in... |
pm4.77 824 | Theorem *4.77 of [Whitehea... |
pm2.63 825 | Theorem *2.63 of [Whitehea... |
pm2.64 826 | Theorem *2.64 of [Whitehea... |
pm2.61ian 827 | Elimination of an antecede... |
pm2.61dan 828 | Elimination of an antecede... |
pm2.61ddan 829 | Elimination of two anteced... |
pm2.61dda 830 | Elimination of two anteced... |
condan 831 | Proof by contradiction. (... |
abai 832 | Introduce one conjunct as ... |
pm5.53 833 | Theorem *5.53 of [Whitehea... |
an12 834 | Swap two conjuncts. Note ... |
an32 835 | A rearrangement of conjunc... |
an13 836 | A rearrangement of conjunc... |
an31 837 | A rearrangement of conjunc... |
bianass 838 | An inference to merge two ... |
an12s 839 | Swap two conjuncts in ante... |
ancom2s 840 | Inference commuting a nest... |
an13s 841 | Swap two conjuncts in ante... |
an32s 842 | Swap two conjuncts in ante... |
ancom1s 843 | Inference commuting a nest... |
an31s 844 | Swap two conjuncts in ante... |
anass1rs 845 | Commutative-associative la... |
anabs1 846 | Absorption into embedded c... |
anabs5 847 | Absorption into embedded c... |
anabs7 848 | Absorption into embedded c... |
a2and 849 | Deduction distributing a c... |
anabsan 850 | Absorption of antecedent w... |
anabss1 851 | Absorption of antecedent i... |
anabss4 852 | Absorption of antecedent i... |
anabss5 853 | Absorption of antecedent i... |
anabsi5 854 | Absorption of antecedent i... |
anabsi6 855 | Absorption of antecedent i... |
anabsi7 856 | Absorption of antecedent i... |
anabsi8 857 | Absorption of antecedent i... |
anabss7 858 | Absorption of antecedent i... |
anabsan2 859 | Absorption of antecedent w... |
anabss3 860 | Absorption of antecedent i... |
an4 861 | Rearrangement of 4 conjunc... |
an42 862 | Rearrangement of 4 conjunc... |
an43 863 | Rearrangement of 4 conjunc... |
an3 864 | A rearrangement of conjunc... |
an4s 865 | Inference rearranging 4 co... |
an42s 866 | Inference rearranging 4 co... |
anandi 867 | Distribution of conjunctio... |
anandir 868 | Distribution of conjunctio... |
anandis 869 | Inference that undistribut... |
anandirs 870 | Inference that undistribut... |
syl2an2 871 | ~ syl2an with antecedents ... |
syl2an2r 872 | ~ syl2anr with antecedents... |
impbida 873 | Deduce an equivalence from... |
pm3.48 874 | Theorem *3.48 of [Whitehea... |
pm3.45 875 | Theorem *3.45 (Fact) of [W... |
im2anan9 876 | Deduction joining nested i... |
im2anan9r 877 | Deduction joining nested i... |
anim12dan 878 | Conjoin antecedents and co... |
orim12d 879 | Disjoin antecedents and co... |
orim1d 880 | Disjoin antecedents and co... |
orim2d 881 | Disjoin antecedents and co... |
orim2 882 | Axiom *1.6 (Sum) of [White... |
pm2.38 883 | Theorem *2.38 of [Whitehea... |
pm2.36 884 | Theorem *2.36 of [Whitehea... |
pm2.37 885 | Theorem *2.37 of [Whitehea... |
pm2.73 886 | Theorem *2.73 of [Whitehea... |
pm2.74 887 | Theorem *2.74 of [Whitehea... |
orimdi 888 | Disjunction distributes ov... |
pm2.76 889 | Theorem *2.76 of [Whitehea... |
pm2.75 890 | Theorem *2.75 of [Whitehea... |
pm2.8 891 | Theorem *2.8 of [Whitehead... |
pm2.81 892 | Theorem *2.81 of [Whitehea... |
pm2.82 893 | Theorem *2.82 of [Whitehea... |
pm2.85 894 | Theorem *2.85 of [Whitehea... |
pm3.2ni 895 | Infer negated disjunction ... |
orabs 896 | Absorption of redundant in... |
oranabs 897 | Absorb a disjunct into a c... |
pm5.1 898 | Two propositions are equiv... |
pm5.21 899 | Two propositions are equiv... |
norbi 900 | If neither of two proposit... |
nbior 901 | If two propositions are no... |
pm3.43 902 | Theorem *3.43 (Comp) of [W... |
jcab 903 | Distributive law for impli... |
ordi 904 | Distributive law for disju... |
ordir 905 | Distributive law for disju... |
pm4.76 906 | Theorem *4.76 of [Whitehea... |
andi 907 | Distributive law for conju... |
andir 908 | Distributive law for conju... |
orddi 909 | Double distributive law fo... |
anddi 910 | Double distributive law fo... |
pm4.39 911 | Theorem *4.39 of [Whitehea... |
pm4.38 912 | Theorem *4.38 of [Whitehea... |
bi2anan9 913 | Deduction joining two equi... |
bi2anan9r 914 | Deduction joining two equi... |
bi2bian9 915 | Deduction joining two bico... |
pm4.72 916 | Implication in terms of bi... |
imimorb 917 | Simplify an implication be... |
pm5.33 918 | Theorem *5.33 of [Whitehea... |
pm5.36 919 | Theorem *5.36 of [Whitehea... |
bianabs 920 | Absorb a hypothesis into t... |
oibabs 921 | Absorption of disjunction ... |
pm3.24 922 | Law of noncontradiction. ... |
pm2.26 923 | Theorem *2.26 of [Whitehea... |
pm5.11 924 | Theorem *5.11 of [Whitehea... |
pm5.12 925 | Theorem *5.12 of [Whitehea... |
pm5.14 926 | Theorem *5.14 of [Whitehea... |
pm5.13 927 | Theorem *5.13 of [Whitehea... |
pm5.17 928 | Theorem *5.17 of [Whitehea... |
pm5.15 929 | Theorem *5.15 of [Whitehea... |
pm5.16 930 | Theorem *5.16 of [Whitehea... |
xor 931 | Two ways to express "exclu... |
nbi2 932 | Two ways to express "exclu... |
dfbi3 933 | An alternate definition of... |
pm5.24 934 | Theorem *5.24 of [Whitehea... |
xordi 935 | Conjunction distributes ov... |
biort 936 | A wff disjoined with truth... |
pm5.55 937 | Theorem *5.55 of [Whitehea... |
ornld 938 | Selecting one statement fr... |
pm5.21nd 939 | Eliminate an antecedent im... |
pm5.35 940 | Theorem *5.35 of [Whitehea... |
pm5.54 941 | Theorem *5.54 of [Whitehea... |
baib 942 | Move conjunction outside o... |
baibr 943 | Move conjunction outside o... |
rbaibr 944 | Move conjunction outside o... |
rbaib 945 | Move conjunction outside o... |
baibd 946 | Move conjunction outside o... |
rbaibd 947 | Move conjunction outside o... |
pm5.44 948 | Theorem *5.44 of [Whitehea... |
pm5.6 949 | Conjunction in antecedent ... |
orcanai 950 | Change disjunction in cons... |
intnan 951 | Introduction of conjunct i... |
intnanr 952 | Introduction of conjunct i... |
intnand 953 | Introduction of conjunct i... |
intnanrd 954 | Introduction of conjunct i... |
mpbiran 955 | Detach truth from conjunct... |
mpbiran2 956 | Detach truth from conjunct... |
mpbir2an 957 | Detach a conjunction of tr... |
mpbi2and 958 | Detach a conjunction of tr... |
mpbir2and 959 | Detach a conjunction of tr... |
pm5.62 960 | Theorem *5.62 of [Whitehea... |
pm5.63 961 | Theorem *5.63 of [Whitehea... |
bianfi 962 | A wff conjoined with false... |
bianfd 963 | A wff conjoined with false... |
pm4.43 964 | Theorem *4.43 of [Whitehea... |
pm4.82 965 | Theorem *4.82 of [Whitehea... |
pm4.83 966 | Theorem *4.83 of [Whitehea... |
pclem6 967 | Negation inferred from emb... |
biantr 968 | A transitive law of equiva... |
orbidi 969 | Disjunction distributes ov... |
biluk 970 | Lukasiewicz's shortest axi... |
pm5.7 971 | Disjunction distributes ov... |
bigolden 972 | Dijkstra-Scholten's Golden... |
pm5.71 973 | Theorem *5.71 of [Whitehea... |
pm5.75 974 | Theorem *5.75 of [Whitehea... |
pm5.75OLD 975 | Obsolete proof of ~ pm5.75... |
bimsc1 976 | Removal of conjunct from o... |
4exmid 977 | The disjunction of the fou... |
ecase2d 978 | Deduction for elimination ... |
ecase3 979 | Inference for elimination ... |
ecase 980 | Inference for elimination ... |
ecase3d 981 | Deduction for elimination ... |
ecased 982 | Deduction for elimination ... |
ecase3ad 983 | Deduction for elimination ... |
ccase 984 | Inference for combining ca... |
ccased 985 | Deduction for combining ca... |
ccase2 986 | Inference for combining ca... |
4cases 987 | Inference eliminating two ... |
4casesdan 988 | Deduction eliminating two ... |
niabn 989 | Miscellaneous inference re... |
consensus 990 | The consensus theorem. Th... |
dedlem0a 991 | Lemma for an alternate ver... |
dedlem0b 992 | Lemma for an alternate ver... |
dedlema 993 | Lemma for weak deduction t... |
dedlemb 994 | Lemma for weak deduction t... |
pm4.42 995 | Theorem *4.42 of [Whitehea... |
ninba 996 | Miscellaneous inference re... |
prlem1 997 | A specialized lemma for se... |
prlem2 998 | A specialized lemma for se... |
oplem1 999 | A specialized lemma for se... |
dn1 1000 | A single axiom for Boolean... |
bianir 1001 | If a wff is equivalent to ... |
jaoi2 1002 | Inference removing a negat... |
jaoi3 1003 | Inference separating a dis... |
cases 1004 | Case disjunction according... |
cases2 1005 | Case disjunction according... |
dfifp2 1008 | Alternate definition of th... |
dfifp3 1009 | Alternate definition of th... |
dfifp4 1010 | Alternate definition of th... |
dfifp5 1011 | Alternate definition of th... |
dfifp6 1012 | Alternate definition of th... |
dfifp7 1013 | Alternate definition of th... |
anifp 1014 | The conditional operator i... |
ifpor 1015 | The conditional operator i... |
ifpn 1016 | Conditional operator for t... |
ifptru 1017 | Value of the conditional o... |
ifpfal 1018 | Value of the conditional o... |
ifpid 1019 | Value of the conditional o... |
casesifp 1020 | Version of ~ cases express... |
ifpbi123d 1021 | Equality deduction for con... |
ifpimpda 1022 | Separation of the values o... |
1fpid3 1023 | The value of the condition... |
elimh 1024 | Hypothesis builder for the... |
dedt 1025 | The weak deduction theorem... |
con3ALT 1026 | Proof of ~ con3 from its a... |
elimhOLD 1027 | Old version of ~ elimh . ... |
dedtOLD 1028 | Old version of ~ dedt . O... |
con3OLD 1029 | Old version of ~ con3ALT .... |
3orass 1034 | Associative law for triple... |
3anass 1035 | Associative law for triple... |
3anrot 1036 | Rotation law for triple co... |
3orrot 1037 | Rotation law for triple di... |
3ancoma 1038 | Commutation law for triple... |
3orcoma 1039 | Commutation law for triple... |
3ancomb 1040 | Commutation law for triple... |
3orcomb 1041 | Commutation law for triple... |
3anrev 1042 | Reversal law for triple co... |
3anan32 1043 | Convert triple conjunction... |
3anan12 1044 | Convert triple conjunction... |
anandi3 1045 | Distribution of triple con... |
anandi3r 1046 | Distribution of triple con... |
3anor 1047 | Triple conjunction express... |
3ianor 1048 | Negated triple conjunction... |
3ioran 1049 | Negated triple disjunction... |
3oran 1050 | Triple disjunction in term... |
3simpa 1051 | Simplification of triple c... |
3simpb 1052 | Simplification of triple c... |
3simpc 1053 | Simplification of triple c... |
simp1 1054 | Simplification of triple c... |
simp2 1055 | Simplification of triple c... |
simp3 1056 | Simplification of triple c... |
simpl1 1057 | Simplification rule. (Con... |
simpl2 1058 | Simplification rule. (Con... |
simpl3 1059 | Simplification rule. (Con... |
simpr1 1060 | Simplification rule. (Con... |
simpr2 1061 | Simplification rule. (Con... |
simpr3 1062 | Simplification rule. (Con... |
simp1i 1063 | Infer a conjunct from a tr... |
simp2i 1064 | Infer a conjunct from a tr... |
simp3i 1065 | Infer a conjunct from a tr... |
simp1d 1066 | Deduce a conjunct from a t... |
simp2d 1067 | Deduce a conjunct from a t... |
simp3d 1068 | Deduce a conjunct from a t... |
simp1bi 1069 | Deduce a conjunct from a t... |
simp2bi 1070 | Deduce a conjunct from a t... |
simp3bi 1071 | Deduce a conjunct from a t... |
3adant1 1072 | Deduction adding a conjunc... |
3adant2 1073 | Deduction adding a conjunc... |
3adant3 1074 | Deduction adding a conjunc... |
3ad2ant1 1075 | Deduction adding conjuncts... |
3ad2ant2 1076 | Deduction adding conjuncts... |
3ad2ant3 1077 | Deduction adding conjuncts... |
simp1l 1078 | Simplification of triple c... |
simp1r 1079 | Simplification of triple c... |
simp2l 1080 | Simplification of triple c... |
simp2r 1081 | Simplification of triple c... |
simp3l 1082 | Simplification of triple c... |
simp3r 1083 | Simplification of triple c... |
simp11 1084 | Simplification of doubly t... |
simp12 1085 | Simplification of doubly t... |
simp13 1086 | Simplification of doubly t... |
simp21 1087 | Simplification of doubly t... |
simp22 1088 | Simplification of doubly t... |
simp23 1089 | Simplification of doubly t... |
simp31 1090 | Simplification of doubly t... |
simp32 1091 | Simplification of doubly t... |
simp33 1092 | Simplification of doubly t... |
simpll1 1093 | Simplification of conjunct... |
simpll2 1094 | Simplification of conjunct... |
simpll3 1095 | Simplification of conjunct... |
simplr1 1096 | Simplification of conjunct... |
simplr2 1097 | Simplification of conjunct... |
simplr3 1098 | Simplification of conjunct... |
simprl1 1099 | Simplification of conjunct... |
simprl2 1100 | Simplification of conjunct... |
simprl3 1101 | Simplification of conjunct... |
simprr1 1102 | Simplification of conjunct... |
simprr2 1103 | Simplification of conjunct... |
simprr3 1104 | Simplification of conjunct... |
simpl1l 1105 | Simplification of conjunct... |
simpl1r 1106 | Simplification of conjunct... |
simpl2l 1107 | Simplification of conjunct... |
simpl2r 1108 | Simplification of conjunct... |
simpl3l 1109 | Simplification of conjunct... |
simpl3r 1110 | Simplification of conjunct... |
simpr1l 1111 | Simplification of conjunct... |
simpr1r 1112 | Simplification of conjunct... |
simpr2l 1113 | Simplification of conjunct... |
simpr2r 1114 | Simplification of conjunct... |
simpr3l 1115 | Simplification of conjunct... |
simpr3r 1116 | Simplification of conjunct... |
simp1ll 1117 | Simplification of conjunct... |
simp1lr 1118 | Simplification of conjunct... |
simp1rl 1119 | Simplification of conjunct... |
simp1rr 1120 | Simplification of conjunct... |
simp2ll 1121 | Simplification of conjunct... |
simp2lr 1122 | Simplification of conjunct... |
simp2rl 1123 | Simplification of conjunct... |
simp2rr 1124 | Simplification of conjunct... |
simp3ll 1125 | Simplification of conjunct... |
simp3lr 1126 | Simplification of conjunct... |
simp3rl 1127 | Simplification of conjunct... |
simp3rr 1128 | Simplification of conjunct... |
simpl11 1129 | Simplification of conjunct... |
simpl12 1130 | Simplification of conjunct... |
simpl13 1131 | Simplification of conjunct... |
simpl21 1132 | Simplification of conjunct... |
simpl22 1133 | Simplification of conjunct... |
simpl23 1134 | Simplification of conjunct... |
simpl31 1135 | Simplification of conjunct... |
simpl32 1136 | Simplification of conjunct... |
simpl33 1137 | Simplification of conjunct... |
simpr11 1138 | Simplification of conjunct... |
simpr12 1139 | Simplification of conjunct... |
simpr13 1140 | Simplification of conjunct... |
simpr21 1141 | Simplification of conjunct... |
simpr22 1142 | Simplification of conjunct... |
simpr23 1143 | Simplification of conjunct... |
simpr31 1144 | Simplification of conjunct... |
simpr32 1145 | Simplification of conjunct... |
simpr33 1146 | Simplification of conjunct... |
simp1l1 1147 | Simplification of conjunct... |
simp1l2 1148 | Simplification of conjunct... |
simp1l3 1149 | Simplification of conjunct... |
simp1r1 1150 | Simplification of conjunct... |
simp1r2 1151 | Simplification of conjunct... |
simp1r3 1152 | Simplification of conjunct... |
simp2l1 1153 | Simplification of conjunct... |
simp2l2 1154 | Simplification of conjunct... |
simp2l3 1155 | Simplification of conjunct... |
simp2r1 1156 | Simplification of conjunct... |
simp2r2 1157 | Simplification of conjunct... |
simp2r3 1158 | Simplification of conjunct... |
simp3l1 1159 | Simplification of conjunct... |
simp3l2 1160 | Simplification of conjunct... |
simp3l3 1161 | Simplification of conjunct... |
simp3r1 1162 | Simplification of conjunct... |
simp3r2 1163 | Simplification of conjunct... |
simp3r3 1164 | Simplification of conjunct... |
simp11l 1165 | Simplification of conjunct... |
simp11r 1166 | Simplification of conjunct... |
simp12l 1167 | Simplification of conjunct... |
simp12r 1168 | Simplification of conjunct... |
simp13l 1169 | Simplification of conjunct... |
simp13r 1170 | Simplification of conjunct... |
simp21l 1171 | Simplification of conjunct... |
simp21r 1172 | Simplification of conjunct... |
simp22l 1173 | Simplification of conjunct... |
simp22r 1174 | Simplification of conjunct... |
simp23l 1175 | Simplification of conjunct... |
simp23r 1176 | Simplification of conjunct... |
simp31l 1177 | Simplification of conjunct... |
simp31r 1178 | Simplification of conjunct... |
simp32l 1179 | Simplification of conjunct... |
simp32r 1180 | Simplification of conjunct... |
simp33l 1181 | Simplification of conjunct... |
simp33r 1182 | Simplification of conjunct... |
simp111 1183 | Simplification of conjunct... |
simp112 1184 | Simplification of conjunct... |
simp113 1185 | Simplification of conjunct... |
simp121 1186 | Simplification of conjunct... |
simp122 1187 | Simplification of conjunct... |
simp123 1188 | Simplification of conjunct... |
simp131 1189 | Simplification of conjunct... |
simp132 1190 | Simplification of conjunct... |
simp133 1191 | Simplification of conjunct... |
simp211 1192 | Simplification of conjunct... |
simp212 1193 | Simplification of conjunct... |
simp213 1194 | Simplification of conjunct... |
simp221 1195 | Simplification of conjunct... |
simp222 1196 | Simplification of conjunct... |
simp223 1197 | Simplification of conjunct... |
simp231 1198 | Simplification of conjunct... |
simp232 1199 | Simplification of conjunct... |
simp233 1200 | Simplification of conjunct... |
simp311 1201 | Simplification of conjunct... |
simp312 1202 | Simplification of conjunct... |
simp313 1203 | Simplification of conjunct... |
simp321 1204 | Simplification of conjunct... |
simp322 1205 | Simplification of conjunct... |
simp323 1206 | Simplification of conjunct... |
simp331 1207 | Simplification of conjunct... |
simp332 1208 | Simplification of conjunct... |
simp333 1209 | Simplification of conjunct... |
3adantl1 1210 | Deduction adding a conjunc... |
3adantl2 1211 | Deduction adding a conjunc... |
3adantl3 1212 | Deduction adding a conjunc... |
3adantr1 1213 | Deduction adding a conjunc... |
3adantr2 1214 | Deduction adding a conjunc... |
3adantr3 1215 | Deduction adding a conjunc... |
3ad2antl1 1216 | Deduction adding conjuncts... |
3ad2antl2 1217 | Deduction adding conjuncts... |
3ad2antl3 1218 | Deduction adding conjuncts... |
3ad2antr1 1219 | Deduction adding conjuncts... |
3ad2antr2 1220 | Deduction adding conjuncts... |
3ad2antr3 1221 | Deduction adding conjuncts... |
3anibar 1222 | Remove a hypothesis from t... |
3mix1 1223 | Introduction in triple dis... |
3mix2 1224 | Introduction in triple dis... |
3mix3 1225 | Introduction in triple dis... |
3mix1i 1226 | Introduction in triple dis... |
3mix2i 1227 | Introduction in triple dis... |
3mix3i 1228 | Introduction in triple dis... |
3mix1d 1229 | Deduction introducing trip... |
3mix2d 1230 | Deduction introducing trip... |
3mix3d 1231 | Deduction introducing trip... |
3pm3.2i 1232 | Infer conjunction of premi... |
pm3.2an3 1233 | Version of ~ pm3.2 for a t... |
pm3.2an3OLD 1234 | Obsolete proof of ~ pm3.2a... |
3jca 1235 | Join consequents with conj... |
3jcad 1236 | Deduction conjoining the c... |
mpbir3an 1237 | Detach a conjunction of tr... |
mpbir3and 1238 | Detach a conjunction of tr... |
syl3anbrc 1239 | Syllogism inference. (Con... |
3anim123i 1240 | Join antecedents and conse... |
3anim1i 1241 | Add two conjuncts to antec... |
3anim2i 1242 | Add two conjuncts to antec... |
3anim3i 1243 | Add two conjuncts to antec... |
3anbi123i 1244 | Join 3 biconditionals with... |
3orbi123i 1245 | Join 3 biconditionals with... |
3anbi1i 1246 | Inference adding two conju... |
3anbi2i 1247 | Inference adding two conju... |
3anbi3i 1248 | Inference adding two conju... |
3imp 1249 | Importation inference. (C... |
3imp31 1250 | The importation inference ... |
3impa 1251 | Importation from double to... |
3impb 1252 | Importation from double to... |
3impia 1253 | Importation to triple conj... |
3impib 1254 | Importation to triple conj... |
ex3 1255 | Apply ~ ex to a hypothesis... |
3exp 1256 | Exportation inference. (C... |
3expa 1257 | Exportation from triple to... |
3expb 1258 | Exportation from triple to... |
3expia 1259 | Exportation from triple co... |
3expib 1260 | Exportation from triple co... |
3com12 1261 | Commutation in antecedent.... |
3com13 1262 | Commutation in antecedent.... |
3com23 1263 | Commutation in antecedent.... |
3coml 1264 | Commutation in antecedent.... |
3comr 1265 | Commutation in antecedent.... |
3adant3r1 1266 | Deduction adding a conjunc... |
3adant3r2 1267 | Deduction adding a conjunc... |
3adant3r3 1268 | Deduction adding a conjunc... |
3imp21 1269 | The importation inference ... |
3imp3i2an 1270 | An elimination deduction. ... |
3an1rs 1271 | Swap conjuncts. (Contribu... |
3imp1 1272 | Importation to left triple... |
3impd 1273 | Importation deduction for ... |
3imp2 1274 | Importation to right tripl... |
3exp1 1275 | Exportation from left trip... |
3expd 1276 | Exportation deduction for ... |
3exp2 1277 | Exportation from right tri... |
exp5o 1278 | A triple exportation infer... |
exp516 1279 | A triple exportation infer... |
exp520 1280 | A triple exportation infer... |
3impexp 1281 | Version of ~ impexp for a ... |
3anassrs 1282 | Associative law for conjun... |
3an4anass 1283 | Associative law for four c... |
ad4ant13 1284 | Deduction adding conjuncts... |
ad4ant14 1285 | Deduction adding conjuncts... |
ad4ant123 1286 | Deduction adding conjuncts... |
ad4ant124 1287 | Deduction adding conjuncts... |
ad4ant134 1288 | Deduction adding conjuncts... |
ad4ant23 1289 | Deduction adding conjuncts... |
ad4ant24 1290 | Deduction adding conjuncts... |
ad4ant234 1291 | Deduction adding conjuncts... |
ad5ant12 1292 | Deduction adding conjuncts... |
ad5ant13 1293 | Deduction adding conjuncts... |
ad5ant14 1294 | Deduction adding conjuncts... |
ad5ant15 1295 | Deduction adding conjuncts... |
ad5ant23 1296 | Deduction adding conjuncts... |
ad5ant24 1297 | Deduction adding conjuncts... |
ad5ant25 1298 | Deduction adding conjuncts... |
ad5ant245 1299 | Deduction adding conjuncts... |
ad5ant234 1300 | Deduction adding conjuncts... |
ad5ant235 1301 | Deduction adding conjuncts... |
ad5ant123 1302 | Deduction adding conjuncts... |
ad5ant124 1303 | Deduction adding conjuncts... |
ad5ant125 1304 | Deduction adding conjuncts... |
ad5ant134 1305 | Deduction adding conjuncts... |
ad5ant135 1306 | Deduction adding conjuncts... |
ad5ant145 1307 | Deduction adding conjuncts... |
ad5ant1345 1308 | Deduction adding conjuncts... |
ad5ant2345 1309 | Deduction adding conjuncts... |
3adant1l 1310 | Deduction adding a conjunc... |
3adant1r 1311 | Deduction adding a conjunc... |
3adant2l 1312 | Deduction adding a conjunc... |
3adant2r 1313 | Deduction adding a conjunc... |
3adant3l 1314 | Deduction adding a conjunc... |
3adant3r 1315 | Deduction adding a conjunc... |
syl12anc 1316 | Syllogism combined with co... |
syl21anc 1317 | Syllogism combined with co... |
syl3anc 1318 | Syllogism combined with co... |
syl22anc 1319 | Syllogism combined with co... |
syl13anc 1320 | Syllogism combined with co... |
syl31anc 1321 | Syllogism combined with co... |
syl112anc 1322 | Syllogism combined with co... |
syl121anc 1323 | Syllogism combined with co... |
syl211anc 1324 | Syllogism combined with co... |
syl23anc 1325 | Syllogism combined with co... |
syl32anc 1326 | Syllogism combined with co... |
syl122anc 1327 | Syllogism combined with co... |
syl212anc 1328 | Syllogism combined with co... |
syl221anc 1329 | Syllogism combined with co... |
syl113anc 1330 | Syllogism combined with co... |
syl131anc 1331 | Syllogism combined with co... |
syl311anc 1332 | Syllogism combined with co... |
syl33anc 1333 | Syllogism combined with co... |
syl222anc 1334 | Syllogism combined with co... |
syl123anc 1335 | Syllogism combined with co... |
syl132anc 1336 | Syllogism combined with co... |
syl213anc 1337 | Syllogism combined with co... |
syl231anc 1338 | Syllogism combined with co... |
syl312anc 1339 | Syllogism combined with co... |
syl321anc 1340 | Syllogism combined with co... |
syl133anc 1341 | Syllogism combined with co... |
syl313anc 1342 | Syllogism combined with co... |
syl331anc 1343 | Syllogism combined with co... |
syl223anc 1344 | Syllogism combined with co... |
syl232anc 1345 | Syllogism combined with co... |
syl322anc 1346 | Syllogism combined with co... |
syl233anc 1347 | Syllogism combined with co... |
syl323anc 1348 | Syllogism combined with co... |
syl332anc 1349 | Syllogism combined with co... |
syl333anc 1350 | A syllogism inference comb... |
syl3an1 1351 | A syllogism inference. (C... |
syl3an2 1352 | A syllogism inference. (C... |
syl3an3 1353 | A syllogism inference. (C... |
syl3an1b 1354 | A syllogism inference. (C... |
syl3an2b 1355 | A syllogism inference. (C... |
syl3an3b 1356 | A syllogism inference. (C... |
syl3an1br 1357 | A syllogism inference. (C... |
syl3an2br 1358 | A syllogism inference. (C... |
syl3an3br 1359 | A syllogism inference. (C... |
syl3an 1360 | A triple syllogism inferen... |
syl3anb 1361 | A triple syllogism inferen... |
syl3anbr 1362 | A triple syllogism inferen... |
syld3an3 1363 | A syllogism inference. (C... |
syld3an1 1364 | A syllogism inference. (C... |
syld3an2 1365 | A syllogism inference. (C... |
syl3anl1 1366 | A syllogism inference. (C... |
syl3anl2 1367 | A syllogism inference. (C... |
syl3anl3 1368 | A syllogism inference. (C... |
syl3anl 1369 | A triple syllogism inferen... |
syl3anr1 1370 | A syllogism inference. (C... |
syl3anr2 1371 | A syllogism inference. (C... |
syl3anr3 1372 | A syllogism inference. (C... |
3impdi 1373 | Importation inference (und... |
3impdir 1374 | Importation inference (und... |
3anidm12 1375 | Inference from idempotent ... |
3anidm13 1376 | Inference from idempotent ... |
3anidm23 1377 | Inference from idempotent ... |
syl2an3an 1378 | ~ syl3an with antecedents ... |
syl2an23an 1379 | Deduction related to ~ syl... |
3ori 1380 | Infer implication from tri... |
3jao 1381 | Disjunction of three antec... |
3jaob 1382 | Disjunction of three antec... |
3jaoi 1383 | Disjunction of three antec... |
3jaod 1384 | Disjunction of three antec... |
3jaoian 1385 | Disjunction of three antec... |
3jaodan 1386 | Disjunction of three antec... |
mpjao3dan 1387 | Eliminate a three-way disj... |
3jaao 1388 | Inference conjoining and d... |
syl3an9b 1389 | Nested syllogism inference... |
3orbi123d 1390 | Deduction joining 3 equiva... |
3anbi123d 1391 | Deduction joining 3 equiva... |
3anbi12d 1392 | Deduction conjoining and a... |
3anbi13d 1393 | Deduction conjoining and a... |
3anbi23d 1394 | Deduction conjoining and a... |
3anbi1d 1395 | Deduction adding conjuncts... |
3anbi2d 1396 | Deduction adding conjuncts... |
3anbi3d 1397 | Deduction adding conjuncts... |
3anim123d 1398 | Deduction joining 3 implic... |
3orim123d 1399 | Deduction joining 3 implic... |
an6 1400 | Rearrangement of 6 conjunc... |
3an6 1401 | Analogue of ~ an4 for trip... |
3or6 1402 | Analogue of ~ or4 for trip... |
mp3an1 1403 | An inference based on modu... |
mp3an2 1404 | An inference based on modu... |
mp3an3 1405 | An inference based on modu... |
mp3an12 1406 | An inference based on modu... |
mp3an13 1407 | An inference based on modu... |
mp3an23 1408 | An inference based on modu... |
mp3an1i 1409 | An inference based on modu... |
mp3anl1 1410 | An inference based on modu... |
mp3anl2 1411 | An inference based on modu... |
mp3anl3 1412 | An inference based on modu... |
mp3anr1 1413 | An inference based on modu... |
mp3anr2 1414 | An inference based on modu... |
mp3anr3 1415 | An inference based on modu... |
mp3an 1416 | An inference based on modu... |
mpd3an3 1417 | An inference based on modu... |
mpd3an23 1418 | An inference based on modu... |
mp3and 1419 | A deduction based on modus... |
mp3an12i 1420 | ~ mp3an with antecedents i... |
mp3an2i 1421 | ~ mp3an with antecedents i... |
mp3an3an 1422 | ~ mp3an with antecedents i... |
mp3an2ani 1423 | An elimination deduction. ... |
biimp3a 1424 | Infer implication from a l... |
biimp3ar 1425 | Infer implication from a l... |
3anandis 1426 | Inference that undistribut... |
3anandirs 1427 | Inference that undistribut... |
ecase23d 1428 | Deduction for elimination ... |
3ecase 1429 | Inference for elimination ... |
3bior1fd 1430 | A disjunction is equivalen... |
3bior1fand 1431 | A disjunction is equivalen... |
3bior2fd 1432 | A wff is equivalent to its... |
3biant1d 1433 | A conjunction is equivalen... |
intn3an1d 1434 | Introduction of a triple c... |
intn3an2d 1435 | Introduction of a triple c... |
intn3an3d 1436 | Introduction of a triple c... |
an3andi 1437 | Distribution of conjunctio... |
an33rean 1438 | Rearrange a 9-fold conjunc... |
nanan 1441 | Write 'and' in terms of 'n... |
nancom 1442 | The 'nand' operator commut... |
nannan 1443 | Lemma for handling nested ... |
nanim 1444 | Show equivalence between i... |
nannot 1445 | Show equivalence between n... |
nanbi 1446 | Show equivalence between t... |
nanbi1 1447 | Introduce a right anti-con... |
nanbi2 1448 | Introduce a left anti-conj... |
nanbi12 1449 | Join two logical equivalen... |
nanbi1i 1450 | Introduce a right anti-con... |
nanbi2i 1451 | Introduce a left anti-conj... |
nanbi12i 1452 | Join two logical equivalen... |
nanbi1d 1453 | Introduce a right anti-con... |
nanbi2d 1454 | Introduce a left anti-conj... |
nanbi12d 1455 | Join two logical equivalen... |
xnor 1458 | Two ways to write XNOR. (C... |
xorcom 1459 | The connector ` \/_ ` is c... |
xorass 1460 | The connector ` \/_ ` is a... |
excxor 1461 | This tautology shows that ... |
xor2 1462 | Two ways to express "exclu... |
xoror 1463 | XOR implies OR. (Contribut... |
xornan 1464 | XOR implies NAND. (Contrib... |
xornan2 1465 | XOR implies NAND (written ... |
xorneg2 1466 | The connector ` \/_ ` is n... |
xorneg1 1467 | The connector ` \/_ ` is n... |
xorneg 1468 | The connector ` \/_ ` is u... |
xorbi12i 1469 | Equality property for XOR.... |
xorbi12d 1470 | Equality property for XOR.... |
anxordi 1471 | Conjunction distributes ov... |
xorexmid 1472 | Exclusive-or variant of th... |
trujust 1477 | Soundness justification th... |
tru 1479 | The truth value ` T. ` is ... |
fal 1482 | The truth value ` F. ` is ... |
dftru2 1483 | An alternate definition of... |
trud 1484 | Eliminate ` T. ` as an ant... |
tbtru 1485 | A proposition is equivalen... |
nbfal 1486 | The negation of a proposit... |
bitru 1487 | A theorem is equivalent to... |
bifal 1488 | A contradiction is equival... |
falim 1489 | The truth value ` F. ` imp... |
falimd 1490 | The truth value ` F. ` imp... |
a1tru 1491 | Anything implies ` T. ` . ... |
truan 1492 | True can be removed from a... |
dfnot 1493 | Given falsum ` F. ` , we c... |
inegd 1494 | Negation introduction rule... |
efald 1495 | Deduction based on reducti... |
pm2.21fal 1496 | If a wff and its negation ... |
truantru 1497 | A ` /\ ` identity. (Contr... |
truanfal 1498 | A ` /\ ` identity. (Contr... |
falantru 1499 | A ` /\ ` identity. (Contr... |
falanfal 1500 | A ` /\ ` identity. (Contr... |
truortru 1501 | A ` \/ ` identity. (Contr... |
truorfal 1502 | A ` \/ ` identity. (Contr... |
falortru 1503 | A ` \/ ` identity. (Contr... |
falorfal 1504 | A ` \/ ` identity. (Contr... |
truimtru 1505 | A ` -> ` identity. (Contr... |
truimfal 1506 | A ` -> ` identity. (Contr... |
falimtru 1507 | A ` -> ` identity. (Contr... |
falimfal 1508 | A ` -> ` identity. (Contr... |
nottru 1509 | A ` -. ` identity. (Contr... |
notfal 1510 | A ` -. ` identity. (Contr... |
trubitru 1511 | A ` <-> ` identity. (Cont... |
falbitru 1512 | A ` <-> ` identity. (Cont... |
trubifal 1513 | A ` <-> ` identity. (Cont... |
falbifal 1514 | A ` <-> ` identity. (Cont... |
trunantru 1515 | A ` -/\ ` identity. (Cont... |
trunanfal 1516 | A ` -/\ ` identity. (Cont... |
falnantru 1517 | A ` -/\ ` identity. (Cont... |
falnanfal 1518 | A ` -/\ ` identity. (Cont... |
truxortru 1519 | A ` \/_ ` identity. (Cont... |
truxorfal 1520 | A ` \/_ ` identity. (Cont... |
falxortru 1521 | A ` \/_ ` identity. (Cont... |
falxorfal 1522 | A ` \/_ ` identity. (Cont... |
hadbi123d 1525 | Equality theorem for the a... |
hadbi123i 1526 | Equality theorem for the a... |
hadass 1527 | Associative law for the ad... |
hadbi 1528 | The adder sum is the same ... |
hadcoma 1529 | Commutative law for the ad... |
hadcomb 1530 | Commutative law for the ad... |
hadrot 1531 | Rotation law for the adder... |
hadnot 1532 | The adder sum distributes ... |
had1 1533 | If the first input is true... |
had0 1534 | If the first input is fals... |
hadifp 1535 | The value of the adder sum... |
cador 1538 | The adder carry in disjunc... |
cadan 1539 | The adder carry in conjunc... |
cadbi123d 1540 | Equality theorem for the a... |
cadbi123i 1541 | Equality theorem for the a... |
cadcoma 1542 | Commutative law for the ad... |
cadcomb 1543 | Commutative law for the ad... |
cadrot 1544 | Rotation law for the adder... |
cadnot 1545 | The adder carry distribute... |
cad1 1546 | If one input is true, then... |
cad0 1547 | If one input is false, the... |
cadifp 1548 | The value of the carry is,... |
cad11 1549 | If (at least) two inputs a... |
cadtru 1550 | The adder carry is true as... |
minimp 1551 | A single axiom for minimal... |
minimp-sylsimp 1552 | Derivation of sylsimp ( ~ ... |
minimp-ax1 1553 | Derivation of ~ ax-1 from ... |
minimp-ax2c 1554 | Derivation of a commuted f... |
minimp-ax2 1555 | Derivation of ~ ax-2 from ... |
minimp-pm2.43 1556 | Derivation of ~ pm2.43 (al... |
meredith 1557 | Carew Meredith's sole axio... |
merlem1 1558 | Step 3 of Meredith's proof... |
merlem2 1559 | Step 4 of Meredith's proof... |
merlem3 1560 | Step 7 of Meredith's proof... |
merlem4 1561 | Step 8 of Meredith's proof... |
merlem5 1562 | Step 11 of Meredith's proo... |
merlem6 1563 | Step 12 of Meredith's proo... |
merlem7 1564 | Between steps 14 and 15 of... |
merlem8 1565 | Step 15 of Meredith's proo... |
merlem9 1566 | Step 18 of Meredith's proo... |
merlem10 1567 | Step 19 of Meredith's proo... |
merlem11 1568 | Step 20 of Meredith's proo... |
merlem12 1569 | Step 28 of Meredith's proo... |
merlem13 1570 | Step 35 of Meredith's proo... |
luk-1 1571 | 1 of 3 axioms for proposit... |
luk-2 1572 | 2 of 3 axioms for proposit... |
luk-3 1573 | 3 of 3 axioms for proposit... |
luklem1 1574 | Used to rederive standard ... |
luklem2 1575 | Used to rederive standard ... |
luklem3 1576 | Used to rederive standard ... |
luklem4 1577 | Used to rederive standard ... |
luklem5 1578 | Used to rederive standard ... |
luklem6 1579 | Used to rederive standard ... |
luklem7 1580 | Used to rederive standard ... |
luklem8 1581 | Used to rederive standard ... |
ax1 1582 | Standard propositional axi... |
ax2 1583 | Standard propositional axi... |
ax3 1584 | Standard propositional axi... |
nic-dfim 1585 | Define implication in term... |
nic-dfneg 1586 | Define negation in terms o... |
nic-mp 1587 | Derive Nicod's rule of mod... |
nic-mpALT 1588 | A direct proof of ~ nic-mp... |
nic-ax 1589 | Nicod's axiom derived from... |
nic-axALT 1590 | A direct proof of ~ nic-ax... |
nic-imp 1591 | Inference for ~ nic-mp usi... |
nic-idlem1 1592 | Lemma for ~ nic-id . (Con... |
nic-idlem2 1593 | Lemma for ~ nic-id . Infe... |
nic-id 1594 | Theorem ~ id expressed wit... |
nic-swap 1595 | The connector ` -/\ ` is s... |
nic-isw1 1596 | Inference version of ~ nic... |
nic-isw2 1597 | Inference for swapping nes... |
nic-iimp1 1598 | Inference version of ~ nic... |
nic-iimp2 1599 | Inference version of ~ nic... |
nic-idel 1600 | Inference to remove the tr... |
nic-ich 1601 | Chained inference. (Contr... |
nic-idbl 1602 | Double the terms. Since d... |
nic-bijust 1603 | Biconditional justificatio... |
nic-bi1 1604 | Inference to extract one s... |
nic-bi2 1605 | Inference to extract the o... |
nic-stdmp 1606 | Derive the standard modus ... |
nic-luk1 1607 | Proof of ~ luk-1 from ~ ni... |
nic-luk2 1608 | Proof of ~ luk-2 from ~ ni... |
nic-luk3 1609 | Proof of ~ luk-3 from ~ ni... |
lukshef-ax1 1610 | This alternative axiom for... |
lukshefth1 1611 | Lemma for ~ renicax . (Co... |
lukshefth2 1612 | Lemma for ~ renicax . (Co... |
renicax 1613 | A rederivation of ~ nic-ax... |
tbw-bijust 1614 | Justification for ~ tbw-ne... |
tbw-negdf 1615 | The definition of negation... |
tbw-ax1 1616 | The first of four axioms i... |
tbw-ax2 1617 | The second of four axioms ... |
tbw-ax3 1618 | The third of four axioms i... |
tbw-ax4 1619 | The fourth of four axioms ... |
tbwsyl 1620 | Used to rederive the Lukas... |
tbwlem1 1621 | Used to rederive the Lukas... |
tbwlem2 1622 | Used to rederive the Lukas... |
tbwlem3 1623 | Used to rederive the Lukas... |
tbwlem4 1624 | Used to rederive the Lukas... |
tbwlem5 1625 | Used to rederive the Lukas... |
re1luk1 1626 | ~ luk-1 derived from the T... |
re1luk2 1627 | ~ luk-2 derived from the T... |
re1luk3 1628 | ~ luk-3 derived from the T... |
merco1 1629 | A single axiom for proposi... |
merco1lem1 1630 | Used to rederive the Tarsk... |
retbwax4 1631 | ~ tbw-ax4 rederived from ~... |
retbwax2 1632 | ~ tbw-ax2 rederived from ~... |
merco1lem2 1633 | Used to rederive the Tarsk... |
merco1lem3 1634 | Used to rederive the Tarsk... |
merco1lem4 1635 | Used to rederive the Tarsk... |
merco1lem5 1636 | Used to rederive the Tarsk... |
merco1lem6 1637 | Used to rederive the Tarsk... |
merco1lem7 1638 | Used to rederive the Tarsk... |
retbwax3 1639 | ~ tbw-ax3 rederived from ~... |
merco1lem8 1640 | Used to rederive the Tarsk... |
merco1lem9 1641 | Used to rederive the Tarsk... |
merco1lem10 1642 | Used to rederive the Tarsk... |
merco1lem11 1643 | Used to rederive the Tarsk... |
merco1lem12 1644 | Used to rederive the Tarsk... |
merco1lem13 1645 | Used to rederive the Tarsk... |
merco1lem14 1646 | Used to rederive the Tarsk... |
merco1lem15 1647 | Used to rederive the Tarsk... |
merco1lem16 1648 | Used to rederive the Tarsk... |
merco1lem17 1649 | Used to rederive the Tarsk... |
merco1lem18 1650 | Used to rederive the Tarsk... |
retbwax1 1651 | ~ tbw-ax1 rederived from ~... |
merco2 1652 | A single axiom for proposi... |
mercolem1 1653 | Used to rederive the Tarsk... |
mercolem2 1654 | Used to rederive the Tarsk... |
mercolem3 1655 | Used to rederive the Tarsk... |
mercolem4 1656 | Used to rederive the Tarsk... |
mercolem5 1657 | Used to rederive the Tarsk... |
mercolem6 1658 | Used to rederive the Tarsk... |
mercolem7 1659 | Used to rederive the Tarsk... |
mercolem8 1660 | Used to rederive the Tarsk... |
re1tbw1 1661 | ~ tbw-ax1 rederived from ~... |
re1tbw2 1662 | ~ tbw-ax2 rederived from ~... |
re1tbw3 1663 | ~ tbw-ax3 rederived from ~... |
re1tbw4 1664 | ~ tbw-ax4 rederived from ~... |
rb-bijust 1665 | Justification for ~ rb-imd... |
rb-imdf 1666 | The definition of implicat... |
anmp 1667 | Modus ponens for ` \/ ` ` ... |
rb-ax1 1668 | The first of four axioms i... |
rb-ax2 1669 | The second of four axioms ... |
rb-ax3 1670 | The third of four axioms i... |
rb-ax4 1671 | The fourth of four axioms ... |
rbsyl 1672 | Used to rederive the Lukas... |
rblem1 1673 | Used to rederive the Lukas... |
rblem2 1674 | Used to rederive the Lukas... |
rblem3 1675 | Used to rederive the Lukas... |
rblem4 1676 | Used to rederive the Lukas... |
rblem5 1677 | Used to rederive the Lukas... |
rblem6 1678 | Used to rederive the Lukas... |
rblem7 1679 | Used to rederive the Lukas... |
re1axmp 1680 | ~ ax-mp derived from Russe... |
re2luk1 1681 | ~ luk-1 derived from Russe... |
re2luk2 1682 | ~ luk-2 derived from Russe... |
re2luk3 1683 | ~ luk-3 derived from Russe... |
mptnan 1684 | Modus ponendo tollens 1, o... |
mptxor 1685 | Modus ponendo tollens 2, o... |
mtpor 1686 | Modus tollendo ponens (inc... |
mtpxor 1687 | Modus tollendo ponens (ori... |
stoic1a 1688 | Stoic logic Thema 1 (part ... |
stoic1b 1689 | Stoic logic Thema 1 (part ... |
stoic2a 1690 | Stoic logic Thema 2 versio... |
stoic2b 1691 | Stoic logic Thema 2 versio... |
stoic3 1692 | Stoic logic Thema 3. Stat... |
stoic4a 1693 | Stoic logic Thema 4 versio... |
stoic4b 1694 | Stoic logic Thema 4 versio... |
alnex 1697 | Theorem 19.7 of [Margaris]... |
eximal 1698 | A utility theorem. An int... |
nf2 1702 | Alternate definition of no... |
nf3 1703 | Alternate definition of no... |
nf4 1704 | Alternate definition of no... |
nfi 1705 | Deduce that ` x ` is not f... |
nfri 1706 | Consequence of the definit... |
nfd 1707 | Deduce that ` x ` is not f... |
nfrd 1708 | Consequence of the definit... |
nftht0 1709 | Closed form of ~ nfth . (... |
nfntht 1710 | Closed form of ~ nfnth . ... |
nfntht2 1711 | Closed form of ~ nfnth . ... |
gen2 1714 | Generalization applied twi... |
mpg 1715 | Modus ponens combined with... |
mpgbi 1716 | Modus ponens on biconditio... |
mpgbir 1717 | Modus ponens on biconditio... |
nfth 1718 | No variable is (effectivel... |
nfnth 1719 | No variable is (effectivel... |
hbth 1720 | No variable is (effectivel... |
nftru 1721 | The true constant has no f... |
nex 1722 | Generalization rule for ne... |
nffal 1723 | The false constant has no ... |
sptruw 1724 | Version of ~ sp when ` ph ... |
nfiOLD 1725 | Obsolete proof of ~ nf5i a... |
nfthOLD 1726 | Obsolete proof of ~ nfth a... |
nfnthOLD 1727 | Obsolete proof of ~ nfnth ... |
alim 1729 | Restatement of Axiom ~ ax-... |
alimi 1730 | Inference quantifying both... |
2alimi 1731 | Inference doubly quantifyi... |
al2im 1732 | Closed form of ~ al2imi . ... |
al2imi 1733 | Inference quantifying ante... |
alanimi 1734 | Variant of ~ al2imi with c... |
alimdh 1735 | Deduction form of Theorem ... |
albi 1736 | Theorem 19.15 of [Margaris... |
albii 1737 | Inference adding universal... |
2albii 1738 | Inference adding two unive... |
sylgt 1739 | Closed form of ~ sylg . (... |
sylg 1740 | A syllogism combined with ... |
alrimih 1741 | Inference form of Theorem ... |
hbxfrbi 1742 | A utility lemma to transfe... |
alex 1743 | Theorem 19.6 of [Margaris]... |
exnal 1744 | Theorem 19.14 of [Margaris... |
2nalexn 1745 | Part of theorem *11.5 in [... |
2exnaln 1746 | Theorem *11.22 in [Whitehe... |
2nexaln 1747 | Theorem *11.25 in [Whitehe... |
alimex 1748 | A utility theorem. An int... |
aleximi 1749 | A variant of ~ al2imi : in... |
alexbii 1750 | Biconditional form of ~ al... |
exim 1751 | Theorem 19.22 of [Margaris... |
eximi 1752 | Inference adding existenti... |
2eximi 1753 | Inference adding two exist... |
eximii 1754 | Inference associated with ... |
ala1 1755 | Add an antecedent in a uni... |
exa1 1756 | Add an antecedent in an ex... |
19.38 1757 | Theorem 19.38 of [Margaris... |
imnang 1758 | Quantified implication in ... |
alinexa 1759 | A transformation of quanti... |
alexn 1760 | A relationship between two... |
2exnexn 1761 | Theorem *11.51 in [Whitehe... |
exbi 1762 | Theorem 19.18 of [Margaris... |
exbiOLD 1763 | Obsolete proof of ~ exbi a... |
exbii 1764 | Inference adding existenti... |
2exbii 1765 | Inference adding two exist... |
3exbii 1766 | Inference adding three exi... |
nfnt 1767 | If ` x ` is not free in ` ... |
nfn 1768 | Inference associated with ... |
nfnd 1769 | Deduction associated with ... |
nfbii 1770 | Equality theorem for not-f... |
nfxfr 1771 | A utility lemma to transfe... |
nfxfrd 1772 | A utility lemma to transfe... |
exanali 1773 | A transformation of quanti... |
exancom 1774 | Commutation of conjunction... |
exan 1775 | Place a conjunct in the sc... |
exanOLD 1776 | Obsolete proof of ~ exan a... |
alrimdh 1777 | Deduction form of Theorem ... |
eximdh 1778 | Deduction from Theorem 19.... |
nexdh 1779 | Deduction for generalizati... |
albidh 1780 | Formula-building rule for ... |
exbidh 1781 | Formula-building rule for ... |
exbidhOLD 1782 | Obsolete proof of ~ exbidh... |
exsimpl 1783 | Simplification of an exist... |
exsimpr 1784 | Simplification of an exist... |
19.40 1785 | Theorem 19.40 of [Margaris... |
19.26 1786 | Theorem 19.26 of [Margaris... |
19.26-2 1787 | Theorem ~ 19.26 with two q... |
19.26-3an 1788 | Theorem ~ 19.26 with tripl... |
19.29 1789 | Theorem 19.29 of [Margaris... |
19.29r 1790 | Variation of ~ 19.29 . (C... |
19.29rOLD 1791 | Obsolete proof of ~ 19.29r... |
19.29r2 1792 | Variation of ~ 19.29r with... |
19.29x 1793 | Variation of ~ 19.29 with ... |
19.35 1794 | Theorem 19.35 of [Margaris... |
19.35i 1795 | Inference associated with ... |
19.35ri 1796 | Inference associated with ... |
19.25 1797 | Theorem 19.25 of [Margaris... |
19.30 1798 | Theorem 19.30 of [Margaris... |
19.43 1799 | Theorem 19.43 of [Margaris... |
19.43OLD 1800 | Obsolete proof of ~ 19.43 ... |
19.33 1801 | Theorem 19.33 of [Margaris... |
19.33b 1802 | The antecedent provides a ... |
19.40-2 1803 | Theorem *11.42 in [Whitehe... |
19.40b 1804 | The antecedent provides a ... |
19.40bOLD 1805 | Obsolete proof of ~ 19.40b... |
albiim 1806 | Split a biconditional and ... |
2albiim 1807 | Split a biconditional and ... |
exintrbi 1808 | Add/remove a conjunct in t... |
exintrbiOLD 1809 | Obsolete proof of ~ exintr... |
exintr 1810 | Introduce a conjunct in th... |
alsyl 1811 | Theorem *10.3 in [Whitehea... |
nfimd 1812 | If in a context ` x ` is n... |
nfim 1813 | If ` x ` is not free in ` ... |
nfand 1814 | If in a context ` x ` is n... |
nf3and 1815 | Deduction form of bound-va... |
nfan 1816 | If ` x ` is not free in ` ... |
nfanOLD 1817 | Obsolete proof of ~ nfan a... |
nfnan 1818 | If ` x ` is not free in ` ... |
nf3an 1819 | If ` x ` is not free in ` ... |
nfbid 1820 | If in a context ` x ` is n... |
nfbi 1821 | If ` x ` is not free in ` ... |
nfor 1822 | If ` x ` is not free in ` ... |
nf3or 1823 | If ` x ` is not free in ` ... |
nfbiiOLD 1824 | Obsolete proof of ~ nfbii ... |
nfxfrOLD 1825 | Obsolete proof of ~ nfxfr ... |
nfxfrdOLD 1826 | Obsolete proof of ~ nfxfrd... |
ax5d 1828 | ~ ax-5 with antecedent. U... |
ax5e 1829 | A rephrasing of ~ ax-5 usi... |
nfv 1830 | If ` x ` is not present in... |
nfvd 1831 | ~ nfv with antecedent. Us... |
alimdv 1832 | Deduction form of Theorem ... |
eximdv 1833 | Deduction form of Theorem ... |
2alimdv 1834 | Deduction form of Theorem ... |
2eximdv 1835 | Deduction form of Theorem ... |
albidv 1836 | Formula-building rule for ... |
exbidv 1837 | Formula-building rule for ... |
2albidv 1838 | Formula-building rule for ... |
2exbidv 1839 | Formula-building rule for ... |
3exbidv 1840 | Formula-building rule for ... |
4exbidv 1841 | Formula-building rule for ... |
alrimiv 1842 | Inference form of Theorem ... |
alrimivv 1843 | Inference form of Theorem ... |
alrimdv 1844 | Deduction form of Theorem ... |
exlimiv 1845 | Inference form of Theorem ... |
exlimiiv 1846 | Inference associated with ... |
exlimivv 1847 | Inference form of Theorem ... |
exlimdv 1848 | Deduction form of Theorem ... |
exlimdvv 1849 | Deduction form of Theorem ... |
exlimddv 1850 | Existential elimination ru... |
nexdv 1851 | Deduction for generalizati... |
nexdvOLD 1852 | Obsolete proof of ~ nexdv ... |
2ax5 1853 | Quantification of two vari... |
stdpc5v 1854 | Version of ~ stdpc5 with a... |
19.21v 1855 | Version of ~ 19.21 with a ... |
19.32v 1856 | Version of ~ 19.32 with a ... |
19.31v 1857 | Version of ~ 19.31 with a ... |
nfvOLD 1858 | Obsolete proof of ~ nfv as... |
nfvdOLD 1859 | Obsolete proof of ~ nfvd a... |
nfdvOLD 1860 | Obsolete proof of ~ nf5dv ... |
weq 1861 | Extend wff definition to i... |
equs3 1862 | Lemma used in proofs of su... |
speimfw 1863 | Specialization, with addit... |
speimfwALT 1864 | Alternate proof of ~ speim... |
spimfw 1865 | Specialization, with addit... |
ax12i 1866 | Inference that has ~ ax-12... |
sbequ2 1869 | An equality theorem for su... |
sb1 1870 | One direction of a simplif... |
spsbe 1871 | A specialization theorem. ... |
sbequ8 1872 | Elimination of equality fr... |
sbimi 1873 | Infer substitution into an... |
sbbii 1874 | Infer substitution into bo... |
ax6v 1876 | Axiom B7 of [Tarski] p. 75... |
ax6ev 1877 | At least one individual ex... |
exiftru 1878 | Rule of existential genera... |
19.2 1879 | Theorem 19.2 of [Margaris]... |
19.2d 1880 | Deduction associated with ... |
19.8w 1881 | Weak version of ~ 19.8a an... |
19.8v 1882 | Version of ~ 19.8a with a ... |
19.9v 1883 | Version of ~ 19.9 with a d... |
19.3v 1884 | Version of ~ 19.3 with a d... |
spvw 1885 | Version of ~ sp when ` x `... |
19.39 1886 | Theorem 19.39 of [Margaris... |
19.24 1887 | Theorem 19.24 of [Margaris... |
19.34 1888 | Theorem 19.34 of [Margaris... |
19.23v 1889 | Version of ~ 19.23 with a ... |
19.23vv 1890 | Theorem ~ 19.23v extended ... |
19.36v 1891 | Version of ~ 19.36 with a ... |
19.36iv 1892 | Inference associated with ... |
pm11.53v 1893 | Version of ~ pm11.53 with ... |
19.12vvv 1894 | Version of ~ 19.12vv with ... |
19.27v 1895 | Version of ~ 19.27 with a ... |
19.28v 1896 | Version of ~ 19.28 with a ... |
19.37v 1897 | Version of ~ 19.37 with a ... |
19.37iv 1898 | Inference associated with ... |
19.44v 1899 | Version of ~ 19.44 with a ... |
19.45v 1900 | Version of ~ 19.45 with a ... |
19.41v 1901 | Version of ~ 19.41 with a ... |
19.41vv 1902 | Version of ~ 19.41 with tw... |
19.41vvv 1903 | Version of ~ 19.41 with th... |
19.41vvvv 1904 | Version of ~ 19.41 with fo... |
19.42v 1905 | Version of ~ 19.42 with a ... |
exdistr 1906 | Distribution of existentia... |
19.42vv 1907 | Version of ~ 19.42 with tw... |
19.42vvv 1908 | Version of ~ 19.42 with th... |
exdistr2 1909 | Distribution of existentia... |
3exdistr 1910 | Distribution of existentia... |
4exdistr 1911 | Distribution of existentia... |
spimeh 1912 | Existential introduction, ... |
spimw 1913 | Specialization. Lemma 8 o... |
spimvw 1914 | Specialization. Lemma 8 o... |
spnfw 1915 | Weak version of ~ sp . Us... |
spfalw 1916 | Version of ~ sp when ` ph ... |
equs4v 1917 | Version of ~ equs4 with a ... |
equsalvw 1918 | Version of ~ equsal with t... |
equsexvw 1919 | Version of ~ equsexv with ... |
cbvaliw 1920 | Change bound variable. Us... |
cbvalivw 1921 | Change bound variable. Us... |
ax7v 1923 | Weakened version of ~ ax-7... |
ax7v1 1924 | First of two weakened vers... |
ax7v2 1925 | Second of two weakened ver... |
equid 1926 | Identity law for equality.... |
nfequid 1927 | Bound-variable hypothesis ... |
equcomiv 1928 | Weaker form of ~ equcomi w... |
ax6evr 1929 | A commuted form of ~ ax6ev... |
ax7 1930 | Proof of ~ ax-7 from ~ ax7... |
equcomi 1931 | Commutative law for equali... |
equcom 1932 | Commutative law for equali... |
equcomd 1933 | Deduction form of ~ equcom... |
equcoms 1934 | An inference commuting equ... |
equtr 1935 | A transitive law for equal... |
equtrr 1936 | A transitive law for equal... |
equeuclr 1937 | Commuted version of ~ eque... |
equeucl 1938 | Equality is a left-Euclide... |
equequ1 1939 | An equivalence law for equ... |
equequ2 1940 | An equivalence law for equ... |
equtr2 1941 | Equality is a left-Euclide... |
equequ2OLD 1942 | Obsolete proof of ~ equequ... |
equtr2OLD 1943 | Obsolete proof of ~ equtr2... |
stdpc6 1944 | One of the two equality ax... |
stdpc7 1945 | One of the two equality ax... |
equvinv 1946 | A variable introduction la... |
equviniva 1947 | A modified version of the ... |
equvinivOLD 1948 | The forward implication of... |
equvinvOLD 1949 | Obsolete version of ~ equv... |
equvelv 1950 | A specialized version of ~... |
ax13b 1951 | An equivalence between two... |
spfw 1952 | Weak version of ~ sp . Us... |
spfwOLD 1953 | Obsolete proof of ~ spfw a... |
spw 1954 | Weak version of the specia... |
cbvalw 1955 | Change bound variable. Us... |
cbvalvw 1956 | Change bound variable. Us... |
cbvexvw 1957 | Change bound variable. Us... |
alcomiw 1958 | Weak version of ~ alcom . ... |
hbn1fw 1959 | Weak version of ~ ax-10 fr... |
hbn1w 1960 | Weak version of ~ hbn1 . ... |
hba1w 1961 | Weak version of ~ hba1 . ... |
hba1wOLD 1962 | Obsolete proof of ~ hba1w ... |
hbe1w 1963 | Weak version of ~ hbe1 . ... |
hbalw 1964 | Weak version of ~ hbal . ... |
spaev 1965 | A special instance of ~ sp... |
cbvaev 1966 | Change bound variable in a... |
aevlem0 1967 | Lemma for ~ aevlem . Inst... |
aevlem 1968 | Lemma for ~ aev and ~ axc1... |
aeveq 1969 | The antecedent ` A. x x = ... |
aev 1970 | A "distinctor elimination"... |
hbaevg 1971 | Generalization of ~ hbaev ... |
hbaev 1972 | Version of ~ hbae with a D... |
aev2 1973 | A version of ~ aev with tw... |
aev2ALT 1974 | Alternate proof of ~ aev2 ... |
axc11nlemOLD2 1975 | Lemma for ~ axc11n . Chan... |
aevlemOLD 1976 | Old proof of ~ aevlem . O... |
wel 1978 | Extend wff definition to i... |
ax8v 1980 | Weakened version of ~ ax-8... |
ax8v1 1981 | First of two weakened vers... |
ax8v2 1982 | Second of two weakened ver... |
ax8 1983 | Proof of ~ ax-8 from ~ ax8... |
elequ1 1984 | An identity law for the no... |
cleljust 1985 | When the class variables i... |
ax9v 1987 | Weakened version of ~ ax-9... |
ax9v1 1988 | First of two weakened vers... |
ax9v2 1989 | Second of two weakened ver... |
ax9 1990 | Proof of ~ ax-9 from ~ ax9... |
elequ2 1991 | An identity law for the no... |
ax6dgen 1992 | Tarski's system uses the w... |
ax10w 1993 | Weak version of ~ ax-10 fr... |
ax11w 1994 | Weak version of ~ ax-11 fr... |
ax11dgen 1995 | Degenerate instance of ~ a... |
ax12wlem 1996 | Lemma for weak version of ... |
ax12w 1997 | Weak version of ~ ax-12 fr... |
ax12dgen 1998 | Degenerate instance of ~ a... |
ax12wdemo 1999 | Example of an application ... |
ax13w 2000 | Weak version (principal in... |
ax13dgen1 2001 | Degenerate instance of ~ a... |
ax13dgen2 2002 | Degenerate instance of ~ a... |
ax13dgen3 2003 | Degenerate instance of ~ a... |
ax13dgen4 2004 | Degenerate instance of ~ a... |
ax13dgen4OLD 2005 | Obsolete proof of ~ ax13dg... |
hbn1 2007 | Alias for ~ ax-10 to be us... |
hbe1 2008 | The setvar ` x ` is not fr... |
hbe1a 2009 | Dual statement of ~ hbe1 .... |
nf5-1 2010 | One direction of ~ nf5 can... |
nf5i 2011 | Deduce that ` x ` is not f... |
nf5dv 2012 | Apply the definition of no... |
nf5dh 2013 | Deduce that ` x ` is not f... |
nfe1 2014 | The setvar ` x ` is not fr... |
nfa1 2015 | The setvar ` x ` is not fr... |
nfna1 2016 | A convenience theorem part... |
nfia1 2017 | Lemma 23 of [Monk2] p. 114... |
nfnf1 2018 | The setvar ` x ` is not fr... |
modal-5 2019 | The analogue in our predic... |
nfe1OLD 2020 | Obsolete proof of ~ nfe1 a... |
alcoms 2022 | Swap quantifiers in an ant... |
hbal 2023 | If ` x ` is not free in ` ... |
alcom 2024 | Theorem 19.5 of [Margaris]... |
alrot3 2025 | Theorem *11.21 in [Whitehe... |
alrot4 2026 | Rotate four universal quan... |
nfa2 2027 | Lemma 24 of [Monk2] p. 114... |
hbald 2028 | Deduction form of bound-va... |
excom 2029 | Theorem 19.11 of [Margaris... |
excomim 2030 | One direction of Theorem 1... |
excom13 2031 | Swap 1st and 3rd existenti... |
exrot3 2032 | Rotate existential quantif... |
exrot4 2033 | Rotate existential quantif... |
ax12v 2035 | This is essentially axiom ... |
ax12v2 2036 | It is possible to remove a... |
ax12vOLD 2037 | Obsolete proof of ~ ax12v2... |
ax12vOLDOLD 2038 | Obsolete proof of ~ ax12v ... |
19.8a 2039 | If a wff is true, it is tr... |
19.8aOLD 2040 | Obsolete proof of ~ 19.8a ... |
sp 2041 | Specialization. A univers... |
spi 2042 | Inference rule reversing g... |
sps 2043 | Generalization of antecede... |
2sp 2044 | A double specialization (s... |
spsd 2045 | Deduction generalizing ant... |
19.2g 2046 | Theorem 19.2 of [Margaris]... |
19.21bi 2047 | Inference form of ~ 19.21 ... |
19.21bbi 2048 | Inference removing double ... |
19.23bi 2049 | Inference form of Theorem ... |
nexr 2050 | Inference associated with ... |
qexmid 2051 | Quantified excluded middle... |
nf5r 2052 | Consequence of the definit... |
nf5ri 2053 | Consequence of the definit... |
nf5rd 2054 | Consequence of the definit... |
nfim1 2055 | A closed form of ~ nfim . ... |
nfan1 2056 | A closed form of ~ nfan . ... |
19.3 2057 | A wff may be quantified wi... |
19.9d 2058 | A deduction version of one... |
19.9t 2059 | A closed version of ~ 19.9... |
19.9 2060 | A wff may be existentially... |
19.21t 2061 | Closed form of Theorem 19.... |
19.21 2062 | Theorem 19.21 of [Margaris... |
stdpc5 2063 | An axiom scheme of standar... |
stdpc5OLD 2064 | Obsolete proof of ~ stdpc5... |
19.21-2 2065 | Version of ~ 19.21 with tw... |
19.23t 2066 | Closed form of Theorem 197... |
19.23 2067 | Theorem 19.23 of [Margaris... |
alimd 2068 | Deduction form of Theorem ... |
alrimi 2069 | Inference form of Theorem ... |
alrimdd 2070 | Deduction form of Theorem ... |
alrimd 2071 | Deduction form of Theorem ... |
eximd 2072 | Deduction form of Theorem ... |
exlimi 2073 | Inference associated with ... |
exlimd 2074 | Deduction form of Theorem ... |
exlimdd 2075 | Existential elimination ru... |
nexd 2076 | Deduction for generalizati... |
albid 2077 | Formula-building rule for ... |
exbid 2078 | Formula-building rule for ... |
nfbidf 2079 | An equality theorem for ef... |
19.16 2080 | Theorem 19.16 of [Margaris... |
19.17 2081 | Theorem 19.17 of [Margaris... |
19.27 2082 | Theorem 19.27 of [Margaris... |
19.28 2083 | Theorem 19.28 of [Margaris... |
19.19 2084 | Theorem 19.19 of [Margaris... |
19.36 2085 | Theorem 19.36 of [Margaris... |
19.36i 2086 | Inference associated with ... |
19.37 2087 | Theorem 19.37 of [Margaris... |
19.32 2088 | Theorem 19.32 of [Margaris... |
19.31 2089 | Theorem 19.31 of [Margaris... |
19.41 2090 | Theorem 19.41 of [Margaris... |
19.42-1 2091 | One direction of ~ 19.42 .... |
19.42 2092 | Theorem 19.42 of [Margaris... |
19.44 2093 | Theorem 19.44 of [Margaris... |
19.45 2094 | Theorem 19.45 of [Margaris... |
equsexv 2095 | Version of ~ equsex with a... |
sbequ1 2096 | An equality theorem for su... |
sbequ12 2097 | An equality theorem for su... |
sbequ12r 2098 | An equality theorem for su... |
sbequ12a 2099 | An equality theorem for su... |
sbid 2100 | An identity theorem for su... |
spimv1 2101 | Version of ~ spim with a d... |
nf5 2102 | Alternate definition of ~ ... |
nf6 2103 | An alternate definition of... |
nf5d 2104 | Deduce that ` x ` is not f... |
nf5di 2105 | Since the converse holds b... |
19.9h 2106 | A wff may be existentially... |
19.21h 2107 | Theorem 19.21 of [Margaris... |
19.23h 2108 | Theorem 19.23 of [Margaris... |
equsalhw 2109 | Weaker version of ~ equsal... |
hbim1 2110 | A closed form of ~ hbim . ... |
hbimd 2111 | Deduction form of bound-va... |
hbim 2112 | If ` x ` is not free in ` ... |
hban 2113 | If ` x ` is not free in ` ... |
hb3an 2114 | If ` x ` is not free in ` ... |
axc4 2115 | Show that the original axi... |
axc4i 2116 | Inference version of ~ axc... |
axc7 2117 | Show that the original axi... |
axc7e 2118 | Abbreviated version of ~ a... |
axc16g 2119 | Generalization of ~ axc16 ... |
axc16 2120 | Proof of older axiom ~ ax-... |
axc16gb 2121 | Biconditional strengthenin... |
axc16nf 2122 | If ~ dtru is false, then t... |
axc11v 2123 | Version of ~ axc11 with a ... |
axc11rv 2124 | Version of ~ axc11r with a... |
axc11rvOLD 2125 | Obsolete proof of ~ axc11r... |
axc11vOLD 2126 | Obsolete proof of ~ axc11v... |
modal-b 2127 | The analogue in our predic... |
19.9ht 2128 | A closed version of ~ 19.9... |
hbnt 2129 | Closed theorem version of ... |
hbntOLD 2130 | Obsolete proof of ~ hbnt a... |
hbn 2131 | If ` x ` is not free in ` ... |
hbnd 2132 | Deduction form of bound-va... |
exlimih 2133 | Inference associated with ... |
exlimdh 2134 | Deduction form of Theorem ... |
equsexhv 2135 | Version of ~ equsexh with ... |
sb56 2136 | Two equivalent ways of exp... |
hba1 2137 | The setvar ` x ` is not fr... |
hbexOLD 2138 | Obsolete proof of ~ hbex a... |
nfal 2139 | If ` x ` is not free in ` ... |
nfex 2140 | If ` x ` is not free in ` ... |
nfexOLD 2141 | Obsolete proof of ~ nfex a... |
hbex 2142 | If ` x ` is not free in ` ... |
nfa1OLD 2143 | Obsolete proof of ~ nfa1 a... |
nfnf 2144 | If ` x ` is not free in ` ... |
nfnf1OLD 2145 | Obsolete proof of ~ nfnf1 ... |
axc11nlemOLD 2146 | Obsolete proof of ~ axc11n... |
axc16gOLD 2147 | Obsolete proof of ~ axc16g... |
aevOLD 2148 | Obsolete proof of ~ aev as... |
axc16nfOLD 2149 | Obsolete proof of ~ axc16n... |
19.12 2150 | Theorem 19.12 of [Margaris... |
nfald 2151 | Deduction form of ~ nfal .... |
nfaldOLD 2152 | Obsolete proof of ~ nfald ... |
nfexd 2153 | If ` x ` is not free in ` ... |
nfa2OLD 2154 | Obsolete proof of ~ nfa2 a... |
exanOLDOLD 2155 | Obsolete proof of ~ exan a... |
aaan 2156 | Rearrange universal quanti... |
eeor 2157 | Rearrange existential quan... |
cbv3v 2158 | Version of ~ cbv3 with a d... |
dvelimhw 2159 | Proof of ~ dvelimh without... |
cbv3hv 2160 | Version of ~ cbv3h with a ... |
cbv3hvOLD 2161 | Obsolete proof of ~ cbv3hv... |
cbv3hvOLDOLD 2162 | Obsolete proof of ~ cbv3hv... |
cbvalv1 2163 | Version of ~ cbval with a ... |
cbvexv1 2164 | Version of ~ cbvex with a ... |
equs5aALT 2165 | Alternate proof of ~ equs5... |
equs5eALT 2166 | Alternate proof of ~ equs5... |
pm11.53 2167 | Theorem *11.53 in [Whitehe... |
19.12vv 2168 | Special case of ~ 19.12 wh... |
eean 2169 | Rearrange existential quan... |
eeanv 2170 | Rearrange existential quan... |
eeeanv 2171 | Rearrange existential quan... |
ee4anv 2172 | Rearrange existential quan... |
cleljustALT 2173 | Alternate proof of ~ clelj... |
cleljustALT2 2174 | Alternate proof of ~ clelj... |
axc11r 2175 | Same as ~ axc11 but with r... |
nfrOLD 2176 | Obsolete proof of ~ nf5r a... |
nfriOLD 2177 | Obsolete proof of ~ nf5ri ... |
nfrdOLD 2178 | Obsolete proof of ~ nf5rd ... |
alimdOLD 2179 | Obsolete proof of ~ alimd ... |
alrimiOLD 2180 | Obsolete proof of ~ alrimi... |
nfdOLD 2181 | Obsolete proof of ~ nf5d a... |
nfdhOLD 2182 | Obsolete proof of ~ nf5dh ... |
alrimddOLD 2183 | Obsolete proof of ~ alrimd... |
alrimdOLD 2184 | Obsolete proof of ~ alrimd... |
eximdOLD 2185 | Obsolete proof of ~ eximd ... |
nexdOLD 2186 | Obsolete proof of ~ nexd a... |
albidOLD 2187 | Obsolete proof of ~ albid ... |
exbidOLD 2188 | Obsolete proof of ~ exbid ... |
nfbidfOLD 2189 | Obsolete proof of ~ nfbidf... |
19.3OLD 2190 | Obsolete proof of ~ 19.3 a... |
19.9dOLD 2191 | Obsolete proof of ~ 19.9d ... |
19.9tOLD 2192 | Obsolete proof of ~ 19.9t ... |
19.9OLD 2193 | Obsolete proof of ~ 19.9 a... |
19.9hOLD 2194 | Obsolete proof of ~ 19.9h ... |
nfa1OLDOLD 2195 | Obsolete proof of ~ nfa1 a... |
nfnf1OLDOLD 2196 | Obsolete proof of ~ nfnf1 ... |
nfntOLD 2197 | Obsolete proof of ~ nfnt a... |
nfnOLD 2198 | Obsolete proof of ~ nfn as... |
nfndOLD 2199 | Obsolete proof of ~ nfnd a... |
19.21t-1OLD 2200 | One direction of the bi-co... |
19.21tOLD 2201 | Obsolete proof of ~ 19.21t... |
19.21OLD 2202 | Obsolete proof of ~ 19.21 ... |
19.21-2OLD 2203 | Obsolete proof of ~ 19.21-... |
19.21hOLD 2204 | Obsolete proof of ~ 19.21h... |
stdpc5OLDOLD 2205 | Obsolete proof of ~ stdpc5... |
19.23tOLD 2206 | Obsolete proof of ~ 19.23t... |
19.23OLD 2207 | Obsolete proof of ~ 19.23 ... |
19.23hOLD 2208 | Obsolete proof of ~ 19.23h... |
exlimiOLD 2209 | Obsolete proof of ~ exlimi... |
exlimihOLD 2210 | Obsolete proof of ~ exlimi... |
exlimdOLD 2211 | Obsolete proof of ~ exlimd... |
exlimdhOLD 2212 | Obsolete proof of ~ exlimd... |
nfdiOLD 2213 | Obsolete proof of ~ nf5di ... |
nfimdOLD 2214 | Obsolete proof of ~ nfimd ... |
hbim1OLD 2215 | Obsolete proof of ~ hbim a... |
nfim1OLD 2216 | Obsolete proof of ~ nfim1 ... |
nfimOLD 2217 | Obsolete proof of ~ nfim a... |
hbimdOLD 2218 | Obsolete proof of ~ hbimd ... |
hbimOLD 2219 | Obsolete proof of ~ hbim a... |
nfandOLD 2220 | Obsolete proof of ~ nfand ... |
nf3andOLD 2221 | Obsolete proof of ~ nf3and... |
19.27OLD 2222 | Obsolete proof of ~ 19.27 ... |
19.28OLD 2223 | Obsolete proof of ~ 19.28 ... |
nfan1OLD 2224 | Obsolete proof of ~ nfan1 ... |
nfanOLDOLD 2225 | Obsolete proof of ~ nfan a... |
nfnanOLD 2226 | Obsolete proof of ~ nfnan ... |
nf3anOLD 2227 | Obsolete proof of ~ nf3an ... |
hbanOLD 2228 | Obsolete proof of ~ hban a... |
hb3anOLD 2229 | Obsolete proof of ~ hb3an ... |
nfbidOLD 2230 | Obsolete proof of ~ nfbid ... |
nfbiOLD 2231 | Obsolete proof of ~ nfbi a... |
nforOLD 2232 | Obsolete proof of ~ nfor a... |
nf3orOLD 2233 | Obsolete proof of ~ nf3or ... |
ax13v 2235 | A weaker version of ~ ax-1... |
ax13lem1 2236 | A version of ~ ax13v with ... |
ax13 2237 | Derive ~ ax-13 from ~ ax13... |
ax6e 2238 | At least one individual ex... |
ax6 2239 | Theorem showing that ~ ax-... |
axc10 2240 | Show that the original axi... |
spimt 2241 | Closed theorem form of ~ s... |
spim 2242 | Specialization, using impl... |
spimed 2243 | Deduction version of ~ spi... |
spime 2244 | Existential introduction, ... |
spimv 2245 | A version of ~ spim with a... |
spimvALT 2246 | Alternate proof of ~ spimv... |
spimev 2247 | Distinct-variable version ... |
spv 2248 | Specialization, using impl... |
spei 2249 | Inference from existential... |
chvar 2250 | Implicit substitution of `... |
chvarv 2251 | Implicit substitution of `... |
chvarvOLD 2252 | Obsolete proof of ~ chvarv... |
cbv3 2253 | Rule used to change bound ... |
cbv3h 2254 | Rule used to change bound ... |
cbv1 2255 | Rule used to change bound ... |
cbv1h 2256 | Rule used to change bound ... |
cbv2h 2257 | Rule used to change bound ... |
cbv2 2258 | Rule used to change bound ... |
cbval 2259 | Rule used to change bound ... |
cbvex 2260 | Rule used to change bound ... |
cbvalv 2261 | Rule used to change bound ... |
cbvalvOLD 2262 | Obsolete proof of ~ cbvalv... |
cbvexv 2263 | Rule used to change bound ... |
cbvexvOLD 2264 | Obsolete proof of ~ cbvexv... |
cbvald 2265 | Deduction used to change b... |
cbvexd 2266 | Deduction used to change b... |
cbval2 2267 | Rule used to change bound ... |
cbvex2 2268 | Rule used to change bound ... |
cbvaldva 2269 | Rule used to change the bo... |
cbvaldvaOLD 2270 | Obsolete proof of ~ cbvald... |
cbvexdva 2271 | Rule used to change the bo... |
cbvexdvaOLD 2272 | Obsolete proof of ~ cbvexd... |
cbval2v 2273 | Rule used to change bound ... |
cbval2vOLD 2274 | Obsolete proof of ~ cbval2... |
cbvex2v 2275 | Rule used to change bound ... |
cbvex2vOLD 2276 | Obsolete proof of ~ cbvex2... |
cbvex4v 2277 | Rule used to change bound ... |
equs4 2278 | Lemma used in proofs of im... |
equsal 2279 | An equivalence related to ... |
equsalh 2280 | An equivalence related to ... |
equsex 2281 | An equivalence related to ... |
equsexALT 2282 | Alternate proof of ~ equse... |
equsexh 2283 | An equivalence related to ... |
ax13lem2 2284 | Lemma for ~ nfeqf2 . This... |
nfeqf2 2285 | An equation between setvar... |
dveeq2 2286 | Quantifier introduction wh... |
nfeqf1 2287 | An equation between setvar... |
dveeq1 2288 | Quantifier introduction wh... |
nfeqf 2289 | A variable is effectively ... |
axc9 2290 | Derive set.mm's original ~... |
axc15 2291 | Derivation of set.mm's ori... |
ax12 2292 | Rederivation of axiom ~ ax... |
ax13OLD 2293 | Obsolete proof of ~ ax13 a... |
axc11nlemALT 2294 | Alternate version of ~ axc... |
axc11n 2295 | Derive set.mm's original ~... |
axc11nOLD 2296 | Obsolete proof of ~ axc11n... |
axc11nOLDOLD 2297 | Old proof of ~ axc11n . O... |
axc11nALT 2298 | Alternate proof of ~ axc11... |
aecom 2299 | Commutation law for identi... |
aecoms 2300 | A commutation rule for ide... |
naecoms 2301 | A commutation rule for dis... |
axc11 2302 | Show that ~ ax-c11 can be ... |
hbae 2303 | All variables are effectiv... |
nfae 2304 | All variables are effectiv... |
hbnae 2305 | All variables are effectiv... |
nfnae 2306 | All variables are effectiv... |
hbnaes 2307 | Rule that applies ~ hbnae ... |
aevlemALTOLD 2308 | Older alternate version of... |
aevALTOLD 2309 | Older alternate proof of ~... |
axc16i 2310 | Inference with ~ axc16 as ... |
axc16nfALT 2311 | Alternate proof of ~ axc16... |
dral2 2312 | Formula-building lemma for... |
dral1 2313 | Formula-building lemma for... |
dral1ALT 2314 | Alternate proof of ~ dral1... |
drex1 2315 | Formula-building lemma for... |
drex2 2316 | Formula-building lemma for... |
drnf1 2317 | Formula-building lemma for... |
drnf2 2318 | Formula-building lemma for... |
nfald2 2319 | Variation on ~ nfald which... |
nfexd2 2320 | Variation on ~ nfexd which... |
exdistrf 2321 | Distribution of existentia... |
dvelimf 2322 | Version of ~ dvelimv witho... |
dvelimdf 2323 | Deduction form of ~ dvelim... |
dvelimh 2324 | Version of ~ dvelim withou... |
dvelim 2325 | This theorem can be used t... |
dvelimv 2326 | Similar to ~ dvelim with f... |
dvelimnf 2327 | Version of ~ dvelim using ... |
dveeq2ALT 2328 | Alternate proof of ~ dveeq... |
ax12OLD 2329 | Obsolete proof of ~ ax12 a... |
ax12v2OLD 2330 | Obsolete proof of ~ ax12v ... |
ax12a2OLD 2331 | Obsolete proof of ~ ax12v ... |
axc15OLD 2332 | Obsolete proof of ~ axc15 ... |
ax12b 2333 | A bidirectional version of... |
equvini 2334 | A variable introduction la... |
equvel 2335 | A variable elimination law... |
equs5a 2336 | A property related to subs... |
equs5e 2337 | A property related to subs... |
equs45f 2338 | Two ways of expressing sub... |
equs5 2339 | Lemma used in proofs of su... |
sb2 2340 | One direction of a simplif... |
stdpc4 2341 | The specialization axiom o... |
2stdpc4 2342 | A double specialization us... |
sb3 2343 | One direction of a simplif... |
sb4 2344 | One direction of a simplif... |
sb4a 2345 | A version of ~ sb4 that do... |
sb4b 2346 | Simplified definition of s... |
hbsb2 2347 | Bound-variable hypothesis ... |
nfsb2 2348 | Bound-variable hypothesis ... |
hbsb2a 2349 | Special case of a bound-va... |
sb4e 2350 | One direction of a simplif... |
hbsb2e 2351 | Special case of a bound-va... |
hbsb3 2352 | If ` y ` is not free in ` ... |
nfs1 2353 | If ` y ` is not free in ` ... |
axc16ALT 2354 | Alternate proof of ~ axc16... |
axc16gALT 2355 | Alternate proof of ~ axc16... |
equsb1 2356 | Substitution applied to an... |
equsb2 2357 | Substitution applied to an... |
dveel1 2358 | Quantifier introduction wh... |
dveel2 2359 | Quantifier introduction wh... |
axc14 2360 | Axiom ~ ax-c14 is redundan... |
dfsb2 2361 | An alternate definition of... |
dfsb3 2362 | An alternate definition of... |
sbequi 2363 | An equality theorem for su... |
sbequ 2364 | An equality theorem for su... |
drsb1 2365 | Formula-building lemma for... |
drsb2 2366 | Formula-building lemma for... |
sbft 2367 | Substitution has no effect... |
sbf 2368 | Substitution for a variabl... |
sbh 2369 | Substitution for a variabl... |
sbf2 2370 | Substitution has no effect... |
nfs1f 2371 | If ` x ` is not free in ` ... |
sb6x 2372 | Equivalence involving subs... |
sb6f 2373 | Equivalence for substituti... |
sb5f 2374 | Equivalence for substituti... |
sbequ5 2375 | Substitution does not chan... |
sbequ6 2376 | Substitution does not chan... |
nfsb4t 2377 | A variable not free remain... |
nfsb4 2378 | A variable not free remain... |
sbn 2379 | Negation inside and outsid... |
sbi1 2380 | Removal of implication fro... |
sbi2 2381 | Introduction of implicatio... |
spsbim 2382 | Specialization of implicat... |
sbim 2383 | Implication inside and out... |
sbrim 2384 | Substitution with a variab... |
sblim 2385 | Substitution with a variab... |
sbor 2386 | Logical OR inside and outs... |
sban 2387 | Conjunction inside and out... |
sb3an 2388 | Conjunction inside and out... |
sbbi 2389 | Equivalence inside and out... |
spsbbi 2390 | Specialization of bicondit... |
sbbid 2391 | Deduction substituting bot... |
sblbis 2392 | Introduce left bicondition... |
sbrbis 2393 | Introduce right biconditio... |
sbrbif 2394 | Introduce right biconditio... |
sbequ8ALT 2395 | Alternate proof of ~ sbequ... |
sbie 2396 | Conversion of implicit sub... |
sbied 2397 | Conversion of implicit sub... |
sbiedv 2398 | Conversion of implicit sub... |
sbcom3 2399 | Substituting ` y ` for ` x... |
sbco 2400 | A composition law for subs... |
sbid2 2401 | An identity law for substi... |
sbidm 2402 | An idempotent law for subs... |
sbco2 2403 | A composition law for subs... |
sbco2d 2404 | A composition law for subs... |
sbco3 2405 | A composition law for subs... |
sbcom 2406 | A commutativity law for su... |
sbt 2407 | A substitution into a theo... |
sbtrt 2408 | Partially closed form of ~... |
sbtr 2409 | A partial converse to ~ sb... |
sb5rf 2410 | Reversed substitution. (C... |
sb6rf 2411 | Reversed substitution. (C... |
sb8 2412 | Substitution of variable i... |
sb8e 2413 | Substitution of variable i... |
sb9 2414 | Commutation of quantificat... |
sb9i 2415 | Commutation of quantificat... |
ax12vALT 2416 | Alternate proof of ~ ax12v... |
sb6 2417 | Equivalence for substituti... |
sb5 2418 | Equivalence for substituti... |
equsb3lem 2419 | Lemma for ~ equsb3 . (Con... |
equsb3 2420 | Substitution applied to an... |
equsb3ALT 2421 | Alternate proof of ~ equsb... |
elsb3 2422 | Substitution applied to an... |
elsb4 2423 | Substitution applied to an... |
hbs1 2424 | The setvar ` x ` is not fr... |
nfs1v 2425 | The setvar ` x ` is not fr... |
sbhb 2426 | Two ways of expressing " `... |
sbnf2 2427 | Two ways of expressing " `... |
nfsb 2428 | If ` z ` is not free in ` ... |
hbsb 2429 | If ` z ` is not free in ` ... |
nfsbd 2430 | Deduction version of ~ nfs... |
2sb5 2431 | Equivalence for double sub... |
2sb6 2432 | Equivalence for double sub... |
sbcom2 2433 | Commutativity law for subs... |
sbcom4 2434 | Commutativity law for subs... |
pm11.07 2435 | Axiom *11.07 in [Whitehead... |
sb6a 2436 | Equivalence for substituti... |
2ax6elem 2437 | We can always find values ... |
2ax6e 2438 | We can always find values ... |
2sb5rf 2439 | Reversed double substituti... |
2sb6rf 2440 | Reversed double substituti... |
sb7f 2441 | This version of ~ dfsb7 do... |
sb7h 2442 | This version of ~ dfsb7 do... |
dfsb7 2443 | An alternate definition of... |
sb10f 2444 | Hao Wang's identity axiom ... |
sbid2v 2445 | An identity law for substi... |
sbelx 2446 | Elimination of substitutio... |
sbel2x 2447 | Elimination of double subs... |
sbal1 2448 | A theorem used in eliminat... |
sbal2 2449 | Move quantifier in and out... |
sbal 2450 | Move universal quantifier ... |
sbex 2451 | Move existential quantifie... |
sbalv 2452 | Quantify with new variable... |
sbco4lem 2453 | Lemma for ~ sbco4 . It re... |
sbco4 2454 | Two ways of exchanging two... |
2sb8e 2455 | An equivalent expression f... |
exsb 2456 | An equivalent expression f... |
2exsb 2457 | An equivalent expression f... |
eujust 2460 | A soundness justification ... |
eujustALT 2461 | Alternate proof of ~ eujus... |
euequ1 2464 | Equality has existential u... |
mo2v 2465 | Alternate definition of "a... |
euf 2466 | A version of the existenti... |
mo2 2467 | Alternate definition of "a... |
nfeu1 2468 | Bound-variable hypothesis ... |
nfmo1 2469 | Bound-variable hypothesis ... |
nfeud2 2470 | Bound-variable hypothesis ... |
nfmod2 2471 | Bound-variable hypothesis ... |
nfeud 2472 | Deduction version of ~ nfe... |
nfmod 2473 | Bound-variable hypothesis ... |
nfeu 2474 | Bound-variable hypothesis ... |
nfmo 2475 | Bound-variable hypothesis ... |
eubid 2476 | Formula-building rule for ... |
mobid 2477 | Formula-building rule for ... |
eubidv 2478 | Formula-building rule for ... |
mobidv 2479 | Formula-building rule for ... |
eubii 2480 | Introduce uniqueness quant... |
mobii 2481 | Formula-building rule for ... |
euex 2482 | Existential uniqueness imp... |
exmo 2483 | Something exists or at mos... |
eu5 2484 | Uniqueness in terms of "at... |
exmoeu2 2485 | Existence implies "at most... |
eu3v 2486 | An alternate way to expres... |
eumo 2487 | Existential uniqueness imp... |
eumoi 2488 | "At most one" inferred fro... |
moabs 2489 | Absorption of existence co... |
exmoeu 2490 | Existence in terms of "at ... |
sb8eu 2491 | Variable substitution in u... |
sb8mo 2492 | Variable substitution for ... |
cbveu 2493 | Rule used to change bound ... |
cbvmo 2494 | Rule used to change bound ... |
mo3 2495 | Alternate definition of "a... |
mo 2496 | Equivalent definitions of ... |
eu2 2497 | An alternate way of defini... |
eu1 2498 | An alternate way to expres... |
euexALT 2499 | Alternate proof of ~ euex ... |
euor 2500 | Introduce a disjunct into ... |
euorv 2501 | Introduce a disjunct into ... |
euor2 2502 | Introduce or eliminate a d... |
sbmo 2503 | Substitution into "at most... |
mo4f 2504 | "At most one" expressed us... |
mo4 2505 | "At most one" expressed us... |
eu4 2506 | Uniqueness using implicit ... |
moim 2507 | "At most one" reverses imp... |
moimi 2508 | "At most one" reverses imp... |
moa1 2509 | If an implication holds fo... |
euimmo 2510 | Uniqueness implies "at mos... |
euim 2511 | Add existential uniqueness... |
moan 2512 | "At most one" is still the... |
moani 2513 | "At most one" is still tru... |
moor 2514 | "At most one" is still the... |
mooran1 2515 | "At most one" imports disj... |
mooran2 2516 | "At most one" exports disj... |
moanim 2517 | Introduction of a conjunct... |
euan 2518 | Introduction of a conjunct... |
moanimv 2519 | Introduction of a conjunct... |
moanmo 2520 | Nested "at most one" quant... |
moaneu 2521 | Nested "at most one" and u... |
euanv 2522 | Introduction of a conjunct... |
mopick 2523 | "At most one" picks a vari... |
eupick 2524 | Existential uniqueness "pi... |
eupicka 2525 | Version of ~ eupick with c... |
eupickb 2526 | Existential uniqueness "pi... |
eupickbi 2527 | Theorem *14.26 in [Whitehe... |
mopick2 2528 | "At most one" can show the... |
moexex 2529 | "At most one" double quant... |
moexexv 2530 | "At most one" double quant... |
2moex 2531 | Double quantification with... |
2euex 2532 | Double quantification with... |
2eumo 2533 | Double quantification with... |
2eu2ex 2534 | Double existential uniquen... |
2moswap 2535 | A condition allowing swap ... |
2euswap 2536 | A condition allowing swap ... |
2exeu 2537 | Double existential uniquen... |
2mo2 2538 | This theorem extends the i... |
2mo 2539 | Two equivalent expressions... |
2mos 2540 | Double "exists at most one... |
2eu1 2541 | Double existential uniquen... |
2eu2 2542 | Double existential uniquen... |
2eu3 2543 | Double existential uniquen... |
2eu4 2544 | This theorem provides us w... |
2eu5 2545 | An alternate definition of... |
2eu6 2546 | Two equivalent expressions... |
2eu7 2547 | Two equivalent expressions... |
2eu8 2548 | Two equivalent expressions... |
exists1 2549 | Two ways to express "only ... |
exists2 2550 | A condition implying that ... |
barbara 2551 | "Barbara", one of the fund... |
celarent 2552 | "Celarent", one of the syl... |
darii 2553 | "Darii", one of the syllog... |
ferio 2554 | "Ferio" ("Ferioque"), one ... |
barbari 2555 | "Barbari", one of the syll... |
celaront 2556 | "Celaront", one of the syl... |
cesare 2557 | "Cesare", one of the syllo... |
camestres 2558 | "Camestres", one of the sy... |
festino 2559 | "Festino", one of the syll... |
baroco 2560 | "Baroco", one of the syllo... |
cesaro 2561 | "Cesaro", one of the syllo... |
camestros 2562 | "Camestros", one of the sy... |
datisi 2563 | "Datisi", one of the syllo... |
disamis 2564 | "Disamis", one of the syll... |
ferison 2565 | "Ferison", one of the syll... |
bocardo 2566 | "Bocardo", one of the syll... |
felapton 2567 | "Felapton", one of the syl... |
darapti 2568 | "Darapti", one of the syll... |
calemes 2569 | "Calemes", one of the syll... |
dimatis 2570 | "Dimatis", one of the syll... |
fresison 2571 | "Fresison", one of the syl... |
calemos 2572 | "Calemos", one of the syll... |
fesapo 2573 | "Fesapo", one of the syllo... |
bamalip 2574 | "Bamalip", one of the syll... |
axia1 2575 | Left 'and' elimination (in... |
axia2 2576 | Right 'and' elimination (i... |
axia3 2577 | 'And' introduction (intuit... |
axin1 2578 | 'Not' introduction (intuit... |
axin2 2579 | 'Not' elimination (intuiti... |
axio 2580 | Definition of 'or' (intuit... |
axi4 2581 | Specialization (intuitioni... |
axi5r 2582 | Converse of ax-c4 (intuiti... |
axial 2583 | The setvar ` x ` is not fr... |
axie1 2584 | The setvar ` x ` is not fr... |
axie2 2585 | A key property of existent... |
axi9 2586 | Axiom of existence (intuit... |
axi10 2587 | Axiom of Quantifier Substi... |
axi12 2588 | Axiom of Quantifier Introd... |
axbnd 2589 | Axiom of Bundling (intuiti... |
axext2 2591 | The Axiom of Extensionalit... |
axext3 2592 | A generalization of the Ax... |
axext3ALT 2593 | Alternate proof of ~ axext... |
axext4 2594 | A bidirectional version of... |
bm1.1 2595 | Any set defined by a prope... |
abid 2598 | Simplification of class ab... |
hbab1 2599 | Bound-variable hypothesis ... |
nfsab1 2600 | Bound-variable hypothesis ... |
hbab 2601 | Bound-variable hypothesis ... |
nfsab 2602 | Bound-variable hypothesis ... |
dfcleq 2604 | The same as ~ df-cleq with... |
cvjust 2605 | Every set is a class. Pro... |
eqriv 2607 | Infer equality of classes ... |
eqrdv 2608 | Deduce equality of classes... |
eqrdav 2609 | Deduce equality of classes... |
eqid 2610 | Law of identity (reflexivi... |
eqidd 2611 | Class identity law with an... |
eqeq1d 2612 | Deduction from equality to... |
eqeq1dALT 2613 | Shorter proof of ~ eqeq1d ... |
eqeq1 2614 | Equality implies equivalen... |
eqeq1i 2615 | Inference from equality to... |
eqcomd 2616 | Deduction from commutative... |
eqcom 2617 | Commutative law for class ... |
eqcoms 2618 | Inference applying commuta... |
eqcomi 2619 | Inference from commutative... |
eqeq2d 2620 | Deduction from equality to... |
eqeq2 2621 | Equality implies equivalen... |
eqeq2i 2622 | Inference from equality to... |
eqeq12 2623 | Equality relationship amon... |
eqeq12i 2624 | A useful inference for sub... |
eqeq12d 2625 | A useful inference for sub... |
eqeqan12d 2626 | A useful inference for sub... |
eqeqan12dALT 2627 | Alternate proof of ~ eqeqa... |
eqeqan12rd 2628 | A useful inference for sub... |
eqtr 2629 | Transitive law for class e... |
eqtr2 2630 | A transitive law for class... |
eqtr3 2631 | A transitive law for class... |
eqtri 2632 | An equality transitivity i... |
eqtr2i 2633 | An equality transitivity i... |
eqtr3i 2634 | An equality transitivity i... |
eqtr4i 2635 | An equality transitivity i... |
3eqtri 2636 | An inference from three ch... |
3eqtrri 2637 | An inference from three ch... |
3eqtr2i 2638 | An inference from three ch... |
3eqtr2ri 2639 | An inference from three ch... |
3eqtr3i 2640 | An inference from three ch... |
3eqtr3ri 2641 | An inference from three ch... |
3eqtr4i 2642 | An inference from three ch... |
3eqtr4ri 2643 | An inference from three ch... |
eqtrd 2644 | An equality transitivity d... |
eqtr2d 2645 | An equality transitivity d... |
eqtr3d 2646 | An equality transitivity e... |
eqtr4d 2647 | An equality transitivity e... |
3eqtrd 2648 | A deduction from three cha... |
3eqtrrd 2649 | A deduction from three cha... |
3eqtr2d 2650 | A deduction from three cha... |
3eqtr2rd 2651 | A deduction from three cha... |
3eqtr3d 2652 | A deduction from three cha... |
3eqtr3rd 2653 | A deduction from three cha... |
3eqtr4d 2654 | A deduction from three cha... |
3eqtr4rd 2655 | A deduction from three cha... |
syl5eq 2656 | An equality transitivity d... |
syl5req 2657 | An equality transitivity d... |
syl5eqr 2658 | An equality transitivity d... |
syl5reqr 2659 | An equality transitivity d... |
syl6eq 2660 | An equality transitivity d... |
syl6req 2661 | An equality transitivity d... |
syl6eqr 2662 | An equality transitivity d... |
syl6reqr 2663 | An equality transitivity d... |
sylan9eq 2664 | An equality transitivity d... |
sylan9req 2665 | An equality transitivity d... |
sylan9eqr 2666 | An equality transitivity d... |
3eqtr3g 2667 | A chained equality inferen... |
3eqtr3a 2668 | A chained equality inferen... |
3eqtr4g 2669 | A chained equality inferen... |
3eqtr4a 2670 | A chained equality inferen... |
eq2tri 2671 | A compound transitive infe... |
eleq1d 2672 | Deduction from equality to... |
eleq2d 2673 | Deduction from equality to... |
eleq2dOLD 2674 | Obsolete proof of ~ eleq2d... |
eleq2dALT 2675 | Alternate proof of ~ eleq2... |
eleq1 2676 | Equality implies equivalen... |
eleq2 2677 | Equality implies equivalen... |
eleq12 2678 | Equality implies equivalen... |
eleq1i 2679 | Inference from equality to... |
eleq2i 2680 | Inference from equality to... |
eleq12i 2681 | Inference from equality to... |
eleq12d 2682 | Deduction from equality to... |
eleq1a 2683 | A transitive-type law rela... |
eqeltri 2684 | Substitution of equal clas... |
eqeltrri 2685 | Substitution of equal clas... |
eleqtri 2686 | Substitution of equal clas... |
eleqtrri 2687 | Substitution of equal clas... |
eqeltrd 2688 | Substitution of equal clas... |
eqeltrrd 2689 | Deduction that substitutes... |
eleqtrd 2690 | Deduction that substitutes... |
eleqtrrd 2691 | Deduction that substitutes... |
syl5eqel 2692 | A membership and equality ... |
syl5eqelr 2693 | A membership and equality ... |
syl5eleq 2694 | A membership and equality ... |
syl5eleqr 2695 | A membership and equality ... |
syl6eqel 2696 | A membership and equality ... |
syl6eqelr 2697 | A membership and equality ... |
syl6eleq 2698 | A membership and equality ... |
syl6eleqr 2699 | A membership and equality ... |
3eltr3i 2700 | Substitution of equal clas... |
3eltr4i 2701 | Substitution of equal clas... |
3eltr3d 2702 | Substitution of equal clas... |
3eltr4d 2703 | Substitution of equal clas... |
3eltr3g 2704 | Substitution of equal clas... |
3eltr4g 2705 | Substitution of equal clas... |
eleq2s 2706 | Substitution of equal clas... |
eqneltrd 2707 | If a class is not an eleme... |
eqneltrrd 2708 | If a class is not an eleme... |
neleqtrd 2709 | If a class is not an eleme... |
neleqtrrd 2710 | If a class is not an eleme... |
cleqh 2711 | Establish equality between... |
nelneq 2712 | A way of showing two class... |
nelneq2 2713 | A way of showing two class... |
eqsb3lem 2714 | Lemma for ~ eqsb3 . (Cont... |
eqsb3 2715 | Substitution applied to an... |
clelsb3 2716 | Substitution applied to an... |
hbxfreq 2717 | A utility lemma to transfe... |
hblem 2718 | Change the free variable o... |
abeq2 2719 | Equality of a class variab... |
abeq1 2720 | Equality of a class variab... |
abeq2d 2721 | Equality of a class variab... |
abeq2i 2722 | Equality of a class variab... |
abeq1i 2723 | Equality of a class variab... |
abbi 2724 | Equivalent wff's correspon... |
abbi2i 2725 | Equality of a class variab... |
abbii 2726 | Equivalent wff's yield equ... |
abbid 2727 | Equivalent wff's yield equ... |
abbidv 2728 | Equivalent wff's yield equ... |
abbi2dv 2729 | Deduction from a wff to a ... |
abbi1dv 2730 | Deduction from a wff to a ... |
abid1 2731 | Every class is equal to a ... |
abid2 2732 | A simplification of class ... |
cbvab 2733 | Rule used to change bound ... |
cbvabv 2734 | Rule used to change bound ... |
clelab 2735 | Membership of a class vari... |
clabel 2736 | Membership of a class abst... |
sbab 2737 | The right-hand side of the... |
nfcjust 2739 | Justification theorem for ... |
nfci 2741 | Deduce that a class ` A ` ... |
nfcii 2742 | Deduce that a class ` A ` ... |
nfcr 2743 | Consequence of the not-fre... |
nfcrii 2744 | Consequence of the not-fre... |
nfcri 2745 | Consequence of the not-fre... |
nfcd 2746 | Deduce that a class ` A ` ... |
nfceqdf 2747 | An equality theorem for ef... |
nfceqi 2748 | Equality theorem for class... |
nfcxfr 2749 | A utility lemma to transfe... |
nfcxfrd 2750 | A utility lemma to transfe... |
nfcv 2751 | If ` x ` is disjoint from ... |
nfcvd 2752 | If ` x ` is disjoint from ... |
nfab1 2753 | Bound-variable hypothesis ... |
nfnfc1 2754 | The setvar ` x ` is bound ... |
nfab 2755 | Bound-variable hypothesis ... |
nfaba1 2756 | Bound-variable hypothesis ... |
nfcrd 2757 | Consequence of the not-fre... |
nfeqd 2758 | Hypothesis builder for equ... |
nfeld 2759 | Hypothesis builder for ele... |
nfnfc 2760 | Hypothesis builder for ` F... |
nfnfcALT 2761 | Alternate proof of ~ nfnfc... |
nfeq 2762 | Hypothesis builder for equ... |
nfel 2763 | Hypothesis builder for ele... |
nfeq1 2764 | Hypothesis builder for equ... |
nfel1 2765 | Hypothesis builder for ele... |
nfeq2 2766 | Hypothesis builder for equ... |
nfel2 2767 | Hypothesis builder for ele... |
drnfc1 2768 | Formula-building lemma for... |
drnfc2 2769 | Formula-building lemma for... |
nfabd2 2770 | Bound-variable hypothesis ... |
nfabd 2771 | Bound-variable hypothesis ... |
dvelimdc 2772 | Deduction form of ~ dvelim... |
dvelimc 2773 | Version of ~ dvelim for cl... |
nfcvf 2774 | If ` x ` and ` y ` are dis... |
nfcvf2 2775 | If ` x ` and ` y ` are dis... |
cleqf 2776 | Establish equality between... |
abid2f 2777 | A simplification of class ... |
abeq2f 2778 | Equality of a class variab... |
sbabel 2779 | Theorem to move a substitu... |
neii 2784 | Inference associated with ... |
neir 2785 | Inference associated with ... |
nne 2786 | Negation of inequality. (... |
neneqd 2787 | Deduction eliminating ineq... |
neneq 2788 | From inequality to non equ... |
neqned 2789 | If it is not the case that... |
neqne 2790 | From non equality to inequ... |
neirr 2791 | No class is unequal to its... |
exmidne 2792 | Excluded middle with equal... |
eqneqall 2793 | A contradiction concerning... |
nonconne 2794 | Law of noncontradiction wi... |
necon3ad 2795 | Contrapositive law deducti... |
necon3bd 2796 | Contrapositive law deducti... |
necon2ad 2797 | Contrapositive inference f... |
necon2bd 2798 | Contrapositive inference f... |
necon1ad 2799 | Contrapositive deduction f... |
necon1bd 2800 | Contrapositive deduction f... |
necon4ad 2801 | Contrapositive inference f... |
necon4bd 2802 | Contrapositive inference f... |
necon3d 2803 | Contrapositive law deducti... |
necon1d 2804 | Contrapositive law deducti... |
necon2d 2805 | Contrapositive inference f... |
necon4d 2806 | Contrapositive inference f... |
necon3ai 2807 | Contrapositive inference f... |
necon3bi 2808 | Contrapositive inference f... |
necon1ai 2809 | Contrapositive inference f... |
necon1bi 2810 | Contrapositive inference f... |
necon2ai 2811 | Contrapositive inference f... |
necon2bi 2812 | Contrapositive inference f... |
necon4ai 2813 | Contrapositive inference f... |
necon3i 2814 | Contrapositive inference f... |
necon1i 2815 | Contrapositive inference f... |
necon2i 2816 | Contrapositive inference f... |
necon4i 2817 | Contrapositive inference f... |
necon3abid 2818 | Deduction from equality to... |
necon3bbid 2819 | Deduction from equality to... |
necon1abid 2820 | Contrapositive deduction f... |
necon1bbid 2821 | Contrapositive inference f... |
necon4abid 2822 | Contrapositive law deducti... |
necon4bbid 2823 | Contrapositive law deducti... |
necon2abid 2824 | Contrapositive deduction f... |
necon2bbid 2825 | Contrapositive deduction f... |
necon3bid 2826 | Deduction from equality to... |
necon4bid 2827 | Contrapositive law deducti... |
necon3abii 2828 | Deduction from equality to... |
necon3bbii 2829 | Deduction from equality to... |
necon1abii 2830 | Contrapositive inference f... |
necon1bbii 2831 | Contrapositive inference f... |
necon2abii 2832 | Contrapositive inference f... |
necon2bbii 2833 | Contrapositive inference f... |
necon3bii 2834 | Inference from equality to... |
necom 2835 | Commutation of inequality.... |
necomi 2836 | Inference from commutative... |
necomd 2837 | Deduction from commutative... |
nesym 2838 | Characterization of inequa... |
nesymi 2839 | Inference associated with ... |
nesymir 2840 | Inference associated with ... |
neeq1d 2841 | Deduction for inequality. ... |
neeq2d 2842 | Deduction for inequality. ... |
neeq12d 2843 | Deduction for inequality. ... |
neeq1 2844 | Equality theorem for inequ... |
neeq2 2845 | Equality theorem for inequ... |
neeq1i 2846 | Inference for inequality. ... |
neeq2i 2847 | Inference for inequality. ... |
neeq12i 2848 | Inference for inequality. ... |
eqnetrd 2849 | Substitution of equal clas... |
eqnetrrd 2850 | Substitution of equal clas... |
neeqtrd 2851 | Substitution of equal clas... |
eqnetri 2852 | Substitution of equal clas... |
eqnetrri 2853 | Substitution of equal clas... |
neeqtri 2854 | Substitution of equal clas... |
neeqtrri 2855 | Substitution of equal clas... |
neeqtrrd 2856 | Substitution of equal clas... |
syl5eqner 2857 | A chained equality inferen... |
3netr3d 2858 | Substitution of equality i... |
3netr4d 2859 | Substitution of equality i... |
3netr3g 2860 | Substitution of equality i... |
3netr4g 2861 | Substitution of equality i... |
nebi 2862 | Contraposition law for ine... |
pm13.18 2863 | Theorem *13.18 in [Whitehe... |
pm13.181 2864 | Theorem *13.181 in [Whiteh... |
pm2.61ine 2865 | Inference eliminating an i... |
pm2.21ddne 2866 | A contradiction implies an... |
pm2.61ne 2867 | Deduction eliminating an i... |
pm2.61dne 2868 | Deduction eliminating an i... |
pm2.61dane 2869 | Deduction eliminating an i... |
pm2.61da2ne 2870 | Deduction eliminating two ... |
pm2.61da3ne 2871 | Deduction eliminating thre... |
pm2.61iine 2872 | Equality version of ~ pm2.... |
neor 2873 | Logical OR with an equalit... |
neanior 2874 | A De Morgan's law for ineq... |
ne3anior 2875 | A De Morgan's law for ineq... |
neorian 2876 | A De Morgan's law for ineq... |
nemtbir 2877 | An inference from an inequ... |
nelne1 2878 | Two classes are different ... |
nelne2 2879 | Two classes are different ... |
nelelne 2880 | Two classes are different ... |
neneor 2881 | If two classes are differe... |
nfne 2882 | Bound-variable hypothesis ... |
nfned 2883 | Bound-variable hypothesis ... |
nabbi 2884 | Not equivalent wff's corre... |
neli 2885 | Inference associated with ... |
nelir 2886 | Inference associated with ... |
neleq12d 2887 | Equality theorem for negat... |
neleq1 2888 | Equality theorem for negat... |
neleq2 2889 | Equality theorem for negat... |
nfnel 2890 | Bound-variable hypothesis ... |
nfneld 2891 | Bound-variable hypothesis ... |
nnel 2892 | Negation of negated member... |
elnelne1 2893 | Two classes are different ... |
elnelne2 2894 | Two classes are different ... |
nelcon3d 2895 | Contrapositive law deducti... |
rgen 2906 | Generalization rule for re... |
ralel 2907 | All elements of a class ar... |
rgenw 2908 | Generalization rule for re... |
rgen2w 2909 | Generalization rule for re... |
mprg 2910 | Modus ponens combined with... |
mprgbir 2911 | Modus ponens on biconditio... |
alral 2912 | Universal quantification i... |
rsp 2913 | Restricted specialization.... |
rspa 2914 | Restricted specialization.... |
rspec 2915 | Specialization rule for re... |
r19.21bi 2916 | Inference from Theorem 19.... |
r19.21be 2917 | Inference from Theorem 19.... |
rspec2 2918 | Specialization rule for re... |
rspec3 2919 | Specialization rule for re... |
rsp2 2920 | Restricted specialization,... |
r2allem 2921 | Lemma factoring out common... |
r2alf 2922 | Double restricted universa... |
r2al 2923 | Double restricted universa... |
r3al 2924 | Triple restricted universa... |
nfra1 2925 | The setvar ` x ` is not fr... |
hbra1 2926 | The setvar ` x ` is not fr... |
hbral 2927 | Bound-variable hypothesis ... |
nfrald 2928 | Deduction version of ~ nfr... |
nfral 2929 | Bound-variable hypothesis ... |
nfra2 2930 | Similar to Lemma 24 of [Mo... |
ral2imi 2931 | Inference quantifying ante... |
ralim 2932 | Distribution of restricted... |
ralimi2 2933 | Inference quantifying both... |
ralimia 2934 | Inference quantifying both... |
ralimiaa 2935 | Inference quantifying both... |
ralimi 2936 | Inference quantifying both... |
hbralrimi 2937 | Inference from Theorem 19.... |
r19.21t 2938 | Restricted quantifier vers... |
r19.21 2939 | Restricted quantifier vers... |
ralrimi 2940 | Inference from Theorem 19.... |
ralimdaa 2941 | Deduction quantifying both... |
ralrimd 2942 | Inference from Theorem 19.... |
r19.21v 2943 | Restricted quantifier vers... |
ralimdv2 2944 | Inference quantifying both... |
ralimdva 2945 | Deduction quantifying both... |
ralimdv 2946 | Deduction quantifying both... |
ralimdvva 2947 | Deduction doubly quantifyi... |
ralrimiv 2948 | Inference from Theorem 19.... |
ralrimiva 2949 | Inference from Theorem 19.... |
ralrimivw 2950 | Inference from Theorem 19.... |
ralrimdv 2951 | Inference from Theorem 19.... |
ralrimdva 2952 | Inference from Theorem 19.... |
ralrimivv 2953 | Inference from Theorem 19.... |
ralrimivva 2954 | Inference from Theorem 19.... |
ralrimivvva 2955 | Inference from Theorem 19.... |
ralrimdvv 2956 | Inference from Theorem 19.... |
ralrimdvva 2957 | Inference from Theorem 19.... |
rgen2 2958 | Generalization rule for re... |
rgen3 2959 | Generalization rule for re... |
rgen2a 2960 | Generalization rule for re... |
ralbii2 2961 | Inference adding different... |
ralbiia 2962 | Inference adding restricte... |
ralbii 2963 | Inference adding restricte... |
2ralbii 2964 | Inference adding two restr... |
ralbida 2965 | Formula-building rule for ... |
ralbid 2966 | Formula-building rule for ... |
ralbidv2 2967 | Formula-building rule for ... |
ralbidva 2968 | Formula-building rule for ... |
ralbidv 2969 | Formula-building rule for ... |
2ralbida 2970 | Formula-building rule for ... |
2ralbidva 2971 | Formula-building rule for ... |
2ralbidv 2972 | Formula-building rule for ... |
raleqbii 2973 | Equality deduction for res... |
raln 2974 | Restricted universally qua... |
ralnex 2975 | Relationship between restr... |
ralnexOLD 2976 | Obsolete proof of ~ ralnex... |
dfral2 2977 | Relationship between restr... |
rexnal 2978 | Relationship between restr... |
dfrex2 2979 | Relationship between restr... |
ralinexa 2980 | A transformation of restri... |
rexanali 2981 | A transformation of restri... |
nrexralim 2982 | Negation of a complex pred... |
nrex 2983 | Inference adding restricte... |
nrexdv 2984 | Deduction adding restricte... |
rexex 2985 | Restricted existence impli... |
rspe 2986 | Restricted specialization.... |
rsp2e 2987 | Restricted specialization.... |
nfre1 2988 | The setvar ` x ` is not fr... |
nfrexd 2989 | Deduction version of ~ nfr... |
nfrex 2990 | Bound-variable hypothesis ... |
rexim 2991 | Theorem 19.22 of [Margaris... |
reximia 2992 | Inference quantifying both... |
reximi2 2993 | Inference quantifying both... |
reximi 2994 | Inference quantifying both... |
reximdai 2995 | Deduction from Theorem 19.... |
reximd2a 2996 | Deduction quantifying both... |
reximdv2 2997 | Deduction quantifying both... |
reximdvai 2998 | Deduction quantifying both... |
reximdv 2999 | Deduction from Theorem 19.... |
reximdva 3000 | Deduction quantifying both... |
reximddv 3001 | Deduction from Theorem 19.... |
reximddv2 3002 | Double deduction from Theo... |
r19.23t 3003 | Closed theorem form of ~ r... |
r19.23 3004 | Restricted quantifier vers... |
r19.23v 3005 | Restricted quantifier vers... |
rexlimi 3006 | Restricted quantifier vers... |
rexlimd2 3007 | Version of ~ rexlimd with ... |
rexlimd 3008 | Deduction form of ~ rexlim... |
rexlimiv 3009 | Inference from Theorem 19.... |
rexlimiva 3010 | Inference from Theorem 19.... |
rexlimivw 3011 | Weaker version of ~ rexlim... |
rexlimdv 3012 | Inference from Theorem 19.... |
rexlimdva 3013 | Inference from Theorem 19.... |
rexlimdvaa 3014 | Inference from Theorem 19.... |
rexlimdv3a 3015 | Inference from Theorem 19.... |
rexlimdvw 3016 | Inference from Theorem 19.... |
rexlimddv 3017 | Restricted existential eli... |
rexlimivv 3018 | Inference from Theorem 19.... |
rexlimdvv 3019 | Inference from Theorem 19.... |
rexlimdvva 3020 | Inference from Theorem 19.... |
rexbii2 3021 | Inference adding different... |
rexbiia 3022 | Inference adding restricte... |
rexbii 3023 | Inference adding restricte... |
2rexbii 3024 | Inference adding two restr... |
rexnal2 3025 | Relationship between two r... |
rexnal3 3026 | Relationship between three... |
ralnex2 3027 | Relationship between two r... |
ralnex3 3028 | Relationship between three... |
rexbida 3029 | Formula-building rule for ... |
rexbidv2 3030 | Formula-building rule for ... |
rexbidva 3031 | Formula-building rule for ... |
rexbidvaALT 3032 | Alternate proof of ~ rexbi... |
rexbid 3033 | Formula-building rule for ... |
rexbidv 3034 | Formula-building rule for ... |
rexbidvALT 3035 | Alternate proof of ~ rexbi... |
rexeqbii 3036 | Equality deduction for res... |
2rexbiia 3037 | Inference adding two restr... |
2rexbidva 3038 | Formula-building rule for ... |
2rexbidv 3039 | Formula-building rule for ... |
rexralbidv 3040 | Formula-building rule for ... |
r2exlem 3041 | Lemma factoring out common... |
r2exf 3042 | Double restricted existent... |
r2ex 3043 | Double restricted existent... |
risset 3044 | Two ways to say " ` A ` be... |
r19.12 3045 | Restricted quantifier vers... |
r19.26 3046 | Restricted quantifier vers... |
r19.26-2 3047 | Restricted quantifier vers... |
r19.26-3 3048 | Version of ~ r19.26 with t... |
r19.26m 3049 | Version of ~ 19.26 and ~ r... |
ralbi 3050 | Distribute a restricted un... |
ralbiim 3051 | Split a biconditional and ... |
r19.27v 3052 | Restricted quantitifer ver... |
r19.28v 3053 | Restricted quantifier vers... |
r19.29 3054 | Restricted quantifier vers... |
r19.29r 3055 | Restricted quantifier vers... |
r19.29imd 3056 | Theorem 19.29 of [Margaris... |
r19.29af2 3057 | A commonly used pattern ba... |
r19.29af 3058 | A commonly used pattern ba... |
r19.29an 3059 | A commonly used pattern ba... |
r19.29a 3060 | A commonly used pattern ba... |
r19.29d2r 3061 | Theorem 19.29 of [Margaris... |
r19.29vva 3062 | A commonly used pattern ba... |
r19.30 3063 | Restricted quantifier vers... |
r19.32v 3064 | Restricted quantifier vers... |
r19.35 3065 | Restricted quantifier vers... |
r19.36v 3066 | Restricted quantifier vers... |
r19.37 3067 | Restricted quantifier vers... |
r19.37v 3068 | Restricted quantifier vers... |
r19.40 3069 | Restricted quantifier vers... |
r19.41v 3070 | Restricted quantifier vers... |
r19.41 3071 | Restricted quantifier vers... |
r19.41vv 3072 | Version of ~ r19.41v with ... |
r19.42v 3073 | Restricted quantifier vers... |
r19.43 3074 | Restricted quantifier vers... |
r19.44v 3075 | One direction of a restric... |
r19.45v 3076 | Restricted quantifier vers... |
ralcomf 3077 | Commutation of restricted ... |
rexcomf 3078 | Commutation of restricted ... |
ralcom 3079 | Commutation of restricted ... |
rexcom 3080 | Commutation of restricted ... |
rexcom13 3081 | Swap first and third restr... |
rexrot4 3082 | Rotate four restricted exi... |
ralcom2 3083 | Commutation of restricted ... |
ralcom3 3084 | A commutation law for rest... |
reean 3085 | Rearrange restricted exist... |
reeanv 3086 | Rearrange restricted exist... |
3reeanv 3087 | Rearrange three restricted... |
2ralor 3088 | Distribute restricted univ... |
nfreu1 3089 | The setvar ` x ` is not fr... |
nfrmo1 3090 | The setvar ` x ` is not fr... |
nfreud 3091 | Deduction version of ~ nfr... |
nfrmod 3092 | Deduction version of ~ nfr... |
nfreu 3093 | Bound-variable hypothesis ... |
nfrmo 3094 | Bound-variable hypothesis ... |
rabid 3095 | An "identity" law of concr... |
rabid2 3096 | An "identity" law for rest... |
rabbi 3097 | Equivalent wff's correspon... |
rabswap 3098 | Swap with a membership rel... |
nfrab1 3099 | The abstraction variable i... |
nfrab 3100 | A variable not free in a w... |
reubida 3101 | Formula-building rule for ... |
reubidva 3102 | Formula-building rule for ... |
reubidv 3103 | Formula-building rule for ... |
reubiia 3104 | Formula-building rule for ... |
reubii 3105 | Formula-building rule for ... |
rmobida 3106 | Formula-building rule for ... |
rmobidva 3107 | Formula-building rule for ... |
rmobidv 3108 | Formula-building rule for ... |
rmobiia 3109 | Formula-building rule for ... |
rmobii 3110 | Formula-building rule for ... |
raleqf 3111 | Equality theorem for restr... |
rexeqf 3112 | Equality theorem for restr... |
reueq1f 3113 | Equality theorem for restr... |
rmoeq1f 3114 | Equality theorem for restr... |
raleq 3115 | Equality theorem for restr... |
rexeq 3116 | Equality theorem for restr... |
reueq1 3117 | Equality theorem for restr... |
rmoeq1 3118 | Equality theorem for restr... |
raleqi 3119 | Equality inference for res... |
rexeqi 3120 | Equality inference for res... |
raleqdv 3121 | Equality deduction for res... |
rexeqdv 3122 | Equality deduction for res... |
raleqbi1dv 3123 | Equality deduction for res... |
rexeqbi1dv 3124 | Equality deduction for res... |
reueqd 3125 | Equality deduction for res... |
rmoeqd 3126 | Equality deduction for res... |
raleqbid 3127 | Equality deduction for res... |
rexeqbid 3128 | Equality deduction for res... |
raleqbidv 3129 | Equality deduction for res... |
rexeqbidv 3130 | Equality deduction for res... |
raleqbidva 3131 | Equality deduction for res... |
rexeqbidva 3132 | Equality deduction for res... |
raleleq 3133 | All elements of a class ar... |
raleleqALT 3134 | Alternate proof of ~ ralel... |
mormo 3135 | Unrestricted "at most one"... |
reu5 3136 | Restricted uniqueness in t... |
reurex 3137 | Restricted unique existenc... |
reurmo 3138 | Restricted existential uni... |
rmo5 3139 | Restricted "at most one" i... |
nrexrmo 3140 | Nonexistence implies restr... |
cbvralf 3141 | Rule used to change bound ... |
cbvrexf 3142 | Rule used to change bound ... |
cbvral 3143 | Rule used to change bound ... |
cbvrex 3144 | Rule used to change bound ... |
cbvreu 3145 | Change the bound variable ... |
cbvrmo 3146 | Change the bound variable ... |
cbvralv 3147 | Change the bound variable ... |
cbvrexv 3148 | Change the bound variable ... |
cbvreuv 3149 | Change the bound variable ... |
cbvrmov 3150 | Change the bound variable ... |
cbvraldva2 3151 | Rule used to change the bo... |
cbvrexdva2 3152 | Rule used to change the bo... |
cbvraldva 3153 | Rule used to change the bo... |
cbvrexdva 3154 | Rule used to change the bo... |
cbvral2v 3155 | Change bound variables of ... |
cbvrex2v 3156 | Change bound variables of ... |
cbvral3v 3157 | Change bound variables of ... |
cbvralsv 3158 | Change bound variable by u... |
cbvrexsv 3159 | Change bound variable by u... |
sbralie 3160 | Implicit to explicit subst... |
rabbiia 3161 | Equivalent wff's yield equ... |
rabbidva2 3162 | Equivalent wff's yield equ... |
rabbidva 3163 | Equivalent wff's yield equ... |
rabbidv 3164 | Equivalent wff's yield equ... |
rabeqf 3165 | Equality theorem for restr... |
rabeq 3166 | Equality theorem for restr... |
rabeqdv 3167 | Equality of restricted cla... |
rabeqbidv 3168 | Equality of restricted cla... |
rabeqbidva 3169 | Equality of restricted cla... |
rabeq2i 3170 | Inference rule from equali... |
cbvrab 3171 | Rule to change the bound v... |
cbvrabv 3172 | Rule to change the bound v... |
vjust 3174 | Soundness justification th... |
vex 3176 | All setvar variables are s... |
eqvf 3177 | The universe contains ever... |
eqv 3178 | The universe contains ever... |
abv 3179 | The class of sets verifyin... |
isset 3180 | Two ways to say " ` A ` is... |
issetf 3181 | A version of ~ isset that ... |
isseti 3182 | A way to say " ` A ` is a ... |
issetri 3183 | A way to say " ` A ` is a ... |
eqvisset 3184 | A class equal to a variabl... |
elex 3185 | If a class is a member of ... |
elexi 3186 | If a class is a member of ... |
elexd 3187 | If a class is a member of ... |
elisset 3188 | An element of a class exis... |
elex2 3189 | If a class contains anothe... |
elex22 3190 | If two classes each contai... |
prcnel 3191 | A proper class doesn't bel... |
ralv 3192 | A universal quantifier res... |
rexv 3193 | An existential quantifier ... |
reuv 3194 | A uniqueness quantifier re... |
rmov 3195 | A uniqueness quantifier re... |
rabab 3196 | A class abstraction restri... |
ralcom4 3197 | Commutation of restricted ... |
rexcom4 3198 | Commutation of restricted ... |
rexcom4a 3199 | Specialized existential co... |
rexcom4b 3200 | Specialized existential co... |
ceqsalt 3201 | Closed theorem version of ... |
ceqsralt 3202 | Restricted quantifier vers... |
ceqsalg 3203 | A representation of explic... |
ceqsalgALT 3204 | Alternate proof of ~ ceqsa... |
ceqsal 3205 | A representation of explic... |
ceqsalv 3206 | A representation of explic... |
ceqsralv 3207 | Restricted quantifier vers... |
gencl 3208 | Implicit substitution for ... |
2gencl 3209 | Implicit substitution for ... |
3gencl 3210 | Implicit substitution for ... |
cgsexg 3211 | Implicit substitution infe... |
cgsex2g 3212 | Implicit substitution infe... |
cgsex4g 3213 | An implicit substitution i... |
ceqsex 3214 | Elimination of an existent... |
ceqsexv 3215 | Elimination of an existent... |
ceqsexv2d 3216 | Elimination of an existent... |
ceqsex2 3217 | Elimination of two existen... |
ceqsex2v 3218 | Elimination of two existen... |
ceqsex3v 3219 | Elimination of three exist... |
ceqsex4v 3220 | Elimination of four existe... |
ceqsex6v 3221 | Elimination of six existen... |
ceqsex8v 3222 | Elimination of eight exist... |
gencbvex 3223 | Change of bound variable u... |
gencbvex2 3224 | Restatement of ~ gencbvex ... |
gencbval 3225 | Change of bound variable u... |
sbhypf 3226 | Introduce an explicit subs... |
vtoclgft 3227 | Closed theorem form of ~ v... |
vtoclgftOLD 3228 | Obsolete proof of ~ vtoclg... |
vtocldf 3229 | Implicit substitution of a... |
vtocld 3230 | Implicit substitution of a... |
vtoclf 3231 | Implicit substitution of a... |
vtocl 3232 | Implicit substitution of a... |
vtoclALT 3233 | Alternate proof of ~ vtocl... |
vtocl2 3234 | Implicit substitution of c... |
vtocl3 3235 | Implicit substitution of c... |
vtoclb 3236 | Implicit substitution of a... |
vtoclgf 3237 | Implicit substitution of a... |
vtoclg1f 3238 | Version of ~ vtoclgf with ... |
vtoclg 3239 | Implicit substitution of a... |
vtoclbg 3240 | Implicit substitution of a... |
vtocl2gf 3241 | Implicit substitution of a... |
vtocl3gf 3242 | Implicit substitution of a... |
vtocl2g 3243 | Implicit substitution of 2... |
vtoclgaf 3244 | Implicit substitution of a... |
vtoclga 3245 | Implicit substitution of a... |
vtocl2gaf 3246 | Implicit substitution of 2... |
vtocl2ga 3247 | Implicit substitution of 2... |
vtocl3gaf 3248 | Implicit substitution of 3... |
vtocl3ga 3249 | Implicit substitution of 3... |
vtocl4g 3250 | Implicit substitution of 4... |
vtocl4ga 3251 | Implicit substitution of 4... |
vtocleg 3252 | Implicit substitution of a... |
vtoclegft 3253 | Implicit substitution of a... |
vtoclef 3254 | Implicit substitution of a... |
vtocle 3255 | Implicit substitution of a... |
vtoclri 3256 | Implicit substitution of a... |
spcimgft 3257 | A closed version of ~ spci... |
spcgft 3258 | A closed version of ~ spcg... |
spcimgf 3259 | Rule of specialization, us... |
spcimegf 3260 | Existential specialization... |
spcgf 3261 | Rule of specialization, us... |
spcegf 3262 | Existential specialization... |
spcimdv 3263 | Restricted specialization,... |
spcdv 3264 | Rule of specialization, us... |
spcimedv 3265 | Restricted existential spe... |
spcgv 3266 | Rule of specialization, us... |
spcegv 3267 | Existential specialization... |
spc2egv 3268 | Existential specialization... |
spc2gv 3269 | Specialization with two qu... |
spc3egv 3270 | Existential specialization... |
spc3gv 3271 | Specialization with three ... |
spcv 3272 | Rule of specialization, us... |
spcev 3273 | Existential specialization... |
spc2ev 3274 | Existential specialization... |
rspct 3275 | A closed version of ~ rspc... |
rspc 3276 | Restricted specialization,... |
rspce 3277 | Restricted existential spe... |
rspcv 3278 | Restricted specialization,... |
rspccv 3279 | Restricted specialization,... |
rspcva 3280 | Restricted specialization,... |
rspccva 3281 | Restricted specialization,... |
rspcev 3282 | Restricted existential spe... |
rspcimdv 3283 | Restricted specialization,... |
rspcimedv 3284 | Restricted existential spe... |
rspcdv 3285 | Restricted specialization,... |
rspcedv 3286 | Restricted existential spe... |
rspcda 3287 | Restricted specialization,... |
rspcdva 3288 | Restricted specialization,... |
rspcedvd 3289 | Restricted existential spe... |
rspcedeq1vd 3290 | Restricted existential spe... |
rspcedeq2vd 3291 | Restricted existential spe... |
rspc2 3292 | 2-variable restricted spec... |
rspc2v 3293 | 2-variable restricted spec... |
rspc2va 3294 | 2-variable restricted spec... |
rspc2ev 3295 | 2-variable restricted exis... |
rspc3v 3296 | 3-variable restricted spec... |
rspc3ev 3297 | 3-variable restricted exis... |
ralxpxfr2d 3298 | Transfer a universal quant... |
rexraleqim 3299 | Statement following from e... |
eqvinc 3300 | A variable introduction la... |
eqvincf 3301 | A variable introduction la... |
alexeqg 3302 | Two ways to express substi... |
ceqex 3303 | Equality implies equivalen... |
ceqsexg 3304 | A representation of explic... |
ceqsexgv 3305 | Elimination of an existent... |
ceqsrexv 3306 | Elimination of a restricte... |
ceqsrexbv 3307 | Elimination of a restricte... |
ceqsrex2v 3308 | Elimination of a restricte... |
clel2 3309 | An alternate definition of... |
clel3g 3310 | An alternate definition of... |
clel3 3311 | An alternate definition of... |
clel4 3312 | An alternate definition of... |
pm13.183 3313 | Compare theorem *13.183 in... |
rr19.3v 3314 | Restricted quantifier vers... |
rr19.28v 3315 | Restricted quantifier vers... |
elabgt 3316 | Membership in a class abst... |
elabgf 3317 | Membership in a class abst... |
elabf 3318 | Membership in a class abst... |
elab 3319 | Membership in a class abst... |
elabg 3320 | Membership in a class abst... |
elabd 3321 | Explicit demonstration the... |
elab2g 3322 | Membership in a class abst... |
elab2 3323 | Membership in a class abst... |
elab4g 3324 | Membership in a class abst... |
elab3gf 3325 | Membership in a class abst... |
elab3g 3326 | Membership in a class abst... |
elab3 3327 | Membership in a class abst... |
elrabi 3328 | Implication for the member... |
elrabf 3329 | Membership in a restricted... |
elrab3t 3330 | Membership in a restricted... |
elrab 3331 | Membership in a restricted... |
elrab3 3332 | Membership in a restricted... |
elrab2 3333 | Membership in a class abst... |
ralab 3334 | Universal quantification o... |
ralrab 3335 | Universal quantification o... |
rexab 3336 | Existential quantification... |
rexrab 3337 | Existential quantification... |
ralab2 3338 | Universal quantification o... |
ralrab2 3339 | Universal quantification o... |
rexab2 3340 | Existential quantification... |
rexrab2 3341 | Existential quantification... |
abidnf 3342 | Identity used to create cl... |
dedhb 3343 | A deduction theorem for co... |
eqeu 3344 | A condition which implies ... |
eueq 3345 | Equality has existential u... |
eueq1 3346 | Equality has existential u... |
eueq2 3347 | Equality has existential u... |
eueq3 3348 | Equality has existential u... |
moeq 3349 | There is at most one set e... |
moeq3 3350 | "At most one" property of ... |
mosub 3351 | "At most one" remains true... |
mo2icl 3352 | Theorem for inferring "at ... |
mob2 3353 | Consequence of "at most on... |
moi2 3354 | Consequence of "at most on... |
mob 3355 | Equality implied by "at mo... |
moi 3356 | Equality implied by "at mo... |
morex 3357 | Derive membership from uni... |
euxfr2 3358 | Transfer existential uniqu... |
euxfr 3359 | Transfer existential uniqu... |
euind 3360 | Existential uniqueness via... |
reu2 3361 | A way to express restricte... |
reu6 3362 | A way to express restricte... |
reu3 3363 | A way to express restricte... |
reu6i 3364 | A condition which implies ... |
eqreu 3365 | A condition which implies ... |
rmo4 3366 | Restricted "at most one" u... |
reu4 3367 | Restricted uniqueness usin... |
reu7 3368 | Restricted uniqueness usin... |
reu8 3369 | Restricted uniqueness usin... |
reu2eqd 3370 | Deduce equality from restr... |
reueq 3371 | Equality has existential u... |
rmoeq 3372 | Equality's restricted exis... |
rmoan 3373 | Restricted "at most one" s... |
rmoim 3374 | Restricted "at most one" i... |
rmoimia 3375 | Restricted "at most one" i... |
rmoimi2 3376 | Restricted "at most one" i... |
2reuswap 3377 | A condition allowing swap ... |
reuind 3378 | Existential uniqueness via... |
2rmorex 3379 | Double restricted quantifi... |
2reu5lem1 3380 | Lemma for ~ 2reu5 . Note ... |
2reu5lem2 3381 | Lemma for ~ 2reu5 . (Cont... |
2reu5lem3 3382 | Lemma for ~ 2reu5 . This ... |
2reu5 3383 | Double restricted existent... |
nelrdva 3384 | Deduce negative membership... |
cdeqi 3387 | Deduce conditional equalit... |
cdeqri 3388 | Property of conditional eq... |
cdeqth 3389 | Deduce conditional equalit... |
cdeqnot 3390 | Distribute conditional equ... |
cdeqal 3391 | Distribute conditional equ... |
cdeqab 3392 | Distribute conditional equ... |
cdeqal1 3393 | Distribute conditional equ... |
cdeqab1 3394 | Distribute conditional equ... |
cdeqim 3395 | Distribute conditional equ... |
cdeqcv 3396 | Conditional equality for s... |
cdeqeq 3397 | Distribute conditional equ... |
cdeqel 3398 | Distribute conditional equ... |
nfcdeq 3399 | If we have a conditional e... |
nfccdeq 3400 | Variation of ~ nfcdeq for ... |
ru 3401 | Russell's Paradox. Propos... |
dfsbcq 3404 | Proper substitution of a c... |
dfsbcq2 3405 | This theorem, which is sim... |
sbsbc 3406 | Show that ~ df-sb and ~ df... |
sbceq1d 3407 | Equality theorem for class... |
sbceq1dd 3408 | Equality theorem for class... |
sbceqbid 3409 | Equality theorem for class... |
sbc8g 3410 | This is the closest we can... |
sbc2or 3411 | The disjunction of two equ... |
sbcex 3412 | By our definition of prope... |
sbceq1a 3413 | Equality theorem for class... |
sbceq2a 3414 | Equality theorem for class... |
spsbc 3415 | Specialization: if a formu... |
spsbcd 3416 | Specialization: if a formu... |
sbcth 3417 | A substitution into a theo... |
sbcthdv 3418 | Deduction version of ~ sbc... |
sbcid 3419 | An identity theorem for su... |
nfsbc1d 3420 | Deduction version of ~ nfs... |
nfsbc1 3421 | Bound-variable hypothesis ... |
nfsbc1v 3422 | Bound-variable hypothesis ... |
nfsbcd 3423 | Deduction version of ~ nfs... |
nfsbc 3424 | Bound-variable hypothesis ... |
sbcco 3425 | A composition law for clas... |
sbcco2 3426 | A composition law for clas... |
sbc5 3427 | An equivalence for class s... |
sbc6g 3428 | An equivalence for class s... |
sbc6 3429 | An equivalence for class s... |
sbc7 3430 | An equivalence for class s... |
cbvsbc 3431 | Change bound variables in ... |
cbvsbcv 3432 | Change the bound variable ... |
sbciegft 3433 | Conversion of implicit sub... |
sbciegf 3434 | Conversion of implicit sub... |
sbcieg 3435 | Conversion of implicit sub... |
sbcie2g 3436 | Conversion of implicit sub... |
sbcie 3437 | Conversion of implicit sub... |
sbciedf 3438 | Conversion of implicit sub... |
sbcied 3439 | Conversion of implicit sub... |
sbcied2 3440 | Conversion of implicit sub... |
elrabsf 3441 | Membership in a restricted... |
eqsbc3 3442 | Substitution applied to an... |
sbcng 3443 | Move negation in and out o... |
sbcimg 3444 | Distribution of class subs... |
sbcan 3445 | Distribution of class subs... |
sbcor 3446 | Distribution of class subs... |
sbcbig 3447 | Distribution of class subs... |
sbcn1 3448 | Move negation in and out o... |
sbcim1 3449 | Distribution of class subs... |
sbcbi1 3450 | Distribution of class subs... |
sbcbi2 3451 | Substituting into equivale... |
sbcal 3452 | Move universal quantifier ... |
sbcex2 3453 | Move existential quantifie... |
sbceqal 3454 | Set theory version of ~ sb... |
sbeqalb 3455 | Theorem *14.121 in [Whiteh... |
sbcbid 3456 | Formula-building deduction... |
sbcbidv 3457 | Formula-building deduction... |
sbcbii 3458 | Formula-building inference... |
eqsbc3r 3459 | ~ eqsbc3 with setvar varia... |
eqsbc3rOLD 3460 | Obsolete proof of ~ eqsbc3... |
sbc3an 3461 | Distribution of class subs... |
sbcel1v 3462 | Class substitution into a ... |
sbcel2gv 3463 | Class substitution into a ... |
sbcel21v 3464 | Class substitution into a ... |
sbcimdv 3465 | Substitution analogue of T... |
sbcimdvOLD 3466 | Obsolete proof of ~ sbcimd... |
sbctt 3467 | Substitution for a variabl... |
sbcgf 3468 | Substitution for a variabl... |
sbc19.21g 3469 | Substitution for a variabl... |
sbcg 3470 | Substitution for a variabl... |
sbc2iegf 3471 | Conversion of implicit sub... |
sbc2ie 3472 | Conversion of implicit sub... |
sbc2iedv 3473 | Conversion of implicit sub... |
sbc3ie 3474 | Conversion of implicit sub... |
sbccomlem 3475 | Lemma for ~ sbccom . (Con... |
sbccom 3476 | Commutative law for double... |
sbcralt 3477 | Interchange class substitu... |
sbcrext 3478 | Interchange class substitu... |
sbcrextOLD 3479 | Obsolete proof of ~ sbcrex... |
sbcralg 3480 | Interchange class substitu... |
sbcrex 3481 | Interchange class substitu... |
sbcreu 3482 | Interchange class substitu... |
sbcabel 3483 | Interchange class substitu... |
rspsbc 3484 | Restricted quantifier vers... |
rspsbca 3485 | Restricted quantifier vers... |
rspesbca 3486 | Existence form of ~ rspsbc... |
spesbc 3487 | Existence form of ~ spsbc ... |
spesbcd 3488 | form of ~ spsbc . (Contri... |
sbcth2 3489 | A substitution into a theo... |
ra4v 3490 | Version of ~ ra4 with a dv... |
ra4 3491 | Restricted quantifier vers... |
rmo2 3492 | Alternate definition of re... |
rmo2i 3493 | Condition implying restric... |
rmo3 3494 | Restricted "at most one" u... |
rmob 3495 | Consequence of "at most on... |
rmoi 3496 | Consequence of "at most on... |
rmob2 3497 | Consequence of "restricted... |
rmoi2 3498 | Consequence of "restricted... |
csb2 3501 | Alternate expression for t... |
csbeq1 3502 | Analogue of ~ dfsbcq for p... |
csbeq2 3503 | Substituting into equivale... |
cbvcsb 3504 | Change bound variables in ... |
cbvcsbv 3505 | Change the bound variable ... |
csbeq1d 3506 | Equality deduction for pro... |
csbid 3507 | Analogue of ~ sbid for pro... |
csbeq1a 3508 | Equality theorem for prope... |
csbco 3509 | Composition law for chaine... |
csbtt 3510 | Substitution doesn't affec... |
csbconstgf 3511 | Substitution doesn't affec... |
csbconstg 3512 | Substitution doesn't affec... |
nfcsb1d 3513 | Bound-variable hypothesis ... |
nfcsb1 3514 | Bound-variable hypothesis ... |
nfcsb1v 3515 | Bound-variable hypothesis ... |
nfcsbd 3516 | Deduction version of ~ nfc... |
nfcsb 3517 | Bound-variable hypothesis ... |
csbhypf 3518 | Introduce an explicit subs... |
csbiebt 3519 | Conversion of implicit sub... |
csbiedf 3520 | Conversion of implicit sub... |
csbieb 3521 | Bidirectional conversion b... |
csbiebg 3522 | Bidirectional conversion b... |
csbiegf 3523 | Conversion of implicit sub... |
csbief 3524 | Conversion of implicit sub... |
csbie 3525 | Conversion of implicit sub... |
csbied 3526 | Conversion of implicit sub... |
csbied2 3527 | Conversion of implicit sub... |
csbie2t 3528 | Conversion of implicit sub... |
csbie2 3529 | Conversion of implicit sub... |
csbie2g 3530 | Conversion of implicit sub... |
cbvralcsf 3531 | A more general version of ... |
cbvrexcsf 3532 | A more general version of ... |
cbvreucsf 3533 | A more general version of ... |
cbvrabcsf 3534 | A more general version of ... |
cbvralv2 3535 | Rule used to change the bo... |
cbvrexv2 3536 | Rule used to change the bo... |
difjust 3542 | Soundness justification th... |
unjust 3544 | Soundness justification th... |
injust 3546 | Soundness justification th... |
dfin5 3548 | Alternate definition for t... |
dfdif2 3549 | Alternate definition of cl... |
eldif 3550 | Expansion of membership in... |
eldifd 3551 | If a class is in one class... |
eldifad 3552 | If a class is in the diffe... |
eldifbd 3553 | If a class is in the diffe... |
dfss 3555 | Variant of subclass defini... |
dfss2 3557 | Alternate definition of th... |
dfss3 3558 | Alternate definition of su... |
dfss2f 3559 | Equivalence for subclass r... |
dfss3f 3560 | Equivalence for subclass r... |
nfss 3561 | If ` x ` is not free in ` ... |
ssel 3562 | Membership relationships f... |
ssel2 3563 | Membership relationships f... |
sseli 3564 | Membership inference from ... |
sselii 3565 | Membership inference from ... |
sseldi 3566 | Membership inference from ... |
sseld 3567 | Membership deduction from ... |
sselda 3568 | Membership deduction from ... |
sseldd 3569 | Membership inference from ... |
ssneld 3570 | If a class is not in anoth... |
ssneldd 3571 | If an element is not in a ... |
ssriv 3572 | Inference rule based on su... |
ssrd 3573 | Deduction rule based on su... |
ssrdv 3574 | Deduction rule based on su... |
sstr2 3575 | Transitivity of subclasses... |
sstr 3576 | Transitivity of subclasses... |
sstri 3577 | Subclass transitivity infe... |
sstrd 3578 | Subclass transitivity dedu... |
syl5ss 3579 | Subclass transitivity dedu... |
syl6ss 3580 | Subclass transitivity dedu... |
sylan9ss 3581 | A subclass transitivity de... |
sylan9ssr 3582 | A subclass transitivity de... |
eqss 3583 | The subclass relationship ... |
eqssi 3584 | Infer equality from two su... |
eqssd 3585 | Equality deduction from tw... |
eqrd 3586 | Deduce equality of classes... |
ssid 3587 | Any class is a subclass of... |
ssv 3588 | Any class is a subclass of... |
sseq1 3589 | Equality theorem for subcl... |
sseq2 3590 | Equality theorem for the s... |
sseq12 3591 | Equality theorem for the s... |
sseq1i 3592 | An equality inference for ... |
sseq2i 3593 | An equality inference for ... |
sseq12i 3594 | An equality inference for ... |
sseq1d 3595 | An equality deduction for ... |
sseq2d 3596 | An equality deduction for ... |
sseq12d 3597 | An equality deduction for ... |
eqsstri 3598 | Substitution of equality i... |
eqsstr3i 3599 | Substitution of equality i... |
sseqtri 3600 | Substitution of equality i... |
sseqtr4i 3601 | Substitution of equality i... |
eqsstrd 3602 | Substitution of equality i... |
eqsstr3d 3603 | Substitution of equality i... |
sseqtrd 3604 | Substitution of equality i... |
sseqtr4d 3605 | Substitution of equality i... |
3sstr3i 3606 | Substitution of equality i... |
3sstr4i 3607 | Substitution of equality i... |
3sstr3g 3608 | Substitution of equality i... |
3sstr4g 3609 | Substitution of equality i... |
3sstr3d 3610 | Substitution of equality i... |
3sstr4d 3611 | Substitution of equality i... |
syl5eqss 3612 | A chained subclass and equ... |
syl5eqssr 3613 | A chained subclass and equ... |
syl6sseq 3614 | A chained subclass and equ... |
syl6sseqr 3615 | A chained subclass and equ... |
syl5sseq 3616 | Subclass transitivity dedu... |
syl5sseqr 3617 | Subclass transitivity dedu... |
syl6eqss 3618 | A chained subclass and equ... |
syl6eqssr 3619 | A chained subclass and equ... |
eqimss 3620 | Equality implies the subcl... |
eqimss2 3621 | Equality implies the subcl... |
eqimssi 3622 | Infer subclass relationshi... |
eqimss2i 3623 | Infer subclass relationshi... |
nssne1 3624 | Two classes are different ... |
nssne2 3625 | Two classes are different ... |
nss 3626 | Negation of subclass relat... |
nelss 3627 | Demonstrate by witnesses t... |
ssrexf 3628 | restricted existential qua... |
ssralv 3629 | Quantification restricted ... |
ssrexv 3630 | Existential quantification... |
ralss 3631 | Restricted universal quant... |
rexss 3632 | Restricted existential qua... |
ss2ab 3633 | Class abstractions in a su... |
abss 3634 | Class abstraction in a sub... |
ssab 3635 | Subclass of a class abstra... |
ssabral 3636 | The relation for a subclas... |
ss2abi 3637 | Inference of abstraction s... |
ss2abdv 3638 | Deduction of abstraction s... |
abssdv 3639 | Deduction of abstraction s... |
abssi 3640 | Inference of abstraction s... |
ss2rab 3641 | Restricted abstraction cla... |
rabss 3642 | Restricted class abstracti... |
ssrab 3643 | Subclass of a restricted c... |
ssrabdv 3644 | Subclass of a restricted c... |
rabssdv 3645 | Subclass of a restricted c... |
ss2rabdv 3646 | Deduction of restricted ab... |
ss2rabi 3647 | Inference of restricted ab... |
rabss2 3648 | Subclass law for restricte... |
ssab2 3649 | Subclass relation for the ... |
ssrab2 3650 | Subclass relation for a re... |
ssrabeq 3651 | If the restricting class o... |
rabssab 3652 | A restricted class is a su... |
uniiunlem 3653 | A subset relationship usef... |
dfpss2 3654 | Alternate definition of pr... |
dfpss3 3655 | Alternate definition of pr... |
psseq1 3656 | Equality theorem for prope... |
psseq2 3657 | Equality theorem for prope... |
psseq1i 3658 | An equality inference for ... |
psseq2i 3659 | An equality inference for ... |
psseq12i 3660 | An equality inference for ... |
psseq1d 3661 | An equality deduction for ... |
psseq2d 3662 | An equality deduction for ... |
psseq12d 3663 | An equality deduction for ... |
pssss 3664 | A proper subclass is a sub... |
pssne 3665 | Two classes in a proper su... |
pssssd 3666 | Deduce subclass from prope... |
pssned 3667 | Proper subclasses are uneq... |
sspss 3668 | Subclass in terms of prope... |
pssirr 3669 | Proper subclass is irrefle... |
pssn2lp 3670 | Proper subclass has no 2-c... |
sspsstri 3671 | Two ways of stating tricho... |
ssnpss 3672 | Partial trichotomy law for... |
psstr 3673 | Transitive law for proper ... |
sspsstr 3674 | Transitive law for subclas... |
psssstr 3675 | Transitive law for subclas... |
psstrd 3676 | Proper subclass inclusion ... |
sspsstrd 3677 | Transitivity involving sub... |
psssstrd 3678 | Transitivity involving sub... |
npss 3679 | A class is not a proper su... |
ssnelpss 3680 | A subclass missing a membe... |
ssnelpssd 3681 | Subclass inclusion with on... |
ssexnelpss 3682 | If there is an element of ... |
difeq1 3683 | Equality theorem for class... |
difeq2 3684 | Equality theorem for class... |
difeq12 3685 | Equality theorem for class... |
difeq1i 3686 | Inference adding differenc... |
difeq2i 3687 | Inference adding differenc... |
difeq12i 3688 | Equality inference for cla... |
difeq1d 3689 | Deduction adding differenc... |
difeq2d 3690 | Deduction adding differenc... |
difeq12d 3691 | Equality deduction for cla... |
difeqri 3692 | Inference from membership ... |
nfdif 3693 | Bound-variable hypothesis ... |
eldifi 3694 | Implication of membership ... |
eldifn 3695 | Implication of membership ... |
elndif 3696 | A set does not belong to a... |
neldif 3697 | Implication of membership ... |
difdif 3698 | Double class difference. ... |
difss 3699 | Subclass relationship for ... |
difssd 3700 | A difference of two classe... |
difss2 3701 | If a class is contained in... |
difss2d 3702 | If a class is contained in... |
ssdifss 3703 | Preservation of a subclass... |
ddif 3704 | Double complement under un... |
ssconb 3705 | Contraposition law for sub... |
sscon 3706 | Contraposition law for sub... |
ssdif 3707 | Difference law for subsets... |
ssdifd 3708 | If ` A ` is contained in `... |
sscond 3709 | If ` A ` is contained in `... |
ssdifssd 3710 | If ` A ` is contained in `... |
ssdif2d 3711 | If ` A ` is contained in `... |
raldifb 3712 | Restricted universal quant... |
complss 3713 | Complementation reverses i... |
compleq 3714 | Two classes are equal if a... |
elun 3715 | Expansion of membership in... |
elunnel1 3716 | A member of a union that i... |
uneqri 3717 | Inference from membership ... |
unidm 3718 | Idempotent law for union o... |
uncom 3719 | Commutative law for union ... |
equncom 3720 | If a class equals the unio... |
equncomi 3721 | Inference form of ~ equnco... |
uneq1 3722 | Equality theorem for the u... |
uneq2 3723 | Equality theorem for the u... |
uneq12 3724 | Equality theorem for the u... |
uneq1i 3725 | Inference adding union to ... |
uneq2i 3726 | Inference adding union to ... |
uneq12i 3727 | Equality inference for the... |
uneq1d 3728 | Deduction adding union to ... |
uneq2d 3729 | Deduction adding union to ... |
uneq12d 3730 | Equality deduction for the... |
nfun 3731 | Bound-variable hypothesis ... |
unass 3732 | Associative law for union ... |
un12 3733 | A rearrangement of union. ... |
un23 3734 | A rearrangement of union. ... |
un4 3735 | A rearrangement of the uni... |
unundi 3736 | Union distributes over its... |
unundir 3737 | Union distributes over its... |
ssun1 3738 | Subclass relationship for ... |
ssun2 3739 | Subclass relationship for ... |
ssun3 3740 | Subclass law for union of ... |
ssun4 3741 | Subclass law for union of ... |
elun1 3742 | Membership law for union o... |
elun2 3743 | Membership law for union o... |
unss1 3744 | Subclass law for union of ... |
ssequn1 3745 | A relationship between sub... |
unss2 3746 | Subclass law for union of ... |
unss12 3747 | Subclass law for union of ... |
ssequn2 3748 | A relationship between sub... |
unss 3749 | The union of two subclasse... |
unssi 3750 | An inference showing the u... |
unssd 3751 | A deduction showing the un... |
unssad 3752 | If ` ( A u. B ) ` is conta... |
unssbd 3753 | If ` ( A u. B ) ` is conta... |
ssun 3754 | A condition that implies i... |
rexun 3755 | Restricted existential qua... |
ralunb 3756 | Restricted quantification ... |
ralun 3757 | Restricted quantification ... |
elin 3758 | Expansion of membership in... |
elini 3759 | Membership in an intersect... |
elind 3760 | Deduce membership in an in... |
elinel1 3761 | Membership in an intersect... |
elinel2 3762 | Membership in an intersect... |
elin2 3763 | Membership in a class defi... |
elin1d 3764 | Elementhood in the first s... |
elin2d 3765 | Elementhood in the first s... |
elin3 3766 | Membership in a class defi... |
incom 3767 | Commutative law for inters... |
ineqri 3768 | Inference from membership ... |
ineq1 3769 | Equality theorem for inter... |
ineq2 3770 | Equality theorem for inter... |
ineq12 3771 | Equality theorem for inter... |
ineq1i 3772 | Equality inference for int... |
ineq2i 3773 | Equality inference for int... |
ineq12i 3774 | Equality inference for int... |
ineq1d 3775 | Equality deduction for int... |
ineq2d 3776 | Equality deduction for int... |
ineq12d 3777 | Equality deduction for int... |
ineqan12d 3778 | Equality deduction for int... |
sseqin2 3779 | A relationship between sub... |
dfss1OLD 3780 | Obsolete as of 22-Jul-2021... |
dfss5OLD 3781 | Obsolete as of 22-Jul-2021... |
nfin 3782 | Bound-variable hypothesis ... |
rabbi2dva 3783 | Deduction from a wff to a ... |
inidm 3784 | Idempotent law for interse... |
inass 3785 | Associative law for inters... |
in12 3786 | A rearrangement of interse... |
in32 3787 | A rearrangement of interse... |
in13 3788 | A rearrangement of interse... |
in31 3789 | A rearrangement of interse... |
inrot 3790 | Rotate the intersection of... |
in4 3791 | Rearrangement of intersect... |
inindi 3792 | Intersection distributes o... |
inindir 3793 | Intersection distributes o... |
sseqin2OLD 3794 | Obsolete proof of ~ sseqin... |
inss1 3795 | The intersection of two cl... |
inss2 3796 | The intersection of two cl... |
ssin 3797 | Subclass of intersection. ... |
ssini 3798 | An inference showing that ... |
ssind 3799 | A deduction showing that a... |
ssrin 3800 | Add right intersection to ... |
sslin 3801 | Add left intersection to s... |
ss2in 3802 | Intersection of subclasses... |
ssinss1 3803 | Intersection preserves sub... |
inss 3804 | Inclusion of an intersecti... |
symdifcom 3807 | Symmetric difference commu... |
symdifeq1 3808 | Equality theorem for symme... |
symdifeq2 3809 | Equality theorem for symme... |
nfsymdif 3810 | Hypothesis builder for sym... |
elsymdif 3811 | Membership in a symmetric ... |
elsymdifxor 3812 | Membership in a symmetric ... |
dfsymdif2 3813 | Alternate definition of th... |
symdif2 3814 | Two ways to express symmet... |
symdifass 3815 | Symmetric difference assoc... |
unabs 3816 | Absorption law for union. ... |
inabs 3817 | Absorption law for interse... |
nssinpss 3818 | Negation of subclass expre... |
nsspssun 3819 | Negation of subclass expre... |
dfss4 3820 | Subclass defined in terms ... |
dfun2 3821 | An alternate definition of... |
dfin2 3822 | An alternate definition of... |
difin 3823 | Difference with intersecti... |
dfun3 3824 | Union defined in terms of ... |
dfin3 3825 | Intersection defined in te... |
dfin4 3826 | Alternate definition of th... |
invdif 3827 | Intersection with universa... |
indif 3828 | Intersection with class di... |
indif2 3829 | Bring an intersection in a... |
indif1 3830 | Bring an intersection in a... |
indifcom 3831 | Commutation law for inters... |
indi 3832 | Distributive law for inter... |
undi 3833 | Distributive law for union... |
indir 3834 | Distributive law for inter... |
undir 3835 | Distributive law for union... |
unineq 3836 | Infer equality from equali... |
uneqin 3837 | Equality of union and inte... |
difundi 3838 | Distributive law for class... |
difundir 3839 | Distributive law for class... |
difindi 3840 | Distributive law for class... |
difindir 3841 | Distributive law for class... |
indifdir 3842 | Distribute intersection ov... |
difdif2 3843 | Class difference by a clas... |
undm 3844 | De Morgan's law for union.... |
indm 3845 | De Morgan's law for inters... |
difun1 3846 | A relationship involving d... |
undif3 3847 | An equality involving clas... |
undif3OLD 3848 | Obsolete proof of ~ undif3... |
difin2 3849 | Represent a class differen... |
dif32 3850 | Swap second and third argu... |
difabs 3851 | Absorption-like law for cl... |
dfsymdif3 3852 | Alternate definition of th... |
unab 3853 | Union of two class abstrac... |
inab 3854 | Intersection of two class ... |
difab 3855 | Difference of two class ab... |
notab 3856 | A class builder defined by... |
unrab 3857 | Union of two restricted cl... |
inrab 3858 | Intersection of two restri... |
inrab2 3859 | Intersection with a restri... |
difrab 3860 | Difference of two restrict... |
dfrab3 3861 | Alternate definition of re... |
dfrab2 3862 | Alternate definition of re... |
notrab 3863 | Complementation of restric... |
dfrab3ss 3864 | Restricted class abstracti... |
rabun2 3865 | Abstraction restricted to ... |
reuss2 3866 | Transfer uniqueness to a s... |
reuss 3867 | Transfer uniqueness to a s... |
reuun1 3868 | Transfer uniqueness to a s... |
reuun2 3869 | Transfer uniqueness to a s... |
reupick 3870 | Restricted uniqueness "pic... |
reupick3 3871 | Restricted uniqueness "pic... |
reupick2 3872 | Restricted uniqueness "pic... |
euelss 3873 | Transfer uniqueness of an ... |
dfnul2 3876 | Alternate definition of th... |
dfnul3 3877 | Alternate definition of th... |
noel 3878 | The empty set has no eleme... |
n0i 3879 | If a set has elements, the... |
ne0i 3880 | If a set has elements, the... |
n0ii 3881 | If a set has elements, the... |
ne0ii 3882 | If a set has elements, the... |
vn0 3883 | The universal class is not... |
eq0f 3884 | The empty set has no eleme... |
neq0f 3885 | A nonempty class has at le... |
n0f 3886 | A nonempty class has at le... |
n0fOLD 3887 | Obsolete proof of ~ n0f as... |
eq0 3888 | The empty set has no eleme... |
neq0 3889 | A nonempty class has at le... |
n0 3890 | A nonempty class has at le... |
reximdva0 3891 | Restricted existence deduc... |
rspn0 3892 | Specialization for restric... |
n0moeu 3893 | A case of equivalence of "... |
rex0 3894 | Vacuous existential quanti... |
0el 3895 | Membership of the empty se... |
ssdif0 3896 | Subclass expressed in term... |
difn0 3897 | If the difference of two s... |
pssdifn0 3898 | A proper subclass has a no... |
pssdif 3899 | A proper subclass has a no... |
difin0ss 3900 | Difference, intersection, ... |
inssdif0 3901 | Intersection, subclass, an... |
difid 3902 | The difference between a c... |
difidALT 3903 | Alternate proof of ~ difid... |
dif0 3904 | The difference between a c... |
ab0 3905 | The class of sets verifyin... |
dfnf5 3906 | Characterization of non-fr... |
ab0orv 3907 | The class builder of a wff... |
abn0 3908 | Nonempty class abstraction... |
rab0 3909 | Any restricted class abstr... |
rab0OLD 3910 | Obsolete proof of ~ rab0 a... |
rabeq0 3911 | Condition for a restricted... |
rabn0 3912 | Nonempty restricted class ... |
rabn0OLD 3913 | Obsolete proof of ~ rabn0 ... |
rabeq0OLD 3914 | Obsolete proof of ~ rabeq0... |
rabxm 3915 | Law of excluded middle, in... |
rabnc 3916 | Law of noncontradiction, i... |
elneldisj 3917 | The set of elements contai... |
elnelun 3918 | The union of the set of el... |
un0 3919 | The union of a class with ... |
in0 3920 | The intersection of a clas... |
0in 3921 | The intersection of the em... |
inv1 3922 | The intersection of a clas... |
unv 3923 | The union of a class with ... |
0ss 3924 | The null set is a subset o... |
ss0b 3925 | Any subset of the empty se... |
ss0 3926 | Any subset of the empty se... |
sseq0 3927 | A subclass of an empty cla... |
ssn0 3928 | A class with a nonempty su... |
0dif 3929 | The difference between the... |
abf 3930 | A class builder with a fal... |
eq0rdv 3931 | Deduction rule for equalit... |
csbprc 3932 | The proper substitution of... |
csbprcOLD 3933 | Obsolete proof of ~ csbprc... |
csb0 3934 | The proper substitution of... |
sbcel12 3935 | Distribute proper substitu... |
sbceqg 3936 | Distribute proper substitu... |
sbcnel12g 3937 | Distribute proper substitu... |
sbcne12 3938 | Distribute proper substitu... |
sbcel1g 3939 | Move proper substitution i... |
sbceq1g 3940 | Move proper substitution t... |
sbcel2 3941 | Move proper substitution i... |
sbceq2g 3942 | Move proper substitution t... |
csbeq2d 3943 | Formula-building deduction... |
csbeq2dv 3944 | Formula-building deduction... |
csbeq2i 3945 | Formula-building inference... |
csbcom 3946 | Commutative law for double... |
sbcnestgf 3947 | Nest the composition of tw... |
csbnestgf 3948 | Nest the composition of tw... |
sbcnestg 3949 | Nest the composition of tw... |
csbnestg 3950 | Nest the composition of tw... |
sbcco3g 3951 | Composition of two substit... |
csbco3g 3952 | Composition of two class s... |
csbnest1g 3953 | Nest the composition of tw... |
csbidm 3954 | Idempotent law for class s... |
csbvarg 3955 | The proper substitution of... |
sbccsb 3956 | Substitution into a wff ex... |
sbccsb2 3957 | Substitution into a wff ex... |
rspcsbela 3958 | Special case related to ~ ... |
sbnfc2 3959 | Two ways of expressing " `... |
csbab 3960 | Move substitution into a c... |
csbun 3961 | Distribution of class subs... |
csbin 3962 | Distribute proper substitu... |
un00 3963 | Two classes are empty iff ... |
vss 3964 | Only the universal class h... |
0pss 3965 | The null set is a proper s... |
npss0 3966 | No set is a proper subset ... |
npss0OLD 3967 | Obsolete proof of ~ npss0 ... |
pssv 3968 | Any non-universal class is... |
disj 3969 | Two ways of saying that tw... |
disjr 3970 | Two ways of saying that tw... |
disj1 3971 | Two ways of saying that tw... |
reldisj 3972 | Two ways of saying that tw... |
disj3 3973 | Two ways of saying that tw... |
disjne 3974 | Members of disjoint sets a... |
disjel 3975 | A set can't belong to both... |
disj2 3976 | Two ways of saying that tw... |
disj4 3977 | Two ways of saying that tw... |
ssdisj 3978 | Intersection with a subcla... |
ssdisjOLD 3979 | Obsolete proof of ~ ssdisj... |
disjpss 3980 | A class is a proper subset... |
undisj1 3981 | The union of disjoint clas... |
undisj2 3982 | The union of disjoint clas... |
ssindif0 3983 | Subclass expressed in term... |
inelcm 3984 | The intersection of classe... |
minel 3985 | A minimum element of a cla... |
minelOLD 3986 | Obsolete proof of ~ minel ... |
undif4 3987 | Distribute union over diff... |
disjssun 3988 | Subset relation for disjoi... |
vdif0 3989 | Universal class equality i... |
difrab0eq 3990 | If the difference between ... |
pssnel 3991 | A proper subclass has a me... |
disjdif 3992 | A class and its relative c... |
difin0 3993 | The difference of a class ... |
unvdif 3994 | The union of a class and i... |
undif1 3995 | Absorption of difference b... |
undif2 3996 | Absorption of difference b... |
undifabs 3997 | Absorption of difference b... |
inundif 3998 | The intersection and class... |
disjdif2 3999 | The difference of a class ... |
difun2 4000 | Absorption of union by dif... |
undif 4001 | Union of complementary par... |
ssdifin0 4002 | A subset of a difference d... |
ssdifeq0 4003 | A class is a subclass of i... |
ssundif 4004 | A condition equivalent to ... |
difcom 4005 | Swap the arguments of a cl... |
pssdifcom1 4006 | Two ways to express overla... |
pssdifcom2 4007 | Two ways to express non-co... |
difdifdir 4008 | Distributive law for class... |
uneqdifeq 4009 | Two ways to say that ` A `... |
uneqdifeqOLD 4010 | Obsolete proof of ~ uneqdi... |
raldifeq 4011 | Equality theorem for restr... |
r19.2z 4012 | Theorem 19.2 of [Margaris]... |
r19.2zb 4013 | A response to the notion t... |
r19.3rz 4014 | Restricted quantification ... |
r19.28z 4015 | Restricted quantifier vers... |
r19.3rzv 4016 | Restricted quantification ... |
r19.9rzv 4017 | Restricted quantification ... |
r19.28zv 4018 | Restricted quantifier vers... |
r19.37zv 4019 | Restricted quantifier vers... |
r19.45zv 4020 | Restricted version of Theo... |
r19.44zv 4021 | Restricted version of Theo... |
r19.27z 4022 | Restricted quantifier vers... |
r19.27zv 4023 | Restricted quantifier vers... |
r19.36zv 4024 | Restricted quantifier vers... |
rzal 4025 | Vacuous quantification is ... |
rexn0 4026 | Restricted existential qua... |
ralidm 4027 | Idempotent law for restric... |
ral0 4028 | Vacuous universal quantifi... |
rgenzOLD 4029 | Obsolete as of 22-Jul-2021... |
ralf0 4030 | The quantification of a fa... |
ralf0OLD 4031 | Obsolete proof of ~ ralf0 ... |
raaan 4032 | Rearrange restricted quant... |
raaanv 4033 | Rearrange restricted quant... |
sbss 4034 | Set substitution into the ... |
sbcssg 4035 | Distribute proper substitu... |
dfif2 4038 | An alternate definition of... |
dfif6 4039 | An alternate definition of... |
ifeq1 4040 | Equality theorem for condi... |
ifeq2 4041 | Equality theorem for condi... |
iftrue 4042 | Value of the conditional o... |
iftruei 4043 | Inference associated with ... |
iftrued 4044 | Value of the conditional o... |
iffalse 4045 | Value of the conditional o... |
iffalsei 4046 | Inference associated with ... |
iffalsed 4047 | Value of the conditional o... |
ifnefalse 4048 | When values are unequal, b... |
ifsb 4049 | Distribute a function over... |
dfif3 4050 | Alternate definition of th... |
dfif4 4051 | Alternate definition of th... |
dfif5 4052 | Alternate definition of th... |
ifeq12 4053 | Equality theorem for condi... |
ifeq1d 4054 | Equality deduction for con... |
ifeq2d 4055 | Equality deduction for con... |
ifeq12d 4056 | Equality deduction for con... |
ifbi 4057 | Equivalence theorem for co... |
ifbid 4058 | Equivalence deduction for ... |
ifbieq1d 4059 | Equivalence/equality deduc... |
ifbieq2i 4060 | Equivalence/equality infer... |
ifbieq2d 4061 | Equivalence/equality deduc... |
ifbieq12i 4062 | Equivalence deduction for ... |
ifbieq12d 4063 | Equivalence deduction for ... |
nfifd 4064 | Deduction version of ~ nfi... |
nfif 4065 | Bound-variable hypothesis ... |
ifeq1da 4066 | Conditional equality. (Co... |
ifeq2da 4067 | Conditional equality. (Co... |
ifeq12da 4068 | Equivalence deduction for ... |
ifbieq12d2 4069 | Equivalence deduction for ... |
ifclda 4070 | Conditional closure. (Con... |
ifeqda 4071 | Separation of the values o... |
elimif 4072 | Elimination of a condition... |
ifbothda 4073 | A wff ` th ` containing a ... |
ifboth 4074 | A wff ` th ` containing a ... |
ifid 4075 | Identical true and false a... |
eqif 4076 | Expansion of an equality w... |
ifval 4077 | Another expression of the ... |
elif 4078 | Membership in a conditiona... |
ifel 4079 | Membership of a conditiona... |
ifcl 4080 | Membership (closure) of a ... |
ifcld 4081 | Membership (closure) of a ... |
ifeqor 4082 | The possible values of a c... |
ifnot 4083 | Negating the first argumen... |
ifan 4084 | Rewrite a conjunction in a... |
ifor 4085 | Rewrite a disjunction in a... |
2if2 4086 | Resolve two nested conditi... |
ifcomnan 4087 | Commute the conditions in ... |
csbif 4088 | Distribute proper substitu... |
dedth 4089 | Weak deduction theorem tha... |
dedth2h 4090 | Weak deduction theorem eli... |
dedth3h 4091 | Weak deduction theorem eli... |
dedth4h 4092 | Weak deduction theorem eli... |
dedth2v 4093 | Weak deduction theorem for... |
dedth3v 4094 | Weak deduction theorem for... |
dedth4v 4095 | Weak deduction theorem for... |
elimhyp 4096 | Eliminate a hypothesis con... |
elimhyp2v 4097 | Eliminate a hypothesis con... |
elimhyp3v 4098 | Eliminate a hypothesis con... |
elimhyp4v 4099 | Eliminate a hypothesis con... |
elimel 4100 | Eliminate a membership hyp... |
elimdhyp 4101 | Version of ~ elimhyp where... |
keephyp 4102 | Transform a hypothesis ` p... |
keephyp2v 4103 | Keep a hypothesis containi... |
keephyp3v 4104 | Keep a hypothesis containi... |
keepel 4105 | Keep a membership hypothes... |
ifex 4106 | Conditional operator exist... |
ifexg 4107 | Conditional operator exist... |
pwjust 4109 | Soundness justification th... |
pweq 4111 | Equality theorem for power... |
pweqi 4112 | Equality inference for pow... |
pweqd 4113 | Equality deduction for pow... |
elpw 4114 | Membership in a power clas... |
selpw 4115 | Setvar variable membership... |
elpwg 4116 | Membership in a power clas... |
elpwi 4117 | Subset relation implied by... |
elpwid 4118 | An element of a power clas... |
elelpwi 4119 | If ` A ` belongs to a part... |
nfpw 4120 | Bound-variable hypothesis ... |
pwidg 4121 | Membership of the original... |
pwid 4122 | A set is a member of its p... |
pwss 4123 | Subclass relationship for ... |
snjust 4124 | Soundness justification th... |
sneq 4135 | Equality theorem for singl... |
sneqi 4136 | Equality inference for sin... |
sneqd 4137 | Equality deduction for sin... |
dfsn2 4138 | Alternate definition of si... |
elsng 4139 | There is exactly one eleme... |
elsn 4140 | There is exactly one eleme... |
velsn 4141 | There is only one element ... |
elsni 4142 | There is only one element ... |
dfpr2 4143 | Alternate definition of un... |
elprg 4144 | A member of an unordered p... |
elpri 4145 | If a class is an element o... |
elpr 4146 | A member of an unordered p... |
elpr2 4147 | A member of an unordered p... |
elpr2OLD 4148 | Obsolete proof of ~ elpr2 ... |
nelpri 4149 | If an element doesn't matc... |
prneli 4150 | If an element doesn't matc... |
nelprd 4151 | If an element doesn't matc... |
eldifpr 4152 | Membership in a set with t... |
snidg 4153 | A set is a member of its s... |
snidb 4154 | A class is a set iff it is... |
snid 4155 | A set is a member of its s... |
vsnid 4156 | A setvar variable is a mem... |
elsn2g 4157 | There is exactly one eleme... |
elsn2 4158 | There is exactly one eleme... |
nelsn 4159 | If a class is not equal to... |
nelsnOLD 4160 | Obsolete proof of ~ nelsn ... |
rabeqsn 4161 | Conditions for a restricte... |
rabsssn 4162 | Conditions for a restricte... |
ralsnsg 4163 | Substitution expressed in ... |
rexsns 4164 | Restricted existential qua... |
ralsng 4165 | Substitution expressed in ... |
rexsng 4166 | Restricted existential qua... |
2ralsng 4167 | Substitution expressed in ... |
exsnrex 4168 | There is a set being the e... |
ralsn 4169 | Convert a quantification o... |
rexsn 4170 | Restricted existential qua... |
elpwunsn 4171 | Membership in an extension... |
eqoreldif 4172 | An element of a set is eit... |
eqoreldifOLD 4173 | Obsolete proof of ~ eqorel... |
eltpg 4174 | Members of an unordered tr... |
eldiftp 4175 | Membership in a set with t... |
eltpi 4176 | A member of an unordered t... |
eltp 4177 | A member of an unordered t... |
dftp2 4178 | Alternate definition of un... |
nfpr 4179 | Bound-variable hypothesis ... |
ifpr 4180 | Membership of a conditiona... |
ralprg 4181 | Convert a quantification o... |
rexprg 4182 | Convert a quantification o... |
raltpg 4183 | Convert a quantification o... |
rextpg 4184 | Convert a quantification o... |
ralpr 4185 | Convert a quantification o... |
rexpr 4186 | Convert an existential qua... |
raltp 4187 | Convert a quantification o... |
rextp 4188 | Convert a quantification o... |
nfsn 4189 | Bound-variable hypothesis ... |
csbsng 4190 | Distribute proper substitu... |
csbprg 4191 | Distribute proper substitu... |
disjsn 4192 | Intersection with the sing... |
disjsn2 4193 | Intersection of distinct s... |
disjpr2 4194 | The intersection of distin... |
disjpr2OLD 4195 | Obsolete proof of ~ disjpr... |
disjprsn 4196 | The disjoint intersection ... |
snprc 4197 | The singleton of a proper ... |
snnzb 4198 | A singleton is nonempty if... |
r19.12sn 4199 | Special case of ~ r19.12 w... |
rabsn 4200 | Condition where a restrict... |
rabsnifsb 4201 | A restricted class abstrac... |
rabsnif 4202 | A restricted class abstrac... |
rabrsn 4203 | A restricted class abstrac... |
euabsn2 4204 | Another way to express exi... |
euabsn 4205 | Another way to express exi... |
reusn 4206 | A way to express restricte... |
absneu 4207 | Restricted existential uni... |
rabsneu 4208 | Restricted existential uni... |
eusn 4209 | Two ways to express " ` A ... |
rabsnt 4210 | Truth implied by equality ... |
prcom 4211 | Commutative law for unorde... |
preq1 4212 | Equality theorem for unord... |
preq2 4213 | Equality theorem for unord... |
preq12 4214 | Equality theorem for unord... |
preq1i 4215 | Equality inference for uno... |
preq2i 4216 | Equality inference for uno... |
preq12i 4217 | Equality inference for uno... |
preq1d 4218 | Equality deduction for uno... |
preq2d 4219 | Equality deduction for uno... |
preq12d 4220 | Equality deduction for uno... |
tpeq1 4221 | Equality theorem for unord... |
tpeq2 4222 | Equality theorem for unord... |
tpeq3 4223 | Equality theorem for unord... |
tpeq1d 4224 | Equality theorem for unord... |
tpeq2d 4225 | Equality theorem for unord... |
tpeq3d 4226 | Equality theorem for unord... |
tpeq123d 4227 | Equality theorem for unord... |
tprot 4228 | Rotation of the elements o... |
tpcoma 4229 | Swap 1st and 2nd members o... |
tpcomb 4230 | Swap 2nd and 3rd members o... |
tpass 4231 | Split off the first elemen... |
qdass 4232 | Two ways to write an unord... |
qdassr 4233 | Two ways to write an unord... |
tpidm12 4234 | Unordered triple ` { A , A... |
tpidm13 4235 | Unordered triple ` { A , B... |
tpidm23 4236 | Unordered triple ` { A , B... |
tpidm 4237 | Unordered triple ` { A , A... |
tppreq3 4238 | An unordered triple is an ... |
prid1g 4239 | An unordered pair contains... |
prid2g 4240 | An unordered pair contains... |
prid1 4241 | An unordered pair contains... |
prid2 4242 | An unordered pair contains... |
prprc1 4243 | A proper class vanishes in... |
prprc2 4244 | A proper class vanishes in... |
prprc 4245 | An unordered pair containi... |
tpid1 4246 | One of the three elements ... |
tpid2 4247 | One of the three elements ... |
tpid3g 4248 | Closed theorem form of ~ t... |
tpid3gOLD 4249 | Obsolete proof of ~ tpid3g... |
tpid3 4250 | One of the three elements ... |
snnzg 4251 | The singleton of a set is ... |
snnz 4252 | The singleton of a set is ... |
prnz 4253 | A pair containing a set is... |
prnzg 4254 | A pair containing a set is... |
prnzgOLD 4255 | Obsolete proof of ~ prnzg ... |
tpnz 4256 | A triplet containing a set... |
tpnzd 4257 | A triplet containing a set... |
raltpd 4258 | Convert a quantification o... |
snss 4259 | The singleton of an elemen... |
eldifsn 4260 | Membership in a set with a... |
eldifsni 4261 | Membership in a set with a... |
neldifsn 4262 | The class ` A ` is not in ... |
neldifsnd 4263 | The class ` A ` is not in ... |
rexdifsn 4264 | Restricted existential qua... |
raldifsni 4265 | Rearrangement of a propert... |
raldifsnb 4266 | Restricted universal quant... |
eldifvsn 4267 | A set is an element of the... |
snssg 4268 | The singleton of an elemen... |
difsn 4269 | An element not in a set ca... |
difprsnss 4270 | Removal of a singleton fro... |
difprsn1 4271 | Removal of a singleton fro... |
difprsn2 4272 | Removal of a singleton fro... |
diftpsn3 4273 | Removal of a singleton fro... |
diftpsn3OLD 4274 | Obsolete proof of ~ diftps... |
difpr 4275 | Removing two elements as p... |
tpprceq3 4276 | An unordered triple is an ... |
tppreqb 4277 | An unordered triple is an ... |
difsnb 4278 | ` ( B \ { A } ) ` equals `... |
difsnpss 4279 | ` ( B \ { A } ) ` is a pro... |
snssi 4280 | The singleton of an elemen... |
snssd 4281 | The singleton of an elemen... |
difsnid 4282 | If we remove a single elem... |
pw0 4283 | Compute the power set of t... |
pwpw0 4284 | Compute the power set of t... |
snsspr1 4285 | A singleton is a subset of... |
snsspr2 4286 | A singleton is a subset of... |
snsstp1 4287 | A singleton is a subset of... |
snsstp2 4288 | A singleton is a subset of... |
snsstp3 4289 | A singleton is a subset of... |
prssg 4290 | A pair of elements of a cl... |
prss 4291 | A pair of elements of a cl... |
prssOLD 4292 | Obsolete proof of ~ prss a... |
prssi 4293 | A pair of elements of a cl... |
prssd 4294 | Deduction version of ~ prs... |
prsspwg 4295 | An unordered pair belongs ... |
ssprss 4296 | A pair as subset of a pair... |
ssprsseq 4297 | A proper pair is a subset ... |
sssn 4298 | The subsets of a singleton... |
ssunsn2 4299 | The property of being sand... |
ssunsn 4300 | Possible values for a set ... |
eqsn 4301 | Two ways to express that a... |
eqsnOLD 4302 | Obsolete proof of ~ eqsn a... |
issn 4303 | A sufficient condition for... |
n0snor2el 4304 | A nonempty set is either a... |
ssunpr 4305 | Possible values for a set ... |
sspr 4306 | The subsets of a pair. (C... |
sstp 4307 | The subsets of a triple. ... |
tpss 4308 | A triplet of elements of a... |
tpssi 4309 | A triple of elements of a ... |
sneqrg 4310 | Closed form of ~ sneqr . ... |
sneqr 4311 | If the singletons of two s... |
snsssn 4312 | If a singleton is a subset... |
sneqrgOLD 4313 | Obsolete proof of ~ sneqrg... |
sneqbg 4314 | Two singletons of sets are... |
snsspw 4315 | The singleton of a class i... |
prsspw 4316 | An unordered pair belongs ... |
preq1b 4317 | Biconditional equality lem... |
preq2b 4318 | Biconditional equality lem... |
preqr1 4319 | Reverse equality lemma for... |
preqr1OLD 4320 | Reverse equality lemma for... |
preqr2 4321 | Reverse equality lemma for... |
preq12b 4322 | Equality relationship for ... |
prel12 4323 | Equality of two unordered ... |
opthpr 4324 | An unordered pair has the ... |
preqr1g 4325 | Reverse equality lemma for... |
preq12bg 4326 | Closed form of ~ preq12b .... |
prel12g 4327 | Closed form of ~ prel12 . ... |
prneimg 4328 | Two pairs are not equal if... |
prnebg 4329 | A (proper) pair is not equ... |
preqsnd 4330 | Equivalence for a pair equ... |
preqsn 4331 | Equivalence for a pair equ... |
preqsnOLD 4332 | Obsolete proof of ~ preqsn... |
elpreqprlem 4333 | Lemma for ~ elpreqpr . (C... |
elpreqpr 4334 | Equality and membership ru... |
elpreqprb 4335 | A set is an element of an ... |
elpr2elpr 4336 | For an element of an unord... |
dfopif 4337 | Rewrite ~ df-op using ` if... |
dfopg 4338 | Value of the ordered pair ... |
dfop 4339 | Value of an ordered pair w... |
opeq1 4340 | Equality theorem for order... |
opeq2 4341 | Equality theorem for order... |
opeq12 4342 | Equality theorem for order... |
opeq1i 4343 | Equality inference for ord... |
opeq2i 4344 | Equality inference for ord... |
opeq12i 4345 | Equality inference for ord... |
opeq1d 4346 | Equality deduction for ord... |
opeq2d 4347 | Equality deduction for ord... |
opeq12d 4348 | Equality deduction for ord... |
oteq1 4349 | Equality theorem for order... |
oteq2 4350 | Equality theorem for order... |
oteq3 4351 | Equality theorem for order... |
oteq1d 4352 | Equality deduction for ord... |
oteq2d 4353 | Equality deduction for ord... |
oteq3d 4354 | Equality deduction for ord... |
oteq123d 4355 | Equality deduction for ord... |
nfop 4356 | Bound-variable hypothesis ... |
nfopd 4357 | Deduction version of bound... |
csbopg 4358 | Distribution of class subs... |
opid 4359 | The ordered pair ` <. A , ... |
ralunsn 4360 | Restricted quantification ... |
2ralunsn 4361 | Double restricted quantifi... |
opprc 4362 | Expansion of an ordered pa... |
opprc1 4363 | Expansion of an ordered pa... |
opprc2 4364 | Expansion of an ordered pa... |
oprcl 4365 | If an ordered pair has an ... |
pwsn 4366 | The power set of a singlet... |
pwsnALT 4367 | Alternate proof of ~ pwsn ... |
pwpr 4368 | The power set of an unorde... |
pwtp 4369 | The power set of an unorde... |
pwpwpw0 4370 | Compute the power set of t... |
pwv 4371 | The power class of the uni... |
dfuni2 4374 | Alternate definition of cl... |
eluni 4375 | Membership in class union.... |
eluni2 4376 | Membership in class union.... |
elunii 4377 | Membership in class union.... |
nfuni 4378 | Bound-variable hypothesis ... |
nfunid 4379 | Deduction version of ~ nfu... |
unieq 4380 | Equality theorem for class... |
unieqi 4381 | Inference of equality of t... |
unieqd 4382 | Deduction of equality of t... |
eluniab 4383 | Membership in union of a c... |
elunirab 4384 | Membership in union of a c... |
unipr 4385 | The union of a pair is the... |
uniprg 4386 | The union of a pair is the... |
unisn 4387 | A set equals the union of ... |
unisng 4388 | A set equals the union of ... |
unisn3 4389 | Union of a singleton in th... |
dfnfc2 4390 | An alternative statement o... |
dfnfc2OLD 4391 | Obsolete proof of ~ dfnfc2... |
uniun 4392 | The class union of the uni... |
uniin 4393 | The class union of the int... |
uniss 4394 | Subclass relationship for ... |
ssuni 4395 | Subclass relationship for ... |
ssuniOLD 4396 | Obsolete proof of ~ ssuni ... |
unissi 4397 | Subclass relationship for ... |
unissd 4398 | Subclass relationship for ... |
uni0b 4399 | The union of a set is empt... |
uni0c 4400 | The union of a set is empt... |
uni0 4401 | The union of the empty set... |
csbuni 4402 | Distribute proper substitu... |
elssuni 4403 | An element of a class is a... |
unissel 4404 | Condition turning a subcla... |
unissb 4405 | Relationship involving mem... |
uniss2 4406 | A subclass condition on th... |
unidif 4407 | If the difference ` A \ B ... |
ssunieq 4408 | Relationship implying unio... |
unimax 4409 | Any member of a class is t... |
dfint2 4412 | Alternate definition of cl... |
inteq 4413 | Equality law for intersect... |
inteqi 4414 | Equality inference for cla... |
inteqd 4415 | Equality deduction for cla... |
elint 4416 | Membership in class inters... |
elint2 4417 | Membership in class inters... |
elintg 4418 | Membership in class inters... |
elintgOLD 4419 | Obsolete proof of ~ elintg... |
elinti 4420 | Membership in class inters... |
nfint 4421 | Bound-variable hypothesis ... |
elintab 4422 | Membership in the intersec... |
elintrab 4423 | Membership in the intersec... |
elintrabg 4424 | Membership in the intersec... |
int0 4425 | The intersection of the em... |
int0OLD 4426 | Obsolete proof of ~ int0 a... |
intss1 4427 | An element of a class incl... |
ssint 4428 | Subclass of a class inters... |
ssintab 4429 | Subclass of the intersecti... |
ssintub 4430 | Subclass of the least uppe... |
ssmin 4431 | Subclass of the minimum va... |
intmin 4432 | Any member of a class is t... |
intss 4433 | Intersection of subclasses... |
intssuni 4434 | The intersection of a none... |
ssintrab 4435 | Subclass of the intersecti... |
unissint 4436 | If the union of a class is... |
intssuni2 4437 | Subclass relationship for ... |
intminss 4438 | Under subset ordering, the... |
intmin2 4439 | Any set is the smallest of... |
intmin3 4440 | Under subset ordering, the... |
intmin4 4441 | Elimination of a conjunct ... |
intab 4442 | The intersection of a spec... |
int0el 4443 | The intersection of a clas... |
intun 4444 | The class intersection of ... |
intpr 4445 | The intersection of a pair... |
intprg 4446 | The intersection of a pair... |
intsng 4447 | Intersection of a singleto... |
intsn 4448 | The intersection of a sing... |
uniintsn 4449 | Two ways to express " ` A ... |
uniintab 4450 | The union and the intersec... |
intunsn 4451 | Theorem joining a singleto... |
rint0 4452 | Relative intersection of a... |
elrint 4453 | Membership in a restricted... |
elrint2 4454 | Membership in a restricted... |
rabasiun 4459 | A class abstraction with a... |
eliun 4460 | Membership in indexed unio... |
eliin 4461 | Membership in indexed inte... |
eliuni 4462 | Membership in an indexed u... |
iuncom 4463 | Commutation of indexed uni... |
iuncom4 4464 | Commutation of union with ... |
iunconst 4465 | Indexed union of a constan... |
iinconst 4466 | Indexed intersection of a ... |
iuniin 4467 | Law combining indexed unio... |
iunss1 4468 | Subclass theorem for index... |
iinss1 4469 | Subclass theorem for index... |
iuneq1 4470 | Equality theorem for index... |
iineq1 4471 | Equality theorem for index... |
ss2iun 4472 | Subclass theorem for index... |
iuneq2 4473 | Equality theorem for index... |
iineq2 4474 | Equality theorem for index... |
iuneq2i 4475 | Equality inference for ind... |
iineq2i 4476 | Equality inference for ind... |
iineq2d 4477 | Equality deduction for ind... |
iuneq2dv 4478 | Equality deduction for ind... |
iineq2dv 4479 | Equality deduction for ind... |
iuneq12df 4480 | Equality deduction for ind... |
iuneq1d 4481 | Equality theorem for index... |
iuneq12d 4482 | Equality deduction for ind... |
iuneq2d 4483 | Equality deduction for ind... |
nfiun 4484 | Bound-variable hypothesis ... |
nfiin 4485 | Bound-variable hypothesis ... |
nfiu1 4486 | Bound-variable hypothesis ... |
nfii1 4487 | Bound-variable hypothesis ... |
dfiun2g 4488 | Alternate definition of in... |
dfiin2g 4489 | Alternate definition of in... |
dfiun2 4490 | Alternate definition of in... |
dfiin2 4491 | Alternate definition of in... |
dfiunv2 4492 | Define double indexed unio... |
cbviun 4493 | Rule used to change the bo... |
cbviin 4494 | Change bound variables in ... |
cbviunv 4495 | Rule used to change the bo... |
cbviinv 4496 | Change bound variables in ... |
iunss 4497 | Subset theorem for an inde... |
ssiun 4498 | Subset implication for an ... |
ssiun2 4499 | Identity law for subset of... |
ssiun2s 4500 | Subset relationship for an... |
iunss2 4501 | A subclass condition on th... |
iunab 4502 | The indexed union of a cla... |
iunrab 4503 | The indexed union of a res... |
iunxdif2 4504 | Indexed union with a class... |
ssiinf 4505 | Subset theorem for an inde... |
ssiin 4506 | Subset theorem for an inde... |
iinss 4507 | Subset implication for an ... |
iinss2 4508 | An indexed intersection is... |
uniiun 4509 | Class union in terms of in... |
intiin 4510 | Class intersection in term... |
iunid 4511 | An indexed union of single... |
iun0 4512 | An indexed union of the em... |
0iun 4513 | An empty indexed union is ... |
0iin 4514 | An empty indexed intersect... |
viin 4515 | Indexed intersection with ... |
iunn0 4516 | There is a nonempty class ... |
iinab 4517 | Indexed intersection of a ... |
iinrab 4518 | Indexed intersection of a ... |
iinrab2 4519 | Indexed intersection of a ... |
iunin2 4520 | Indexed union of intersect... |
iunin1 4521 | Indexed union of intersect... |
iinun2 4522 | Indexed intersection of un... |
iundif2 4523 | Indexed union of class dif... |
2iunin 4524 | Rearrange indexed unions o... |
iindif2 4525 | Indexed intersection of cl... |
iinin2 4526 | Indexed intersection of in... |
iinin1 4527 | Indexed intersection of in... |
iinvdif 4528 | The indexed intersection o... |
elriin 4529 | Elementhood in a relative ... |
riin0 4530 | Relative intersection of a... |
riinn0 4531 | Relative intersection of a... |
riinrab 4532 | Relative intersection of a... |
symdif0 4533 | Symmetric difference with ... |
symdifv 4534 | Symmetric difference with ... |
symdifid 4535 | Symmetric difference with ... |
iinxsng 4536 | A singleton index picks ou... |
iinxprg 4537 | Indexed intersection with ... |
iunxsng 4538 | A singleton index picks ou... |
iunxsn 4539 | A singleton index picks ou... |
iunun 4540 | Separate a union in an ind... |
iunxun 4541 | Separate a union in the in... |
iunxdif3 4542 | An indexed union where som... |
iunxprg 4543 | A pair index picks out two... |
iunxiun 4544 | Separate an indexed union ... |
iinuni 4545 | A relationship involving u... |
iununi 4546 | A relationship involving u... |
sspwuni 4547 | Subclass relationship for ... |
pwssb 4548 | Two ways to express a coll... |
elpwuni 4549 | Relationship for power cla... |
iinpw 4550 | The power class of an inte... |
iunpwss 4551 | Inclusion of an indexed un... |
rintn0 4552 | Relative intersection of a... |
dfdisj2 4555 | Alternate definition for d... |
disjss2 4556 | If each element of a colle... |
disjeq2 4557 | Equality theorem for disjo... |
disjeq2dv 4558 | Equality deduction for dis... |
disjss1 4559 | A subset of a disjoint col... |
disjeq1 4560 | Equality theorem for disjo... |
disjeq1d 4561 | Equality theorem for disjo... |
disjeq12d 4562 | Equality theorem for disjo... |
cbvdisj 4563 | Change bound variables in ... |
cbvdisjv 4564 | Change bound variables in ... |
nfdisj 4565 | Bound-variable hypothesis ... |
nfdisj1 4566 | Bound-variable hypothesis ... |
disjor 4567 | Two ways to say that a col... |
disjors 4568 | Two ways to say that a col... |
disji2 4569 | Property of a disjoint col... |
disji 4570 | Property of a disjoint col... |
invdisj 4571 | If there is a function ` C... |
invdisjrab 4572 | The restricted class abstr... |
disjiun 4573 | A disjoint collection yiel... |
sndisj 4574 | Any collection of singleto... |
0disj 4575 | Any collection of empty se... |
disjxsn 4576 | A singleton collection is ... |
disjx0 4577 | An empty collection is dis... |
disjprg 4578 | A pair collection is disjo... |
disjxiun 4579 | An indexed union of a disj... |
disjxiunOLD 4580 | Obsolete proof of ~ disjxi... |
disjxun 4581 | The union of two disjoint ... |
disjss3 4582 | Expand a disjoint collecti... |
breq 4585 | Equality theorem for binar... |
breq1 4586 | Equality theorem for a bin... |
breq2 4587 | Equality theorem for a bin... |
breq12 4588 | Equality theorem for a bin... |
breqi 4589 | Equality inference for bin... |
breq1i 4590 | Equality inference for a b... |
breq2i 4591 | Equality inference for a b... |
breq12i 4592 | Equality inference for a b... |
breq1d 4593 | Equality deduction for a b... |
breqd 4594 | Equality deduction for a b... |
breq2d 4595 | Equality deduction for a b... |
breq12d 4596 | Equality deduction for a b... |
breq123d 4597 | Equality deduction for a b... |
breqdi 4598 | Equality deduction for a b... |
breqan12d 4599 | Equality deduction for a b... |
breqan12rd 4600 | Equality deduction for a b... |
eqnbrtrd 4601 | Substitution of equal clas... |
nbrne1 4602 | Two classes are different ... |
nbrne2 4603 | Two classes are different ... |
eqbrtri 4604 | Substitution of equal clas... |
eqbrtrd 4605 | Substitution of equal clas... |
eqbrtrri 4606 | Substitution of equal clas... |
eqbrtrrd 4607 | Substitution of equal clas... |
breqtri 4608 | Substitution of equal clas... |
breqtrd 4609 | Substitution of equal clas... |
breqtrri 4610 | Substitution of equal clas... |
breqtrrd 4611 | Substitution of equal clas... |
3brtr3i 4612 | Substitution of equality i... |
3brtr4i 4613 | Substitution of equality i... |
3brtr3d 4614 | Substitution of equality i... |
3brtr4d 4615 | Substitution of equality i... |
3brtr3g 4616 | Substitution of equality i... |
3brtr4g 4617 | Substitution of equality i... |
syl5eqbr 4618 | A chained equality inferen... |
syl5eqbrr 4619 | A chained equality inferen... |
syl5breq 4620 | A chained equality inferen... |
syl5breqr 4621 | A chained equality inferen... |
syl6eqbr 4622 | A chained equality inferen... |
syl6eqbrr 4623 | A chained equality inferen... |
syl6breq 4624 | A chained equality inferen... |
syl6breqr 4625 | A chained equality inferen... |
ssbrd 4626 | Deduction from a subclass ... |
ssbri 4627 | Inference from a subclass ... |
nfbrd 4628 | Deduction version of bound... |
nfbr 4629 | Bound-variable hypothesis ... |
brab1 4630 | Relationship between a bin... |
br0 4631 | The empty binary relation ... |
brne0 4632 | If two sets are in a binar... |
brun 4633 | The union of two binary re... |
brin 4634 | The intersection of two re... |
brdif 4635 | The difference of two bina... |
sbcbr123 4636 | Move substitution in and o... |
sbcbr 4637 | Move substitution in and o... |
sbcbr12g 4638 | Move substitution in and o... |
sbcbr1g 4639 | Move substitution in and o... |
sbcbr2g 4640 | Move substitution in and o... |
brsymdif 4641 | The binary relationship of... |
opabss 4646 | The collection of ordered ... |
opabbid 4647 | Equivalent wff's yield equ... |
opabbidv 4648 | Equivalent wff's yield equ... |
opabbii 4649 | Equivalent wff's yield equ... |
nfopab 4650 | Bound-variable hypothesis ... |
nfopab1 4651 | The first abstraction vari... |
nfopab2 4652 | The second abstraction var... |
cbvopab 4653 | Rule used to change bound ... |
cbvopabv 4654 | Rule used to change bound ... |
cbvopab1 4655 | Change first bound variabl... |
cbvopab2 4656 | Change second bound variab... |
cbvopab1s 4657 | Change first bound variabl... |
cbvopab1v 4658 | Rule used to change the fi... |
cbvopab2v 4659 | Rule used to change the se... |
unopab 4660 | Union of two ordered pair ... |
mpteq12f 4661 | An equality theorem for th... |
mpteq12dva 4662 | An equality inference for ... |
mpteq12dv 4663 | An equality inference for ... |
mpteq12 4664 | An equality theorem for th... |
mpteq1 4665 | An equality theorem for th... |
mpteq1d 4666 | An equality theorem for th... |
mpteq1i 4667 | An equality theorem for th... |
mpteq2ia 4668 | An equality inference for ... |
mpteq2i 4669 | An equality inference for ... |
mpteq12i 4670 | An equality inference for ... |
mpteq2da 4671 | Slightly more general equa... |
mpteq2dva 4672 | Slightly more general equa... |
mpteq2dv 4673 | An equality inference for ... |
nfmpt 4674 | Bound-variable hypothesis ... |
nfmpt1 4675 | Bound-variable hypothesis ... |
cbvmptf 4676 | Rule to change the bound v... |
cbvmpt 4677 | Rule to change the bound v... |
cbvmptv 4678 | Rule to change the bound v... |
mptv 4679 | Function with universal do... |
dftr2 4682 | An alternate way of defini... |
dftr5 4683 | An alternate way of defini... |
dftr3 4684 | An alternate way of defini... |
dftr4 4685 | An alternate way of defini... |
treq 4686 | Equality theorem for the t... |
trel 4687 | In a transitive class, the... |
trel3 4688 | In a transitive class, the... |
trss 4689 | An element of a transitive... |
trssOLD 4690 | Obsolete proof of ~ trss a... |
trin 4691 | The intersection of transi... |
tr0 4692 | The empty set is transitiv... |
trv 4693 | The universe is transitive... |
triun 4694 | The indexed union of a cla... |
truni 4695 | The union of a class of tr... |
trint 4696 | The intersection of a clas... |
trintss 4697 | If ` A ` is transitive and... |
trint0 4698 | Any nonempty transitive cl... |
axrep1 4700 | The version of the Axiom o... |
axrep2 4701 | Axiom of Replacement expre... |
axrep3 4702 | Axiom of Replacement sligh... |
axrep4 4703 | A more traditional version... |
axrep5 4704 | Axiom of Replacement (simi... |
zfrepclf 4705 | An inference rule based on... |
zfrep3cl 4706 | An inference rule based on... |
zfrep4 4707 | A version of Replacement u... |
axsep 4708 | Separation Scheme, which i... |
axsep2 4710 | A less restrictive version... |
zfauscl 4711 | Separation Scheme (Aussond... |
bm1.3ii 4712 | Convert implication to equ... |
ax6vsep 4713 | Derive a weakened version ... |
zfnuleu 4714 | Show the uniqueness of the... |
axnulALT 4715 | Alternate proof of ~ axnul... |
axnul 4716 | The Null Set Axiom of ZF s... |
0ex 4718 | The Null Set Axiom of ZF s... |
sseliALT 4719 | Alternate proof of ~ sseli... |
csbexg 4720 | The existence of proper su... |
csbex 4721 | The existence of proper su... |
unisn2 4722 | A version of ~ unisn witho... |
nalset 4723 | No set contains all sets. ... |
vprc 4724 | The universal class is not... |
nvel 4725 | The universal class doesn'... |
vnex 4726 | The universal class does n... |
inex1 4727 | Separation Scheme (Aussond... |
inex2 4728 | Separation Scheme (Aussond... |
inex1g 4729 | Closed-form, generalized S... |
ssex 4730 | The subset of a set is als... |
ssexi 4731 | The subset of a set is als... |
ssexg 4732 | The subset of a set is als... |
ssexd 4733 | A subclass of a set is a s... |
sselpwd 4734 | Elementhood to a power set... |
difexg 4735 | Existence of a difference.... |
difexi 4736 | Existence of a difference,... |
difexOLD 4737 | Obsolete version of ~ dife... |
zfausab 4738 | Separation Scheme (Aussond... |
rabexg 4739 | Separation Scheme in terms... |
rabex 4740 | Separation Scheme in terms... |
rabexd 4741 | Separation Scheme in terms... |
rabex2 4742 | Separation Scheme in terms... |
rab2ex 4743 | A class abstraction based ... |
rabex2OLD 4744 | Obsolete version of ~ rabe... |
rab2exOLD 4745 | Obsolete version of ~ rabe... |
elssabg 4746 | Membership in a class abst... |
intex 4747 | The intersection of a none... |
intnex 4748 | If a class intersection is... |
intexab 4749 | The intersection of a none... |
intexrab 4750 | The intersection of a none... |
iinexg 4751 | The existence of a class i... |
intabs 4752 | Absorption of a redundant ... |
inuni 4753 | The intersection of a unio... |
elpw2g 4754 | Membership in a power clas... |
elpw2 4755 | Membership in a power clas... |
pwnss 4756 | The power set of a set is ... |
pwne 4757 | No set equals its power se... |
class2set 4758 | Construct, from any class ... |
class2seteq 4759 | Equality theorem based on ... |
0elpw 4760 | Every power class contains... |
pwne0 4761 | A power class is never emp... |
0nep0 4762 | The empty set and its powe... |
0inp0 4763 | Something cannot be equal ... |
unidif0 4764 | The removal of the empty s... |
iin0 4765 | An indexed intersection of... |
notzfaus 4766 | In the Separation Scheme ~... |
intv 4767 | The intersection of the un... |
axpweq 4768 | Two equivalent ways to exp... |
zfpow 4770 | Axiom of Power Sets expres... |
axpow2 4771 | A variant of the Axiom of ... |
axpow3 4772 | A variant of the Axiom of ... |
el 4773 | Every set is an element of... |
pwex 4774 | Power set axiom expressed ... |
vpwex 4775 | The powerset of a setvar i... |
pwexg 4776 | Power set axiom expressed ... |
abssexg 4777 | Existence of a class of su... |
snexALT 4778 | Alternate proof of ~ snex ... |
p0ex 4779 | The power set of the empty... |
p0exALT 4780 | Alternate proof of ~ p0ex ... |
pp0ex 4781 | The power set of the power... |
ord3ex 4782 | The ordinal number 3 is a ... |
dtru 4783 | At least two sets exist (o... |
axc16b 4784 | This theorem shows that ax... |
eunex 4785 | Existential uniqueness imp... |
eusv1 4786 | Two ways to express single... |
eusvnf 4787 | Even if ` x ` is free in `... |
eusvnfb 4788 | Two ways to say that ` A (... |
eusv2i 4789 | Two ways to express single... |
eusv2nf 4790 | Two ways to express single... |
eusv2 4791 | Two ways to express single... |
reusv1 4792 | Two ways to express single... |
reusv1OLD 4793 | Obsolete proof of ~ reusv1... |
reusv2lem1 4794 | Lemma for ~ reusv2 . (Con... |
reusv2lem2 4795 | Lemma for ~ reusv2 . (Con... |
reusv2lem2OLD 4796 | Obsolete proof of ~ reusv2... |
reusv2lem3 4797 | Lemma for ~ reusv2 . (Con... |
reusv2lem4 4798 | Lemma for ~ reusv2 . (Con... |
reusv2lem5 4799 | Lemma for ~ reusv2 . (Con... |
reusv2 4800 | Two ways to express single... |
reusv3i 4801 | Two ways of expressing exi... |
reusv3 4802 | Two ways to express single... |
eusv4 4803 | Two ways to express single... |
alxfr 4804 | Transfer universal quantif... |
ralxfrd 4805 | Transfer universal quantif... |
ralxfrdOLD 4806 | Obsolete proof of ~ ralxfr... |
rexxfrd 4807 | Transfer universal quantif... |
ralxfr2d 4808 | Transfer universal quantif... |
rexxfr2d 4809 | Transfer universal quantif... |
ralxfrd2 4810 | Transfer universal quantif... |
rexxfrd2 4811 | Transfer existence from a ... |
ralxfr 4812 | Transfer universal quantif... |
ralxfrALT 4813 | Alternate proof of ~ ralxf... |
rexxfr 4814 | Transfer existence from a ... |
rabxfrd 4815 | Class builder membership a... |
rabxfr 4816 | Class builder membership a... |
reuxfr2d 4817 | Transfer existential uniqu... |
reuxfr2 4818 | Transfer existential uniqu... |
reuxfrd 4819 | Transfer existential uniqu... |
reuxfr 4820 | Transfer existential uniqu... |
reuhypd 4821 | A theorem useful for elimi... |
reuhyp 4822 | A theorem useful for elimi... |
nfnid 4823 | A setvar variable is not f... |
nfcvb 4824 | The "distinctor" expressio... |
pwuni 4825 | A class is a subclass of t... |
dtruALT 4826 | Alternate proof of ~ dtru ... |
dtrucor 4827 | Corollary of ~ dtru . Thi... |
dtrucor2 4828 | The theorem form of the de... |
dvdemo1 4829 | Demonstration of a theorem... |
dvdemo2 4830 | Demonstration of a theorem... |
zfpair 4831 | The Axiom of Pairing of Ze... |
axpr 4832 | Unabbreviated version of t... |
zfpair2 4834 | Derive the abbreviated ver... |
snex 4835 | A singleton is a set. The... |
prex 4836 | The Axiom of Pairing using... |
elALT 4837 | Alternate proof of ~ el , ... |
dtruALT2 4838 | Alternate proof of ~ dtru ... |
snelpwi 4839 | A singleton of a set belon... |
snelpw 4840 | A singleton of a set belon... |
prelpw 4841 | A pair of two sets belongs... |
prelpwi 4842 | A pair of two sets belongs... |
rext 4843 | A theorem similar to exten... |
sspwb 4844 | Classes are subclasses if ... |
unipw 4845 | A class equals the union o... |
univ 4846 | The union of the universe ... |
pwel 4847 | Membership of a power clas... |
pwtr 4848 | A class is transitive iff ... |
ssextss 4849 | An extensionality-like pri... |
ssext 4850 | An extensionality-like pri... |
nssss 4851 | Negation of subclass relat... |
pweqb 4852 | Classes are equal if and o... |
intid 4853 | The intersection of all se... |
moabex 4854 | "At most one" existence im... |
rmorabex 4855 | Restricted "at most one" e... |
euabex 4856 | The abstraction of a wff w... |
nnullss 4857 | A nonempty class (even if ... |
exss 4858 | Restricted existence in a ... |
opex 4859 | An ordered pair of classes... |
otex 4860 | An ordered triple of class... |
elopg 4861 | Characterization of the el... |
elop 4862 | Characterization of the el... |
elopOLD 4863 | Obsolete version of ~ elop... |
opi1 4864 | One of the two elements in... |
opi2 4865 | One of the two elements of... |
opeluu 4866 | Each member of an ordered ... |
op1stb 4867 | Extract the first member o... |
opnz 4868 | An ordered pair is nonempt... |
opnzi 4869 | An ordered pair is nonempt... |
opth1 4870 | Equality of the first memb... |
opth 4871 | The ordered pair theorem. ... |
opthg 4872 | Ordered pair theorem. ` C ... |
opth1g 4873 | Equality of the first memb... |
opthg2 4874 | Ordered pair theorem. (Co... |
opth2 4875 | Ordered pair theorem. (Co... |
opthneg 4876 | Two ordered pairs are not ... |
opthne 4877 | Two ordered pairs are not ... |
otth2 4878 | Ordered triple theorem, wi... |
otth 4879 | Ordered triple theorem. (... |
otthg 4880 | Ordered triple theorem, cl... |
eqvinop 4881 | A variable introduction la... |
copsexg 4882 | Substitution of class ` A ... |
copsex2t 4883 | Closed theorem form of ~ c... |
copsex2g 4884 | Implicit substitution infe... |
copsex4g 4885 | An implicit substitution i... |
0nelop 4886 | A property of ordered pair... |
opeqex 4887 | Equivalence of existence i... |
oteqex2 4888 | Equivalence of existence i... |
oteqex 4889 | Equivalence of existence i... |
opcom 4890 | An ordered pair commutes i... |
moop2 4891 | "At most one" property of ... |
opeqsn 4892 | Equivalence for an ordered... |
opeqpr 4893 | Equivalence for an ordered... |
snopeqop 4894 | Equivalence for an ordered... |
propeqop 4895 | Equivalence for an ordered... |
propssopi 4896 | If a pair of ordered pairs... |
mosubopt 4897 | "At most one" remains true... |
mosubop 4898 | "At most one" remains true... |
euop2 4899 | Transfer existential uniqu... |
euotd 4900 | Prove existential uniquene... |
opthwiener 4901 | Justification theorem for ... |
uniop 4902 | The union of an ordered pa... |
uniopel 4903 | Ordered pair membership is... |
otsndisj 4904 | The singletons consisting ... |
otiunsndisj 4905 | The union of singletons co... |
iunopeqop 4906 | Implication of an ordered ... |
opabid 4907 | The law of concretion. Sp... |
elopab 4908 | Membership in a class abst... |
opelopabsbALT 4909 | The law of concretion in t... |
opelopabsb 4910 | The law of concretion in t... |
brabsb 4911 | The law of concretion in t... |
opelopabt 4912 | Closed theorem form of ~ o... |
opelopabga 4913 | The law of concretion. Th... |
brabga 4914 | The law of concretion for ... |
opelopab2a 4915 | Ordered pair membership in... |
opelopaba 4916 | The law of concretion. Th... |
braba 4917 | The law of concretion for ... |
opelopabg 4918 | The law of concretion. Th... |
brabg 4919 | The law of concretion for ... |
opelopabgf 4920 | The law of concretion. Th... |
opelopab2 4921 | Ordered pair membership in... |
opelopab 4922 | The law of concretion. Th... |
brab 4923 | The law of concretion for ... |
opelopabaf 4924 | The law of concretion. Th... |
opelopabf 4925 | The law of concretion. Th... |
ssopab2 4926 | Equivalence of ordered pai... |
ssopab2b 4927 | Equivalence of ordered pai... |
ssopab2i 4928 | Inference of ordered pair ... |
ssopab2dv 4929 | Inference of ordered pair ... |
eqopab2b 4930 | Equivalence of ordered pai... |
opabn0 4931 | Nonempty ordered pair clas... |
csbopab 4932 | Move substitution into a c... |
csbopabgALT 4933 | Move substitution into a c... |
csbmpt12 4934 | Move substitution into a m... |
csbmpt2 4935 | Move substitution into the... |
iunopab 4936 | Move indexed union inside ... |
elopabr 4937 | Membership in a class abst... |
elopabran 4938 | Membership in a class abst... |
rbropapd 4939 | Properties of a pair in an... |
rbropap 4940 | Properties of a pair in a ... |
2rbropap 4941 | Properties of a pair in a ... |
pwin 4942 | The power class of the int... |
pwunss 4943 | The power class of the uni... |
pwssun 4944 | The power class of the uni... |
pwundif 4945 | Break up the power class o... |
pwun 4946 | The power class of the uni... |
epelg 4950 | The epsilon relation and m... |
epelc 4951 | The epsilon relationship a... |
epel 4952 | The epsilon relation and t... |
dfid3 4954 | A stronger version of ~ df... |
dfid4 4955 | The identity function usin... |
dfid2 4956 | Alternate definition of th... |
poss 4961 | Subset theorem for the par... |
poeq1 4962 | Equality theorem for parti... |
poeq2 4963 | Equality theorem for parti... |
nfpo 4964 | Bound-variable hypothesis ... |
nfso 4965 | Bound-variable hypothesis ... |
pocl 4966 | Properties of partial orde... |
ispod 4967 | Sufficient conditions for ... |
swopolem 4968 | Perform the substitutions ... |
swopo 4969 | A strict weak order is a p... |
poirr 4970 | A partial order relation i... |
potr 4971 | A partial order relation i... |
po2nr 4972 | A partial order relation h... |
po3nr 4973 | A partial order relation h... |
po0 4974 | Any relation is a partial ... |
pofun 4975 | A function preserves a par... |
sopo 4976 | A strict linear order is a... |
soss 4977 | Subset theorem for the str... |
soeq1 4978 | Equality theorem for the s... |
soeq2 4979 | Equality theorem for the s... |
sonr 4980 | A strict order relation is... |
sotr 4981 | A strict order relation is... |
solin 4982 | A strict order relation is... |
so2nr 4983 | A strict order relation ha... |
so3nr 4984 | A strict order relation ha... |
sotric 4985 | A strict order relation sa... |
sotrieq 4986 | Trichotomy law for strict ... |
sotrieq2 4987 | Trichotomy law for strict ... |
sotr2 4988 | A transitivity relation. ... |
issod 4989 | An irreflexive, transitive... |
issoi 4990 | An irreflexive, transitive... |
isso2i 4991 | Deduce strict ordering fro... |
so0 4992 | Any relation is a strict o... |
somo 4993 | A totally ordered set has ... |
fri 5000 | Property of well-founded r... |
seex 5001 | The ` R ` -preimage of an ... |
exse 5002 | Any relation on a set is s... |
dffr2 5003 | Alternate definition of we... |
frc 5004 | Property of well-founded r... |
frss 5005 | Subset theorem for the wel... |
sess1 5006 | Subset theorem for the set... |
sess2 5007 | Subset theorem for the set... |
freq1 5008 | Equality theorem for the w... |
freq2 5009 | Equality theorem for the w... |
seeq1 5010 | Equality theorem for the s... |
seeq2 5011 | Equality theorem for the s... |
nffr 5012 | Bound-variable hypothesis ... |
nfse 5013 | Bound-variable hypothesis ... |
nfwe 5014 | Bound-variable hypothesis ... |
frirr 5015 | A well-founded relation is... |
fr2nr 5016 | A well-founded relation ha... |
fr0 5017 | Any relation is well-found... |
frminex 5018 | If an element of a well-fo... |
efrirr 5019 | Irreflexivity of the epsil... |
efrn2lp 5020 | A set founded by epsilon c... |
epse 5021 | The epsilon relation is se... |
tz7.2 5022 | Similar to Theorem 7.2 of ... |
dfepfr 5023 | An alternate way of saying... |
epfrc 5024 | A subset of an epsilon-fou... |
wess 5025 | Subset theorem for the wel... |
weeq1 5026 | Equality theorem for the w... |
weeq2 5027 | Equality theorem for the w... |
wefr 5028 | A well-ordering is well-fo... |
weso 5029 | A well-ordering is a stric... |
wecmpep 5030 | The elements of an epsilon... |
wetrep 5031 | An epsilon well-ordering i... |
wefrc 5032 | A nonempty (possibly prope... |
we0 5033 | Any relation is a well-ord... |
wereu 5034 | A subset of a well-ordered... |
wereu2 5035 | All nonempty (possibly pro... |
xpeq1 5052 | Equality theorem for Carte... |
xpeq2 5053 | Equality theorem for Carte... |
elxpi 5054 | Membership in a Cartesian ... |
elxp 5055 | Membership in a Cartesian ... |
elxp2 5056 | Membership in a Cartesian ... |
elxp2OLD 5057 | Obsolete proof of ~ elxp2 ... |
xpeq12 5058 | Equality theorem for Carte... |
xpeq1i 5059 | Equality inference for Car... |
xpeq2i 5060 | Equality inference for Car... |
xpeq12i 5061 | Equality inference for Car... |
xpeq1d 5062 | Equality deduction for Car... |
xpeq2d 5063 | Equality deduction for Car... |
xpeq12d 5064 | Equality deduction for Car... |
sqxpeqd 5065 | Equality deduction for a C... |
nfxp 5066 | Bound-variable hypothesis ... |
0nelxp 5067 | The empty set is not a mem... |
0nelxpOLD 5068 | Obsolete proof of ~ 0nelxp... |
0nelelxp 5069 | A member of a Cartesian pr... |
opelxp 5070 | Ordered pair membership in... |
brxp 5071 | Binary relation on a Carte... |
opelxpi 5072 | Ordered pair membership in... |
opelxpd 5073 | Ordered pair membership in... |
opelxp1 5074 | The first member of an ord... |
opelxp2 5075 | The second member of an or... |
otelxp1 5076 | The first member of an ord... |
otel3xp 5077 | An ordered triple is an el... |
rabxp 5078 | Membership in a class buil... |
brrelex12 5079 | A true binary relation on ... |
brrelex 5080 | A true binary relation on ... |
brrelex2 5081 | A true binary relation on ... |
brrelexi 5082 | The first argument of a bi... |
brrelex2i 5083 | The second argument of a b... |
nprrel 5084 | No proper class is related... |
fconstmpt 5085 | Representation of a consta... |
vtoclr 5086 | Variable to class conversi... |
opelvvg 5087 | Ordered pair membership in... |
opelvv 5088 | Ordered pair membership in... |
opthprc 5089 | Justification theorem for ... |
brel 5090 | Two things in a binary rel... |
brab2a 5091 | Ordered pair membership in... |
elxp3 5092 | Membership in a Cartesian ... |
opeliunxp 5093 | Membership in a union of C... |
xpundi 5094 | Distributive law for Carte... |
xpundir 5095 | Distributive law for Carte... |
xpiundi 5096 | Distributive law for Carte... |
xpiundir 5097 | Distributive law for Carte... |
iunxpconst 5098 | Membership in a union of C... |
xpun 5099 | The Cartesian product of t... |
elvv 5100 | Membership in universal cl... |
elvvv 5101 | Membership in universal cl... |
elvvuni 5102 | An ordered pair contains i... |
brinxp2 5103 | Intersection of binary rel... |
brinxp 5104 | Intersection of binary rel... |
poinxp 5105 | Intersection of partial or... |
soinxp 5106 | Intersection of total orde... |
frinxp 5107 | Intersection of well-found... |
seinxp 5108 | Intersection of set-like r... |
weinxp 5109 | Intersection of well-order... |
posn 5110 | Partial ordering of a sing... |
sosn 5111 | Strict ordering on a singl... |
frsn 5112 | Founded relation on a sing... |
wesn 5113 | Well-ordering of a singlet... |
elopaelxp 5114 | Membership in an ordered p... |
bropaex12 5115 | Two classes related by an ... |
opabssxp 5116 | An abstraction relation is... |
brab2ga 5117 | The law of concretion for ... |
optocl 5118 | Implicit substitution of c... |
2optocl 5119 | Implicit substitution of c... |
3optocl 5120 | Implicit substitution of c... |
opbrop 5121 | Ordered pair membership in... |
0xp 5122 | The Cartesian product with... |
csbxp 5123 | Distribute proper substitu... |
releq 5124 | Equality theorem for the r... |
releqi 5125 | Equality inference for the... |
releqd 5126 | Equality deduction for the... |
nfrel 5127 | Bound-variable hypothesis ... |
sbcrel 5128 | Distribute proper substitu... |
relss 5129 | Subclass theorem for relat... |
ssrel 5130 | A subclass relationship de... |
ssrelOLD 5131 | Obsolete proof of ~ ssrel ... |
eqrel 5132 | Extensionality principle f... |
ssrel2 5133 | A subclass relationship de... |
relssi 5134 | Inference from subclass pr... |
relssdv 5135 | Deduction from subclass pr... |
eqrelriv 5136 | Inference from extensional... |
eqrelriiv 5137 | Inference from extensional... |
eqbrriv 5138 | Inference from extensional... |
eqrelrdv 5139 | Deduce equality of relatio... |
eqbrrdv 5140 | Deduction from extensional... |
eqbrrdiv 5141 | Deduction from extensional... |
eqrelrdv2 5142 | A version of ~ eqrelrdv . ... |
ssrelrel 5143 | A subclass relationship de... |
eqrelrel 5144 | Extensionality principle f... |
elrel 5145 | A member of a relation is ... |
relsn 5146 | A singleton is a relation ... |
relsnop 5147 | A singleton of an ordered ... |
xpss12 5148 | Subset theorem for Cartesi... |
xpss 5149 | A Cartesian product is inc... |
relxp 5150 | A Cartesian product is a r... |
xpss1 5151 | Subset relation for Cartes... |
xpss2 5152 | Subset relation for Cartes... |
copsex2gb 5153 | Implicit substitution infe... |
copsex2ga 5154 | Implicit substitution infe... |
elopaba 5155 | Membership in an ordered p... |
xpsspw 5156 | A Cartesian product is inc... |
unixpss 5157 | The double class union of ... |
relun 5158 | The union of two relations... |
relin1 5159 | The intersection with a re... |
relin2 5160 | The intersection with a re... |
reldif 5161 | A difference cutting down ... |
reliun 5162 | An indexed union is a rela... |
reliin 5163 | An indexed intersection is... |
reluni 5164 | The union of a class is a ... |
relint 5165 | The intersection of a clas... |
rel0 5166 | The empty set is a relatio... |
relopabi 5167 | A class of ordered pairs i... |
relopabiALT 5168 | Alternate proof of ~ relop... |
relopab 5169 | A class of ordered pairs i... |
mptrel 5170 | The maps-to notation alway... |
reli 5171 | The identity relation is a... |
rele 5172 | The membership relation is... |
opabid2 5173 | A relation expressed as an... |
inopab 5174 | Intersection of two ordere... |
difopab 5175 | The difference of two orde... |
inxp 5176 | The intersection of two Ca... |
xpindi 5177 | Distributive law for Carte... |
xpindir 5178 | Distributive law for Carte... |
xpiindi 5179 | Distributive law for Carte... |
xpriindi 5180 | Distributive law for Carte... |
eliunxp 5181 | Membership in a union of C... |
opeliunxp2 5182 | Membership in a union of C... |
raliunxp 5183 | Write a double restricted ... |
rexiunxp 5184 | Write a double restricted ... |
ralxp 5185 | Universal quantification r... |
rexxp 5186 | Existential quantification... |
exopxfr 5187 | Transfer ordered-pair exis... |
exopxfr2 5188 | Transfer ordered-pair exis... |
djussxp 5189 | Disjoint union is a subset... |
ralxpf 5190 | Version of ~ ralxp with bo... |
rexxpf 5191 | Version of ~ rexxp with bo... |
iunxpf 5192 | Indexed union on a Cartesi... |
opabbi2dv 5193 | Deduce equality of a relat... |
relop 5194 | A necessary and sufficient... |
ideqg 5195 | For sets, the identity rel... |
ideq 5196 | For sets, the identity rel... |
ididg 5197 | A set is identical to itse... |
issetid 5198 | Two ways of expressing set... |
coss1 5199 | Subclass theorem for compo... |
coss2 5200 | Subclass theorem for compo... |
coeq1 5201 | Equality theorem for compo... |
coeq2 5202 | Equality theorem for compo... |
coeq1i 5203 | Equality inference for com... |
coeq2i 5204 | Equality inference for com... |
coeq1d 5205 | Equality deduction for com... |
coeq2d 5206 | Equality deduction for com... |
coeq12i 5207 | Equality inference for com... |
coeq12d 5208 | Equality deduction for com... |
nfco 5209 | Bound-variable hypothesis ... |
brcog 5210 | Ordered pair membership in... |
opelco2g 5211 | Ordered pair membership in... |
brcogw 5212 | Ordered pair membership in... |
eqbrrdva 5213 | Deduction from extensional... |
brco 5214 | Binary relation on a compo... |
opelco 5215 | Ordered pair membership in... |
cnvss 5216 | Subset theorem for convers... |
cnvssOLD 5217 | Obsolete proof of ~ cnvss ... |
cnveq 5218 | Equality theorem for conve... |
cnveqi 5219 | Equality inference for con... |
cnveqd 5220 | Equality deduction for con... |
elcnv 5221 | Membership in a converse. ... |
elcnv2 5222 | Membership in a converse. ... |
nfcnv 5223 | Bound-variable hypothesis ... |
opelcnvg 5224 | Ordered-pair membership in... |
brcnvg 5225 | The converse of a binary r... |
opelcnv 5226 | Ordered-pair membership in... |
brcnv 5227 | The converse of a binary r... |
csbcnv 5228 | Move class substitution in... |
csbcnvgALT 5229 | Move class substitution in... |
cnvco 5230 | Distributive law of conver... |
cnvuni 5231 | The converse of a class un... |
dfdm3 5232 | Alternate definition of do... |
dfrn2 5233 | Alternate definition of ra... |
dfrn3 5234 | Alternate definition of ra... |
elrn2g 5235 | Membership in a range. (C... |
elrng 5236 | Membership in a range. (C... |
ssrelrn 5237 | If a relation is a subset ... |
dfdm4 5238 | Alternate definition of do... |
dfdmf 5239 | Definition of domain, usin... |
csbdm 5240 | Distribute proper substitu... |
eldmg 5241 | Domain membership. Theore... |
eldm2g 5242 | Domain membership. Theore... |
eldm 5243 | Membership in a domain. T... |
eldm2 5244 | Membership in a domain. T... |
dmss 5245 | Subset theorem for domain.... |
dmeq 5246 | Equality theorem for domai... |
dmeqi 5247 | Equality inference for dom... |
dmeqd 5248 | Equality deduction for dom... |
opeldmd 5249 | Membership of first of an ... |
opeldm 5250 | Membership of first of an ... |
breldm 5251 | Membership of first of a b... |
breldmg 5252 | Membership of first of a b... |
dmun 5253 | The domain of a union is t... |
dmin 5254 | The domain of an intersect... |
dmiun 5255 | The domain of an indexed u... |
dmuni 5256 | The domain of a union. Pa... |
dmopab 5257 | The domain of a class of o... |
dmopabss 5258 | Upper bound for the domain... |
dmopab3 5259 | The domain of a restricted... |
dm0 5260 | The domain of the empty se... |
dmi 5261 | The domain of the identity... |
dmv 5262 | The domain of the universe... |
dm0rn0 5263 | An empty domain implies an... |
reldm0 5264 | A relation is empty iff it... |
dmxp 5265 | The domain of a Cartesian ... |
dmxpid 5266 | The domain of a square Car... |
dmxpin 5267 | The domain of the intersec... |
xpid11 5268 | The Cartesian product of a... |
dmcnvcnv 5269 | The domain of the double c... |
rncnvcnv 5270 | The range of the double co... |
elreldm 5271 | The first member of an ord... |
rneq 5272 | Equality theorem for range... |
rneqi 5273 | Equality inference for ran... |
rneqd 5274 | Equality deduction for ran... |
rnss 5275 | Subset theorem for range. ... |
brelrng 5276 | The second argument of a b... |
brelrn 5277 | The second argument of a b... |
opelrn 5278 | Membership of second membe... |
releldm 5279 | The first argument of a bi... |
relelrn 5280 | The second argument of a b... |
releldmb 5281 | Membership in a domain. (... |
relelrnb 5282 | Membership in a range. (C... |
releldmi 5283 | The first argument of a bi... |
relelrni 5284 | The second argument of a b... |
dfrnf 5285 | Definition of range, using... |
elrn2 5286 | Membership in a range. (C... |
elrn 5287 | Membership in a range. (C... |
nfdm 5288 | Bound-variable hypothesis ... |
nfrn 5289 | Bound-variable hypothesis ... |
dmiin 5290 | Domain of an intersection.... |
rnopab 5291 | The range of a class of or... |
rnmpt 5292 | The range of a function in... |
elrnmpt 5293 | The range of a function in... |
elrnmpt1s 5294 | Elementhood in an image se... |
elrnmpt1 5295 | Elementhood in an image se... |
elrnmptg 5296 | Membership in the range of... |
elrnmpti 5297 | Membership in the range of... |
rn0 5298 | The range of the empty set... |
dfiun3g 5299 | Alternate definition of in... |
dfiin3g 5300 | Alternate definition of in... |
dfiun3 5301 | Alternate definition of in... |
dfiin3 5302 | Alternate definition of in... |
riinint 5303 | Express a relative indexed... |
relrn0 5304 | A relation is empty iff it... |
dmrnssfld 5305 | The domain and range of a ... |
dmcoss 5306 | Domain of a composition. ... |
rncoss 5307 | Range of a composition. (... |
dmcosseq 5308 | Domain of a composition. ... |
dmcoeq 5309 | Domain of a composition. ... |
rncoeq 5310 | Range of a composition. (... |
reseq1 5311 | Equality theorem for restr... |
reseq2 5312 | Equality theorem for restr... |
reseq1i 5313 | Equality inference for res... |
reseq2i 5314 | Equality inference for res... |
reseq12i 5315 | Equality inference for res... |
reseq1d 5316 | Equality deduction for res... |
reseq2d 5317 | Equality deduction for res... |
reseq12d 5318 | Equality deduction for res... |
nfres 5319 | Bound-variable hypothesis ... |
csbres 5320 | Distribute proper substitu... |
res0 5321 | A restriction to the empty... |
opelres 5322 | Ordered pair membership in... |
brres 5323 | Binary relation on a restr... |
opelresg 5324 | Ordered pair membership in... |
brresg 5325 | Binary relation on a restr... |
opres 5326 | Ordered pair membership in... |
resieq 5327 | A restricted identity rela... |
opelresi 5328 | ` <. A , A >. ` belongs to... |
resres 5329 | The restriction of a restr... |
resundi 5330 | Distributive law for restr... |
resundir 5331 | Distributive law for restr... |
resindi 5332 | Class restriction distribu... |
resindir 5333 | Class restriction distribu... |
inres 5334 | Move intersection into cla... |
resdifcom 5335 | Commutative law for restri... |
resiun1 5336 | Distribution of restrictio... |
resiun1OLD 5337 | Obsolete proof of ~ resiun... |
resiun2 5338 | Distribution of restrictio... |
dmres 5339 | The domain of a restrictio... |
ssdmres 5340 | A domain restricted to a s... |
dmresexg 5341 | The domain of a restrictio... |
resss 5342 | A class includes its restr... |
rescom 5343 | Commutative law for restri... |
ssres 5344 | Subclass theorem for restr... |
ssres2 5345 | Subclass theorem for restr... |
relres 5346 | A restriction is a relatio... |
resabs1 5347 | Absorption law for restric... |
resabs1d 5348 | Absorption law for restric... |
resabs2 5349 | Absorption law for restric... |
residm 5350 | Idempotent law for restric... |
resima 5351 | A restriction to an image.... |
resima2 5352 | Image under a restricted c... |
resima2OLD 5353 | Obsolete proof of ~ resima... |
xpssres 5354 | Restriction of a constant ... |
elres 5355 | Membership in a restrictio... |
elsnres 5356 | Membership in restriction ... |
relssres 5357 | Simplification law for res... |
dmressnsn 5358 | The domain of a restrictio... |
eldmressnsn 5359 | The element of the domain ... |
eldmeldmressn 5360 | An element of the domain (... |
resdm 5361 | A relation restricted to i... |
resexg 5362 | The restriction of a set i... |
resex 5363 | The restriction of a set i... |
resindm 5364 | Class restriction distribu... |
resdmdfsn 5365 | Restricting a function to ... |
resopab 5366 | Restriction of a class abs... |
iss 5367 | A subclass of the identity... |
resopab2 5368 | Restriction of a class abs... |
resmpt 5369 | Restriction of the mapping... |
resmpt3 5370 | Unconditional restriction ... |
resmptd 5371 | Restriction of the mapping... |
dfres2 5372 | Alternate definition of th... |
mptss 5373 | Sufficient condition for i... |
opabresid 5374 | The restricted identity ex... |
mptresid 5375 | The restricted identity ex... |
dmresi 5376 | The domain of a restricted... |
restidsing 5377 | Restriction of the identit... |
restidsingOLD 5378 | Obsolete proof of ~ restid... |
resid 5379 | Any relation restricted to... |
imaeq1 5380 | Equality theorem for image... |
imaeq2 5381 | Equality theorem for image... |
imaeq1i 5382 | Equality theorem for image... |
imaeq2i 5383 | Equality theorem for image... |
imaeq1d 5384 | Equality theorem for image... |
imaeq2d 5385 | Equality theorem for image... |
imaeq12d 5386 | Equality theorem for image... |
dfima2 5387 | Alternate definition of im... |
dfima3 5388 | Alternate definition of im... |
elimag 5389 | Membership in an image. T... |
elima 5390 | Membership in an image. T... |
elima2 5391 | Membership in an image. T... |
elima3 5392 | Membership in an image. T... |
nfima 5393 | Bound-variable hypothesis ... |
nfimad 5394 | Deduction version of bound... |
imadmrn 5395 | The image of the domain of... |
imassrn 5396 | The image of a class is a ... |
imai 5397 | Image under the identity r... |
rnresi 5398 | The range of the restricte... |
resiima 5399 | The image of a restriction... |
ima0 5400 | Image of the empty set. T... |
0ima 5401 | Image under the empty rela... |
csbima12 5402 | Move class substitution in... |
imadisj 5403 | A class whose image under ... |
cnvimass 5404 | A preimage under any class... |
cnvimarndm 5405 | The preimage of the range ... |
imasng 5406 | The image of a singleton. ... |
relimasn 5407 | The image of a singleton. ... |
elrelimasn 5408 | Elementhood in the image o... |
elimasn 5409 | Membership in an image of ... |
elimasng 5410 | Membership in an image of ... |
elimasni 5411 | Membership in an image of ... |
args 5412 | Two ways to express the cl... |
eliniseg 5413 | Membership in an initial s... |
epini 5414 | Any set is equal to its pr... |
iniseg 5415 | An idiom that signifies an... |
inisegn0 5416 | Nonemptyness of an initial... |
dffr3 5417 | Alternate definition of we... |
dfse2 5418 | Alternate definition of se... |
imass1 5419 | Subset theorem for image. ... |
imass2 5420 | Subset theorem for image. ... |
ndmima 5421 | The image of a singleton o... |
relcnv 5422 | A converse is a relation. ... |
relbrcnvg 5423 | When ` R ` is a relation, ... |
eliniseg2 5424 | Eliminate the class existe... |
relbrcnv 5425 | When ` R ` is a relation, ... |
cotrg 5426 | Two ways of saying that th... |
cotr 5427 | Two ways of saying a relat... |
issref 5428 | Two ways to state a relati... |
cnvsym 5429 | Two ways of saying a relat... |
intasym 5430 | Two ways of saying a relat... |
asymref 5431 | Two ways of saying a relat... |
asymref2 5432 | Two ways of saying a relat... |
intirr 5433 | Two ways of saying a relat... |
brcodir 5434 | Two ways of saying that tw... |
codir 5435 | Two ways of saying a relat... |
qfto 5436 | A quantifier-free way of e... |
xpidtr 5437 | A square Cartesian product... |
trin2 5438 | The intersection of two tr... |
poirr2 5439 | A partial order relation i... |
trinxp 5440 | The relation induced by a ... |
soirri 5441 | A strict order relation is... |
sotri 5442 | A strict order relation is... |
son2lpi 5443 | A strict order relation ha... |
sotri2 5444 | A transitivity relation. ... |
sotri3 5445 | A transitivity relation. ... |
poleloe 5446 | Express "less than or equa... |
poltletr 5447 | Transitive law for general... |
somin1 5448 | Property of a minimum in a... |
somincom 5449 | Commutativity of minimum i... |
somin2 5450 | Property of a minimum in a... |
soltmin 5451 | Being less than a minimum,... |
cnvopab 5452 | The converse of a class ab... |
mptcnv 5453 | The converse of a mapping ... |
cnv0 5454 | The converse of the empty ... |
cnv0OLD 5455 | Obsolete version of ~ cnv0... |
cnvi 5456 | The converse of the identi... |
cnvun 5457 | The converse of a union is... |
cnvdif 5458 | Distributive law for conve... |
cnvin 5459 | Distributive law for conve... |
rnun 5460 | Distributive law for range... |
rnin 5461 | The range of an intersecti... |
rniun 5462 | The range of an indexed un... |
rnuni 5463 | The range of a union. Par... |
imaundi 5464 | Distributive law for image... |
imaundir 5465 | The image of a union. (Co... |
dminss 5466 | An upper bound for interse... |
imainss 5467 | An upper bound for interse... |
inimass 5468 | The image of an intersecti... |
inimasn 5469 | The intersection of the im... |
cnvxp 5470 | The converse of a Cartesia... |
xp0 5471 | The Cartesian product with... |
xpnz 5472 | The Cartesian product of n... |
xpeq0 5473 | At least one member of an ... |
xpdisj1 5474 | Cartesian products with di... |
xpdisj2 5475 | Cartesian products with di... |
xpsndisj 5476 | Cartesian products with tw... |
difxp 5477 | Difference of Cartesian pr... |
difxp1 5478 | Difference law for Cartesi... |
difxp2 5479 | Difference law for Cartesi... |
djudisj 5480 | Disjoint unions with disjo... |
xpdifid 5481 | The set of distinct couple... |
resdisj 5482 | A double restriction to di... |
rnxp 5483 | The range of a Cartesian p... |
dmxpss 5484 | The domain of a Cartesian ... |
rnxpss 5485 | The range of a Cartesian p... |
rnxpid 5486 | The range of a square Cart... |
ssxpb 5487 | A Cartesian product subcla... |
xp11 5488 | The Cartesian product of n... |
xpcan 5489 | Cancellation law for Carte... |
xpcan2 5490 | Cancellation law for Carte... |
ssrnres 5491 | Subset of the range of a r... |
rninxp 5492 | Range of the intersection ... |
dminxp 5493 | Domain of the intersection... |
imainrect 5494 | Image of a relation restri... |
xpima 5495 | The image by a constant fu... |
xpima1 5496 | The image by a Cartesian p... |
xpima2 5497 | The image by a Cartesian p... |
xpimasn 5498 | The image of a singleton b... |
sossfld 5499 | The base set of a strict o... |
sofld 5500 | The base set of a nonempty... |
cnvcnv3 5501 | The set of all ordered pai... |
dfrel2 5502 | Alternate definition of re... |
dfrel4v 5503 | A relation can be expresse... |
dfrel4 5504 | A relation can be expresse... |
cnvcnv 5505 | The double converse of a c... |
cnvcnv2 5506 | The double converse of a c... |
cnvcnvss 5507 | The double converse of a c... |
cnveqb 5508 | Equality theorem for conve... |
cnveq0 5509 | A relation empty iff its c... |
dfrel3 5510 | Alternate definition of re... |
dmresv 5511 | The domain of a universal ... |
rnresv 5512 | The range of a universal r... |
dfrn4 5513 | Range defined in terms of ... |
csbrn 5514 | Distribute proper substitu... |
rescnvcnv 5515 | The restriction of the dou... |
cnvcnvres 5516 | The double converse of the... |
imacnvcnv 5517 | The image of the double co... |
dmsnn0 5518 | The domain of a singleton ... |
rnsnn0 5519 | The range of a singleton i... |
dmsn0 5520 | The domain of the singleto... |
cnvsn0 5521 | The converse of the single... |
dmsn0el 5522 | The domain of a singleton ... |
relsn2 5523 | A singleton is a relation ... |
dmsnopg 5524 | The domain of a singleton ... |
dmsnopss 5525 | The domain of a singleton ... |
dmpropg 5526 | The domain of an unordered... |
dmsnop 5527 | The domain of a singleton ... |
dmprop 5528 | The domain of an unordered... |
dmtpop 5529 | The domain of an unordered... |
cnvcnvsn 5530 | Double converse of a singl... |
dmsnsnsn 5531 | The domain of the singleto... |
rnsnopg 5532 | The range of a singleton o... |
rnpropg 5533 | The range of a pair of ord... |
rnsnop 5534 | The range of a singleton o... |
op1sta 5535 | Extract the first member o... |
cnvsn 5536 | Converse of a singleton of... |
op2ndb 5537 | Extract the second member ... |
op2nda 5538 | Extract the second member ... |
cnvsng 5539 | Converse of a singleton of... |
opswap 5540 | Swap the members of an ord... |
cnvresima 5541 | An image under the convers... |
resdm2 5542 | A class restricted to its ... |
resdmres 5543 | Restriction to the domain ... |
imadmres 5544 | The image of the domain of... |
mptpreima 5545 | The preimage of a function... |
mptiniseg 5546 | Converse singleton image o... |
dmmpt 5547 | The domain of the mapping ... |
dmmptss 5548 | The domain of a mapping is... |
dmmptg 5549 | The domain of the mapping ... |
relco 5550 | A composition is a relatio... |
dfco2 5551 | Alternate definition of a ... |
dfco2a 5552 | Generalization of ~ dfco2 ... |
coundi 5553 | Class composition distribu... |
coundir 5554 | Class composition distribu... |
cores 5555 | Restricted first member of... |
resco 5556 | Associative law for the re... |
imaco 5557 | Image of the composition o... |
rnco 5558 | The range of the compositi... |
rnco2 5559 | The range of the compositi... |
dmco 5560 | The domain of a compositio... |
coeq0 5561 | A composition of two relat... |
coiun 5562 | Composition with an indexe... |
cocnvcnv1 5563 | A composition is not affec... |
cocnvcnv2 5564 | A composition is not affec... |
cores2 5565 | Absorption of a reverse (p... |
co02 5566 | Composition with the empty... |
co01 5567 | Composition with the empty... |
coi1 5568 | Composition with the ident... |
coi2 5569 | Composition with the ident... |
coires1 5570 | Composition with a restric... |
coass 5571 | Associative law for class ... |
relcnvtr 5572 | A relation is transitive i... |
relssdmrn 5573 | A relation is included in ... |
cnvssrndm 5574 | The converse is a subset o... |
cossxp 5575 | Composition as a subset of... |
relrelss 5576 | Two ways to describe the s... |
unielrel 5577 | The membership relation fo... |
relfld 5578 | The double union of a rela... |
relresfld 5579 | Restriction of a relation ... |
relcoi2 5580 | Composition with the ident... |
relcoi1 5581 | Composition with the ident... |
unidmrn 5582 | The double union of the co... |
relcnvfld 5583 | if ` R ` is a relation, it... |
dfdm2 5584 | Alternate definition of do... |
unixp 5585 | The double class union of ... |
unixp0 5586 | A Cartesian product is emp... |
unixpid 5587 | Field of a square Cartesia... |
ressn 5588 | Restriction of a class to ... |
cnviin 5589 | The converse of an interse... |
cnvpo 5590 | The converse of a partial ... |
cnvso 5591 | The converse of a strict o... |
xpco 5592 | Composition of two Cartesi... |
xpcoid 5593 | Composition of two square ... |
elsnxp 5594 | Elementhood to a cartesian... |
elsnxpOLD 5595 | Obsolete proof of ~ elsnxp... |
predeq123 5598 | Equality theorem for the p... |
predeq1 5599 | Equality theorem for the p... |
predeq2 5600 | Equality theorem for the p... |
predeq3 5601 | Equality theorem for the p... |
nfpred 5602 | Bound-variable hypothesis ... |
predpredss 5603 | If ` A ` is a subset of ` ... |
predss 5604 | The predecessor class of `... |
sspred 5605 | Another subset/predecessor... |
dfpred2 5606 | An alternate definition of... |
dfpred3 5607 | An alternate definition of... |
dfpred3g 5608 | An alternate definition of... |
elpredim 5609 | Membership in a predecesso... |
elpred 5610 | Membership in a predecesso... |
elpredg 5611 | Membership in a predecesso... |
predasetex 5612 | The predecessor class exis... |
dffr4 5613 | Alternate definition of we... |
predel 5614 | Membership in the predeces... |
predpo 5615 | Property of the precessor ... |
predso 5616 | Property of the predecesso... |
predbrg 5617 | Closed form of ~ elpredim ... |
setlikespec 5618 | If ` R ` is set-like in ` ... |
predidm 5619 | Idempotent law for the pre... |
predin 5620 | Intersection law for prede... |
predun 5621 | Union law for predecessor ... |
preddif 5622 | Difference law for predece... |
predep 5623 | The predecessor under the ... |
preddowncl 5624 | A property of classes that... |
predpoirr 5625 | Given a partial ordering, ... |
predfrirr 5626 | Given a well-founded relat... |
pred0 5627 | The predecessor class over... |
tz6.26 5628 | All nonempty (possibly pro... |
tz6.26i 5629 | All nonempty (possibly pro... |
wfi 5630 | The Principle of Well-Foun... |
wfii 5631 | The Principle of Well-Foun... |
wfisg 5632 | Well-Founded Induction Sch... |
wfis 5633 | Well-Founded Induction Sch... |
wfis2fg 5634 | Well-Founded Induction Sch... |
wfis2f 5635 | Well Founded Induction sch... |
wfis2g 5636 | Well-Founded Induction Sch... |
wfis2 5637 | Well Founded Induction sch... |
wfis3 5638 | Well Founded Induction sch... |
ordeq 5647 | Equality theorem for the o... |
elong 5648 | An ordinal number is an or... |
elon 5649 | An ordinal number is an or... |
eloni 5650 | An ordinal number has the ... |
elon2 5651 | An ordinal number is an or... |
limeq 5652 | Equality theorem for the l... |
ordwe 5653 | Epsilon well-orders every ... |
ordtr 5654 | An ordinal class is transi... |
ordfr 5655 | Epsilon is well-founded on... |
ordelss 5656 | An element of an ordinal c... |
trssord 5657 | A transitive subclass of a... |
ordirr 5658 | Epsilon irreflexivity of o... |
nordeq 5659 | A member of an ordinal cla... |
ordn2lp 5660 | An ordinal class cannot be... |
tz7.5 5661 | A nonempty subclass of an ... |
ordelord 5662 | An element of an ordinal c... |
tron 5663 | The class of all ordinal n... |
ordelon 5664 | An element of an ordinal c... |
onelon 5665 | An element of an ordinal n... |
tz7.7 5666 | A transitive class belongs... |
ordelssne 5667 | For ordinal classes, membe... |
ordelpss 5668 | For ordinal classes, membe... |
ordsseleq 5669 | For ordinal classes, inclu... |
ordin 5670 | The intersection of two or... |
onin 5671 | The intersection of two or... |
ordtri3or 5672 | A trichotomy law for ordin... |
ordtri1 5673 | A trichotomy law for ordin... |
ontri1 5674 | A trichotomy law for ordin... |
ordtri2 5675 | A trichotomy law for ordin... |
ordtri3 5676 | A trichotomy law for ordin... |
ordtri3OLD 5677 | Obsolete proof of ~ ordtri... |
ordtri4 5678 | A trichotomy law for ordin... |
orddisj 5679 | An ordinal class and its s... |
onfr 5680 | The ordinal class is well-... |
onelpss 5681 | Relationship between membe... |
onsseleq 5682 | Relationship between subse... |
onelss 5683 | An element of an ordinal n... |
ordtr1 5684 | Transitive law for ordinal... |
ordtr2 5685 | Transitive law for ordinal... |
ordtr3 5686 | Transitive law for ordinal... |
ordtr3OLD 5687 | Obsolete proof of ~ ordtr3... |
ontr1 5688 | Transitive law for ordinal... |
ontr2 5689 | Transitive law for ordinal... |
ordunidif 5690 | The union of an ordinal st... |
ordintdif 5691 | If ` B ` is smaller than `... |
onintss 5692 | If a property is true for ... |
oneqmini 5693 | A way to show that an ordi... |
ord0 5694 | The empty set is an ordina... |
0elon 5695 | The empty set is an ordina... |
ord0eln0 5696 | A nonempty ordinal contain... |
on0eln0 5697 | An ordinal number contains... |
dflim2 5698 | An alternate definition of... |
inton 5699 | The intersection of the cl... |
nlim0 5700 | The empty set is not a lim... |
limord 5701 | A limit ordinal is ordinal... |
limuni 5702 | A limit ordinal is its own... |
limuni2 5703 | The union of a limit ordin... |
0ellim 5704 | A limit ordinal contains t... |
limelon 5705 | A limit ordinal class that... |
onn0 5706 | The class of all ordinal n... |
suceq 5707 | Equality of successors. (... |
elsuci 5708 | Membership in a successor.... |
elsucg 5709 | Membership in a successor.... |
elsuc2g 5710 | Variant of membership in a... |
elsuc 5711 | Membership in a successor.... |
elsuc2 5712 | Membership in a successor.... |
nfsuc 5713 | Bound-variable hypothesis ... |
elelsuc 5714 | Membership in a successor.... |
sucel 5715 | Membership of a successor ... |
suc0 5716 | The successor of the empty... |
sucprc 5717 | A proper class is its own ... |
unisuc 5718 | A transitive class is equa... |
sssucid 5719 | A class is included in its... |
sucidg 5720 | Part of Proposition 7.23 o... |
sucid 5721 | A set belongs to its succe... |
nsuceq0 5722 | No successor is empty. (C... |
eqelsuc 5723 | A set belongs to the succe... |
iunsuc 5724 | Inductive definition for t... |
suctr 5725 | The successor of a transit... |
suctrOLD 5726 | Obsolete proof of ~ suctr ... |
trsuc 5727 | A set whose successor belo... |
trsucss 5728 | A member of the successor ... |
ordsssuc 5729 | A subset of an ordinal bel... |
onsssuc 5730 | A subset of an ordinal num... |
ordsssuc2 5731 | An ordinal subset of an or... |
onmindif 5732 | When its successor is subt... |
ordnbtwn 5733 | There is no set between an... |
ordnbtwnOLD 5734 | Obsolete proof of ~ ordnbt... |
onnbtwn 5735 | There is no set between an... |
sucssel 5736 | A set whose successor is a... |
orddif 5737 | Ordinal derived from its s... |
orduniss 5738 | An ordinal class includes ... |
ordtri2or 5739 | A trichotomy law for ordin... |
ordtri2or2 5740 | A trichotomy law for ordin... |
ordtri2or3 5741 | A consequence of total ord... |
ordelinel 5742 | The intersection of two or... |
ordelinelOLD 5743 | Obsolete proof of ~ ordeli... |
ordssun 5744 | Property of a subclass of ... |
ordequn 5745 | The maximum (i.e. union) o... |
ordun 5746 | The maximum (i.e. union) o... |
ordunisssuc 5747 | A subclass relationship fo... |
suc11 5748 | The successor operation be... |
onordi 5749 | An ordinal number is an or... |
ontrci 5750 | An ordinal number is a tra... |
onirri 5751 | An ordinal number is not a... |
oneli 5752 | A member of an ordinal num... |
onelssi 5753 | A member of an ordinal num... |
onssneli 5754 | An ordering law for ordina... |
onssnel2i 5755 | An ordering law for ordina... |
onelini 5756 | An element of an ordinal n... |
oneluni 5757 | An ordinal number equals i... |
onunisuci 5758 | An ordinal number is equal... |
onsseli 5759 | Subset is equivalent to me... |
onun2i 5760 | The union of two ordinal n... |
unizlim 5761 | An ordinal equal to its ow... |
on0eqel 5762 | An ordinal number either e... |
snsn0non 5763 | The singleton of the singl... |
onxpdisj 5764 | Ordinal numbers and ordere... |
onnev 5765 | The class of ordinal numbe... |
iotajust 5767 | Soundness justification th... |
dfiota2 5769 | Alternate definition for d... |
nfiota1 5770 | Bound-variable hypothesis ... |
nfiotad 5771 | Deduction version of ~ nfi... |
nfiota 5772 | Bound-variable hypothesis ... |
cbviota 5773 | Change bound variables in ... |
cbviotav 5774 | Change bound variables in ... |
sb8iota 5775 | Variable substitution in d... |
iotaeq 5776 | Equality theorem for descr... |
iotabi 5777 | Equivalence theorem for de... |
uniabio 5778 | Part of Theorem 8.17 in [Q... |
iotaval 5779 | Theorem 8.19 in [Quine] p.... |
iotauni 5780 | Equivalence between two di... |
iotaint 5781 | Equivalence between two di... |
iota1 5782 | Property of iota. (Contri... |
iotanul 5783 | Theorem 8.22 in [Quine] p.... |
iotassuni 5784 | The ` iota ` class is a su... |
iotaex 5785 | Theorem 8.23 in [Quine] p.... |
iota4 5786 | Theorem *14.22 in [Whitehe... |
iota4an 5787 | Theorem *14.23 in [Whitehe... |
iota5 5788 | A method for computing iot... |
iotabidv 5789 | Formula-building deduction... |
iotabii 5790 | Formula-building deduction... |
iotacl 5791 | Membership law for descrip... |
iota2df 5792 | A condition that allows us... |
iota2d 5793 | A condition that allows us... |
iota2 5794 | The unique element such th... |
sniota 5795 | A class abstraction with a... |
dfiota4 5796 | The ` iota ` operation usi... |
csbiota 5797 | Class substitution within ... |
dffun2 5814 | Alternate definition of a ... |
dffun3 5815 | Alternate definition of fu... |
dffun4 5816 | Alternate definition of a ... |
dffun5 5817 | Alternate definition of fu... |
dffun6f 5818 | Definition of function, us... |
dffun6 5819 | Alternate definition of a ... |
funmo 5820 | A function has at most one... |
funrel 5821 | A function is a relation. ... |
funss 5822 | Subclass theorem for funct... |
funeq 5823 | Equality theorem for funct... |
funeqi 5824 | Equality inference for the... |
funeqd 5825 | Equality deduction for the... |
nffun 5826 | Bound-variable hypothesis ... |
sbcfung 5827 | Distribute proper substitu... |
funeu 5828 | There is exactly one value... |
funeu2 5829 | There is exactly one value... |
dffun7 5830 | Alternate definition of a ... |
dffun8 5831 | Alternate definition of a ... |
dffun9 5832 | Alternate definition of a ... |
funfn 5833 | An equivalence for the fun... |
funi 5834 | The identity relation is a... |
nfunv 5835 | The universe is not a func... |
funopg 5836 | A Kuratowski ordered pair ... |
funopab 5837 | A class of ordered pairs i... |
funopabeq 5838 | A class of ordered pairs o... |
funopab4 5839 | A class of ordered pairs o... |
funmpt 5840 | A function in maps-to nota... |
funmpt2 5841 | Functionality of a class g... |
funco 5842 | The composition of two fun... |
funres 5843 | A restriction of a functio... |
funssres 5844 | The restriction of a funct... |
fun2ssres 5845 | Equality of restrictions o... |
funun 5846 | The union of functions wit... |
fununmo 5847 | If the union of classes is... |
fununfun 5848 | If the union of classes is... |
fundif 5849 | A function with removed el... |
funcnvsn 5850 | The converse singleton of ... |
funsng 5851 | A singleton of an ordered ... |
fnsng 5852 | Functionality and domain o... |
funsn 5853 | A singleton of an ordered ... |
funprg 5854 | A set of two pairs is a fu... |
funprgOLD 5855 | Obsolete proof of ~ funprg... |
funtpg 5856 | A set of three pairs is a ... |
funtpgOLD 5857 | Obsolete proof of ~ funtpg... |
funpr 5858 | A function with a domain o... |
funtp 5859 | A function with a domain o... |
fnsn 5860 | Functionality and domain o... |
fnprg 5861 | Function with a domain of ... |
fntpg 5862 | Function with a domain of ... |
fntp 5863 | A function with a domain o... |
funcnvpr 5864 | The converse pair of order... |
funcnvtp 5865 | The converse triple of ord... |
funcnvqp 5866 | The converse quadruple of ... |
funcnvqpOLD 5867 | Obsolete proof of ~ funcnv... |
fun0 5868 | The empty set is a functio... |
funcnv0 5869 | The converse of the empty ... |
funcnvcnv 5870 | The double converse of a f... |
funcnv2 5871 | A simpler equivalence for ... |
funcnv 5872 | The converse of a class is... |
funcnv3 5873 | A condition showing a clas... |
fun2cnv 5874 | The double converse of a c... |
svrelfun 5875 | A single-valued relation i... |
fncnv 5876 | Single-rootedness (see ~ f... |
fun11 5877 | Two ways of stating that `... |
fununi 5878 | The union of a chain (with... |
funin 5879 | The intersection with a fu... |
funres11 5880 | The restriction of a one-t... |
funcnvres 5881 | The converse of a restrict... |
cnvresid 5882 | Converse of a restricted i... |
funcnvres2 5883 | The converse of a restrict... |
funimacnv 5884 | The image of the preimage ... |
funimass1 5885 | A kind of contraposition l... |
funimass2 5886 | A kind of contraposition l... |
imadif 5887 | The image of a difference ... |
imain 5888 | The image of an intersecti... |
funimaexg 5889 | Axiom of Replacement using... |
funimaex 5890 | The image of a set under a... |
isarep1 5891 | Part of a study of the Axi... |
isarep2 5892 | Part of a study of the Axi... |
fneq1 5893 | Equality theorem for funct... |
fneq2 5894 | Equality theorem for funct... |
fneq1d 5895 | Equality deduction for fun... |
fneq2d 5896 | Equality deduction for fun... |
fneq12d 5897 | Equality deduction for fun... |
fneq12 5898 | Equality theorem for funct... |
fneq1i 5899 | Equality inference for fun... |
fneq2i 5900 | Equality inference for fun... |
nffn 5901 | Bound-variable hypothesis ... |
fnfun 5902 | A function with domain is ... |
fnrel 5903 | A function with domain is ... |
fndm 5904 | The domain of a function. ... |
funfni 5905 | Inference to convert a fun... |
fndmu 5906 | A function has a unique do... |
fnbr 5907 | The first argument of bina... |
fnop 5908 | The first argument of an o... |
fneu 5909 | There is exactly one value... |
fneu2 5910 | There is exactly one value... |
fnun 5911 | The union of two functions... |
fnunsn 5912 | Extension of a function wi... |
fnco 5913 | Composition of two functio... |
fnresdm 5914 | A function does not change... |
fnresdisj 5915 | A function restricted to a... |
2elresin 5916 | Membership in two function... |
fnssresb 5917 | Restriction of a function ... |
fnssres 5918 | Restriction of a function ... |
fnresin1 5919 | Restriction of a function'... |
fnresin2 5920 | Restriction of a function'... |
fnres 5921 | An equivalence for functio... |
fnresi 5922 | Functionality and domain o... |
fnima 5923 | The image of a function's ... |
fn0 5924 | A function with empty doma... |
fnimadisj 5925 | A class that is disjoint w... |
fnimaeq0 5926 | Images under a function ne... |
dfmpt3 5927 | Alternate definition for t... |
mptfnf 5928 | The maps-to notation defin... |
fnmptf 5929 | The maps-to notation defin... |
fnopabg 5930 | Functionality and domain o... |
fnopab 5931 | Functionality and domain o... |
mptfng 5932 | The maps-to notation defin... |
fnmpt 5933 | The maps-to notation defin... |
mpt0 5934 | A mapping operation with e... |
fnmpti 5935 | Functionality and domain o... |
dmmpti 5936 | Domain of the mapping oper... |
dmmptd 5937 | The domain of the mapping ... |
mptun 5938 | Union of mappings which ar... |
feq1 5939 | Equality theorem for funct... |
feq2 5940 | Equality theorem for funct... |
feq3 5941 | Equality theorem for funct... |
feq23 5942 | Equality theorem for funct... |
feq1d 5943 | Equality deduction for fun... |
feq2d 5944 | Equality deduction for fun... |
feq3d 5945 | Equality deduction for fun... |
feq12d 5946 | Equality deduction for fun... |
feq123d 5947 | Equality deduction for fun... |
feq123 5948 | Equality theorem for funct... |
feq1i 5949 | Equality inference for fun... |
feq2i 5950 | Equality inference for fun... |
feq12i 5951 | Equality inference for fun... |
feq23i 5952 | Equality inference for fun... |
feq23d 5953 | Equality deduction for fun... |
nff 5954 | Bound-variable hypothesis ... |
sbcfng 5955 | Distribute proper substitu... |
sbcfg 5956 | Distribute proper substitu... |
elimf 5957 | Eliminate a mapping hypoth... |
ffn 5958 | A mapping is a function wi... |
ffnd 5959 | A mapping is a function wi... |
dffn2 5960 | Any function is a mapping ... |
ffun 5961 | A mapping is a function. ... |
ffund 5962 | A mapping is a function, d... |
frel 5963 | A mapping is a relation. ... |
fdm 5964 | The domain of a mapping. ... |
fdmi 5965 | The domain of a mapping. ... |
frn 5966 | The range of a mapping. (... |
dffn3 5967 | A function maps to its ran... |
ffrn 5968 | A function maps to its ran... |
fss 5969 | Expanding the codomain of ... |
fssd 5970 | Expanding the codomain of ... |
fco 5971 | Composition of two mapping... |
fco2 5972 | Functionality of a composi... |
fssxp 5973 | A mapping is a class of or... |
funssxp 5974 | Two ways of specifying a p... |
ffdm 5975 | A mapping is a partial fun... |
ffdmd 5976 | The domain of a function. ... |
fdmrn 5977 | A different way to write `... |
opelf 5978 | The members of an ordered ... |
fun 5979 | The union of two functions... |
fun2 5980 | The union of two functions... |
fun2d 5981 | The union of functions wit... |
fnfco 5982 | Composition of two functio... |
fssres 5983 | Restriction of a function ... |
fssresd 5984 | Restriction of a function ... |
fssres2 5985 | Restriction of a restricte... |
fresin 5986 | An identity for the mappin... |
resasplit 5987 | If two functions agree on ... |
fresaun 5988 | The union of two functions... |
fresaunres2 5989 | From the union of two func... |
fresaunres1 5990 | From the union of two func... |
fcoi1 5991 | Composition of a mapping a... |
fcoi2 5992 | Composition of restricted ... |
feu 5993 | There is exactly one value... |
fimass 5994 | The image of a class is a ... |
fcnvres 5995 | The converse of a restrict... |
fimacnvdisj 5996 | The preimage of a class di... |
fint 5997 | Function into an intersect... |
fin 5998 | Mapping into an intersecti... |
f0 5999 | The empty function. (Cont... |
f00 6000 | A class is a function with... |
f0bi 6001 | A function with empty doma... |
f0dom0 6002 | A function is empty iff it... |
f0rn0 6003 | If there is no element in ... |
fconst 6004 | A Cartesian product with a... |
fconstg 6005 | A Cartesian product with a... |
fnconstg 6006 | A Cartesian product with a... |
fconst6g 6007 | Constant function with loo... |
fconst6 6008 | A constant function as a m... |
f1eq1 6009 | Equality theorem for one-t... |
f1eq2 6010 | Equality theorem for one-t... |
f1eq3 6011 | Equality theorem for one-t... |
nff1 6012 | Bound-variable hypothesis ... |
dff12 6013 | Alternate definition of a ... |
f1f 6014 | A one-to-one mapping is a ... |
f1fn 6015 | A one-to-one mapping is a ... |
f1fun 6016 | A one-to-one mapping is a ... |
f1rel 6017 | A one-to-one onto mapping ... |
f1dm 6018 | The domain of a one-to-one... |
f1ss 6019 | A function that is one-to-... |
f1ssr 6020 | A function that is one-to-... |
f1ssres 6021 | A function that is one-to-... |
f1cnvcnv 6022 | Two ways to express that a... |
f1co 6023 | Composition of one-to-one ... |
foeq1 6024 | Equality theorem for onto ... |
foeq2 6025 | Equality theorem for onto ... |
foeq3 6026 | Equality theorem for onto ... |
nffo 6027 | Bound-variable hypothesis ... |
fof 6028 | An onto mapping is a mappi... |
fofun 6029 | An onto mapping is a funct... |
fofn 6030 | An onto mapping is a funct... |
forn 6031 | The codomain of an onto fu... |
dffo2 6032 | Alternate definition of an... |
foima 6033 | The image of the domain of... |
dffn4 6034 | A function maps onto its r... |
funforn 6035 | A function maps its domain... |
fodmrnu 6036 | An onto function has uniqu... |
fores 6037 | Restriction of an onto fun... |
foco 6038 | Composition of onto functi... |
foconst 6039 | A nonzero constant functio... |
f1oeq1 6040 | Equality theorem for one-t... |
f1oeq2 6041 | Equality theorem for one-t... |
f1oeq3 6042 | Equality theorem for one-t... |
f1oeq23 6043 | Equality theorem for one-t... |
f1eq123d 6044 | Equality deduction for one... |
foeq123d 6045 | Equality deduction for ont... |
f1oeq123d 6046 | Equality deduction for one... |
f1oeq3d 6047 | Equality deduction for one... |
nff1o 6048 | Bound-variable hypothesis ... |
f1of1 6049 | A one-to-one onto mapping ... |
f1of 6050 | A one-to-one onto mapping ... |
f1ofn 6051 | A one-to-one onto mapping ... |
f1ofun 6052 | A one-to-one onto mapping ... |
f1orel 6053 | A one-to-one onto mapping ... |
f1odm 6054 | The domain of a one-to-one... |
dff1o2 6055 | Alternate definition of on... |
dff1o3 6056 | Alternate definition of on... |
f1ofo 6057 | A one-to-one onto function... |
dff1o4 6058 | Alternate definition of on... |
dff1o5 6059 | Alternate definition of on... |
f1orn 6060 | A one-to-one function maps... |
f1f1orn 6061 | A one-to-one function maps... |
f1ocnv 6062 | The converse of a one-to-o... |
f1ocnvb 6063 | A relation is a one-to-one... |
f1ores 6064 | The restriction of a one-t... |
f1orescnv 6065 | The converse of a one-to-o... |
f1imacnv 6066 | Preimage of an image. (Co... |
foimacnv 6067 | A reverse version of ~ f1i... |
foun 6068 | The union of two onto func... |
f1oun 6069 | The union of two one-to-on... |
resdif 6070 | The restriction of a one-t... |
resin 6071 | The restriction of a one-t... |
f1oco 6072 | Composition of one-to-one ... |
f1cnv 6073 | The converse of an injecti... |
funcocnv2 6074 | Composition with the conve... |
fococnv2 6075 | The composition of an onto... |
f1ococnv2 6076 | The composition of a one-t... |
f1cocnv2 6077 | Composition of an injectiv... |
f1ococnv1 6078 | The composition of a one-t... |
f1cocnv1 6079 | Composition of an injectiv... |
funcoeqres 6080 | Re-express a constraint on... |
f10 6081 | The empty set maps one-to-... |
f10d 6082 | The empty set maps one-to-... |
f1o00 6083 | One-to-one onto mapping of... |
fo00 6084 | Onto mapping of the empty ... |
f1o0 6085 | One-to-one onto mapping of... |
f1oi 6086 | A restriction of the ident... |
f1ovi 6087 | The identity relation is a... |
f1osn 6088 | A singleton of an ordered ... |
f1osng 6089 | A singleton of an ordered ... |
f1sng 6090 | A singleton of an ordered ... |
fsnd 6091 | A singleton of an ordered ... |
f1oprswap 6092 | A two-element swap is a bi... |
f1oprg 6093 | An unordered pair of order... |
tz6.12-2 6094 | Function value when ` F ` ... |
fveu 6095 | The value of a function at... |
brprcneu 6096 | If ` A ` is a proper class... |
fvprc 6097 | A function's value at a pr... |
fv2 6098 | Alternate definition of fu... |
dffv3 6099 | A definition of function v... |
dffv4 6100 | The previous definition of... |
elfv 6101 | Membership in a function v... |
fveq1 6102 | Equality theorem for funct... |
fveq2 6103 | Equality theorem for funct... |
fveq1i 6104 | Equality inference for fun... |
fveq1d 6105 | Equality deduction for fun... |
fveq2i 6106 | Equality inference for fun... |
fveq2d 6107 | Equality deduction for fun... |
fveq12i 6108 | Equality deduction for fun... |
fveq12d 6109 | Equality deduction for fun... |
nffv 6110 | Bound-variable hypothesis ... |
nffvmpt1 6111 | Bound-variable hypothesis ... |
nffvd 6112 | Deduction version of bound... |
fvex 6113 | The value of a class exist... |
fvif 6114 | Move a conditional outside... |
iffv 6115 | Move a conditional outside... |
fv3 6116 | Alternate definition of th... |
fvres 6117 | The value of a restricted ... |
fvresd 6118 | The value of a restricted ... |
funssfv 6119 | The value of a member of t... |
tz6.12-1 6120 | Function value. Theorem 6... |
tz6.12 6121 | Function value. Theorem 6... |
tz6.12f 6122 | Function value, using boun... |
tz6.12c 6123 | Corollary of Theorem 6.12(... |
tz6.12i 6124 | Corollary of Theorem 6.12(... |
fvbr0 6125 | Two possibilities for the ... |
fvrn0 6126 | A function value is a memb... |
fvssunirn 6127 | The result of a function v... |
ndmfv 6128 | The value of a class outsi... |
ndmfvrcl 6129 | Reverse closure law for fu... |
elfvdm 6130 | If a function value has a ... |
elfvex 6131 | If a function value has a ... |
elfvexd 6132 | If a function value is non... |
eliman0 6133 | A non-nul function value i... |
nfvres 6134 | The value of a non-member ... |
nfunsn 6135 | If the restriction of a cl... |
fvfundmfvn0 6136 | If a class' value at an ar... |
0fv 6137 | Function value of the empt... |
fv2prc 6138 | A function's value at a fu... |
elfv2ex 6139 | If a function value of a f... |
fveqres 6140 | Equal values imply equal v... |
csbfv12 6141 | Move class substitution in... |
csbfv2g 6142 | Move class substitution in... |
csbfv 6143 | Substitution for a functio... |
funbrfv 6144 | The second argument of a b... |
funopfv 6145 | The second element in an o... |
fnbrfvb 6146 | Equivalence of function va... |
fnopfvb 6147 | Equivalence of function va... |
funbrfvb 6148 | Equivalence of function va... |
funopfvb 6149 | Equivalence of function va... |
funbrfv2b 6150 | Function value in terms of... |
dffn5 6151 | Representation of a functi... |
fnrnfv 6152 | The range of a function ex... |
fvelrnb 6153 | A member of a function's r... |
foelrni 6154 | A member of a surjective f... |
dfimafn 6155 | Alternate definition of th... |
dfimafn2 6156 | Alternate definition of th... |
funimass4 6157 | Membership relation for th... |
fvelima 6158 | Function value in an image... |
feqmptd 6159 | Deduction form of ~ dffn5 ... |
feqresmpt 6160 | Express a restricted funct... |
feqmptdf 6161 | Deduction form of ~ dffn5f... |
dffn5f 6162 | Representation of a functi... |
fvelimab 6163 | Function value in an image... |
fvelimabd 6164 | Deduction form of ~ fvelim... |
fvi 6165 | The value of the identity ... |
fviss 6166 | The value of the identity ... |
fniinfv 6167 | The indexed intersection o... |
fnsnfv 6168 | Singleton of function valu... |
opabiotafun 6169 | Define a function whose va... |
opabiotadm 6170 | Define a function whose va... |
opabiota 6171 | Define a function whose va... |
fnimapr 6172 | The image of a pair under ... |
ssimaex 6173 | The existence of a subimag... |
ssimaexg 6174 | The existence of a subimag... |
funfv 6175 | A simplified expression fo... |
funfv2 6176 | The value of a function. ... |
funfv2f 6177 | The value of a function. ... |
fvun 6178 | Value of the union of two ... |
fvun1 6179 | The value of a union when ... |
fvun2 6180 | The value of a union when ... |
dffv2 6181 | Alternate definition of fu... |
dmfco 6182 | Domains of a function comp... |
fvco2 6183 | Value of a function compos... |
fvco 6184 | Value of a function compos... |
fvco3 6185 | Value of a function compos... |
fvco4i 6186 | Conditions for a compositi... |
fvopab3g 6187 | Value of a function given ... |
fvopab3ig 6188 | Value of a function given ... |
fvmptg 6189 | Value of a function given ... |
fvmpti 6190 | Value of a function given ... |
fvmpt 6191 | Value of a function given ... |
fvmpt2f 6192 | Value of a function given ... |
fvtresfn 6193 | Functionality of a tuple-r... |
fvmpts 6194 | Value of a function given ... |
fvmpt3 6195 | Value of a function given ... |
fvmpt3i 6196 | Value of a function given ... |
fvmptd 6197 | Deduction version of ~ fvm... |
mptrcl 6198 | Reverse closure for a mapp... |
fvmpt2i 6199 | Value of a function given ... |
fvmpt2 6200 | Value of a function given ... |
fvmptss 6201 | If all the values of the m... |
fvmpt2d 6202 | Deduction version of ~ fvm... |
fvmptex 6203 | Express a function ` F ` w... |
fvmptdf 6204 | Alternate deduction versio... |
fvmptdv 6205 | Alternate deduction versio... |
fvmptdv2 6206 | Alternate deduction versio... |
mpteqb 6207 | Bidirectional equality the... |
fvmptt 6208 | Closed theorem form of ~ f... |
fvmptf 6209 | Value of a function given ... |
fvmptnf 6210 | The value of a function gi... |
fvmptn 6211 | This somewhat non-intuitiv... |
fvmptss2 6212 | A mapping always evaluates... |
elfvmptrab1 6213 | Implications for the value... |
elfvmptrab 6214 | Implications for the value... |
fvopab4ndm 6215 | Value of a function given ... |
fvmptndm 6216 | Value of a function given ... |
fvopab5 6217 | The value of a function th... |
fvopab6 6218 | Value of a function given ... |
eqfnfv 6219 | Equality of functions is d... |
eqfnfv2 6220 | Equality of functions is d... |
eqfnfv3 6221 | Derive equality of functio... |
eqfnfvd 6222 | Deduction for equality of ... |
eqfnfv2f 6223 | Equality of functions is d... |
eqfunfv 6224 | Equality of functions is d... |
fvreseq0 6225 | Equality of restricted fun... |
fvreseq1 6226 | Equality of a function res... |
fvreseq 6227 | Equality of restricted fun... |
fnmptfvd 6228 | A function with a given do... |
fndmdif 6229 | Two ways to express the lo... |
fndmdifcom 6230 | The difference set between... |
fndmdifeq0 6231 | The difference set of two ... |
fndmin 6232 | Two ways to express the lo... |
fneqeql 6233 | Two functions are equal if... |
fneqeql2 6234 | Two functions are equal if... |
fnreseql 6235 | Two functions are equal on... |
chfnrn 6236 | The range of a choice func... |
funfvop 6237 | Ordered pair with function... |
funfvbrb 6238 | Two ways to say that ` A `... |
fvimacnvi 6239 | A member of a preimage is ... |
fvimacnv 6240 | The argument of a function... |
funimass3 6241 | A kind of contraposition l... |
funimass5 6242 | A subclass of a preimage i... |
funconstss 6243 | Two ways of specifying tha... |
fvimacnvALT 6244 | Alternate proof of ~ fvima... |
elpreima 6245 | Membership in the preimage... |
fniniseg 6246 | Membership in the preimage... |
fncnvima2 6247 | Inverse images under funct... |
fniniseg2 6248 | Inverse point images under... |
unpreima 6249 | Preimage of a union. (Con... |
inpreima 6250 | Preimage of an intersectio... |
difpreima 6251 | Preimage of a difference. ... |
respreima 6252 | The preimage of a restrict... |
iinpreima 6253 | Preimage of an intersectio... |
intpreima 6254 | Preimage of an intersectio... |
fimacnv 6255 | The preimage of the codoma... |
fimacnvinrn 6256 | Taking the converse image ... |
fimacnvinrn2 6257 | Taking the converse image ... |
fvn0ssdmfun 6258 | If a class' function value... |
fnopfv 6259 | Ordered pair with function... |
fvelrn 6260 | A function's value belongs... |
nelrnfvne 6261 | A function value cannot be... |
fveqdmss 6262 | If the empty set is not co... |
fveqressseq 6263 | If the empty set is not co... |
fnfvelrn 6264 | A function's value belongs... |
ffvelrn 6265 | A function's value belongs... |
ffvelrni 6266 | A function's value belongs... |
ffvelrnda 6267 | A function's value belongs... |
ffvelrnd 6268 | A function's value belongs... |
rexrn 6269 | Restricted existential qua... |
ralrn 6270 | Restricted universal quant... |
elrnrexdm 6271 | For any element in the ran... |
elrnrexdmb 6272 | For any element in the ran... |
eldmrexrn 6273 | For any element in the dom... |
eldmrexrnb 6274 | For any element in the dom... |
fvcofneq 6275 | The values of two function... |
ralrnmpt 6276 | A restricted quantifier ov... |
rexrnmpt 6277 | A restricted quantifier ov... |
f0cli 6278 | Unconditional closure of a... |
dff2 6279 | Alternate definition of a ... |
dff3 6280 | Alternate definition of a ... |
dff4 6281 | Alternate definition of a ... |
dffo3 6282 | An onto mapping expressed ... |
dffo4 6283 | Alternate definition of an... |
dffo5 6284 | Alternate definition of an... |
exfo 6285 | A relation equivalent to t... |
foelrn 6286 | Property of a surjective f... |
foco2 6287 | If a composition of two fu... |
foco2OLD 6288 | Obsolete proof of ~ foco2 ... |
fmpt 6289 | Functionality of the mappi... |
f1ompt 6290 | Express bijection for a ma... |
fmpti 6291 | Functionality of the mappi... |
fmptd 6292 | Domain and codomain of the... |
fmpt3d 6293 | Domain and co-domain of th... |
fmptdf 6294 | A version of ~ fmptd using... |
ffnfv 6295 | A function maps to a class... |
ffnfvf 6296 | A function maps to a class... |
fnfvrnss 6297 | An upper bound for range d... |
frnssb 6298 | A function is a function i... |
rnmptss 6299 | The range of an operation ... |
fmpt2d 6300 | Domain and codomain of the... |
ffvresb 6301 | A necessary and sufficient... |
f1oresrab 6302 | Build a bijection between ... |
fmptco 6303 | Composition of two functio... |
fmptcof 6304 | Version of ~ fmptco where ... |
fmptcos 6305 | Composition of two functio... |
fcompt 6306 | Express composition of two... |
fcoconst 6307 | Composition with a constan... |
fsn 6308 | A function maps a singleto... |
fsn2 6309 | A function that maps a sin... |
fsng 6310 | A function maps a singleto... |
fsn2g 6311 | A function that maps a sin... |
xpsng 6312 | The Cartesian product of t... |
xpsn 6313 | The Cartesian product of t... |
f1o2sn 6314 | A singleton with a nested ... |
residpr 6315 | Restriction of the identit... |
dfmpt 6316 | Alternate definition for t... |
fnasrn 6317 | A function expressed as th... |
funiun 6318 | A function is a union of s... |
funopsn 6319 | If a function is an ordere... |
funop 6320 | An ordered pair is a funct... |
funsndifnop 6321 | A singleton of an ordered ... |
funsneqopsn 6322 | A singleton of an ordered ... |
funsneqop 6323 | A singleton of an ordered ... |
funsneqopb 6324 | A singleton of an ordered ... |
ressnop0 6325 | If ` A ` is not in ` C ` ,... |
fpr 6326 | A function with a domain o... |
fprg 6327 | A function with a domain o... |
ftpg 6328 | A function with a domain o... |
ftp 6329 | A function with a domain o... |
fnressn 6330 | A function restricted to a... |
funressn 6331 | A function restricted to a... |
fressnfv 6332 | The value of a function re... |
fvrnressn 6333 | If the value of a function... |
fvressn 6334 | The value of a function re... |
fvn0fvelrn 6335 | If the value of a function... |
fvconst 6336 | The value of a constant fu... |
fnsnb 6337 | A function whose domain is... |
fmptsn 6338 | Express a singleton functi... |
fmptsng 6339 | Express a singleton functi... |
fmptsnd 6340 | Express a singleton functi... |
fmptap 6341 | Append an additional value... |
fmptapd 6342 | Append an additional value... |
fmptpr 6343 | Express a pair function in... |
fvresi 6344 | The value of a restricted ... |
fninfp 6345 | Express the class of fixed... |
fnelfp 6346 | Property of a fixed point ... |
fndifnfp 6347 | Express the class of non-f... |
fnelnfp 6348 | Property of a non-fixed po... |
fnnfpeq0 6349 | A function is the identity... |
fvunsn 6350 | Remove an ordered pair not... |
fvsn 6351 | The value of a singleton o... |
fvsng 6352 | The value of a singleton o... |
fvsnun1 6353 | The value of a function wi... |
fvsnun2 6354 | The value of a function wi... |
fnsnsplit 6355 | Split a function into a si... |
fsnunf 6356 | Adjoining a point to a fun... |
fsnunf2 6357 | Adjoining a point to a pun... |
fsnunfv 6358 | Recover the added point fr... |
fsnunres 6359 | Recover the original funct... |
funresdfunsn 6360 | Restricting a function to ... |
fvpr1 6361 | The value of a function wi... |
fvpr2 6362 | The value of a function wi... |
fvpr1g 6363 | The value of a function wi... |
fvpr2g 6364 | The value of a function wi... |
fvtp1 6365 | The first value of a funct... |
fvtp2 6366 | The second value of a func... |
fvtp3 6367 | The third value of a funct... |
fvtp1g 6368 | The value of a function wi... |
fvtp2g 6369 | The value of a function wi... |
fvtp3g 6370 | The value of a function wi... |
tpres 6371 | An unordered triple of ord... |
fvconst2g 6372 | The value of a constant fu... |
fconst2g 6373 | A constant function expres... |
fvconst2 6374 | The value of a constant fu... |
fconst2 6375 | A constant function expres... |
fconst5 6376 | Two ways to express that a... |
fnprb 6377 | A function whose domain ha... |
fntpb 6378 | A function whose domain ha... |
fnpr2g 6379 | A function whose domain ha... |
fpr2g 6380 | A function that maps a pai... |
fconstfv 6381 | A constant function expres... |
fconst3 6382 | Two ways to express a cons... |
fconst4 6383 | Two ways to express a cons... |
resfunexg 6384 | The restriction of a funct... |
resiexd 6385 | The restriction of the ide... |
fnex 6386 | If the domain of a functio... |
funex 6387 | If the domain of a functio... |
opabex 6388 | Existence of a function ex... |
mptexg 6389 | If the domain of a functio... |
mptex 6390 | If the domain of a functio... |
mptexd 6391 | If the domain of a functio... |
mptrabex 6392 | If the domain of a functio... |
mptrabexOLD 6393 | Obsolete version of ~ mptr... |
fex 6394 | If the domain of a mapping... |
eufnfv 6395 | A function is uniquely det... |
funfvima 6396 | A function's value in a pr... |
funfvima2 6397 | A function's value in an i... |
resfvresima 6398 | The value of the function ... |
funfvima3 6399 | A class including a functi... |
fnfvima 6400 | The function value of an o... |
rexima 6401 | Existential quantification... |
ralima 6402 | Universal quantification u... |
idref 6403 | TODO: This is the same as... |
fvclss 6404 | Upper bound for the class ... |
elabrex 6405 | Elementhood in an image se... |
abrexco 6406 | Composition of two image m... |
imaiun 6407 | The image of an indexed un... |
imauni 6408 | The image of a union is th... |
fniunfv 6409 | The indexed union of a fun... |
funiunfv 6410 | The indexed union of a fun... |
funiunfvf 6411 | The indexed union of a fun... |
eluniima 6412 | Membership in the union of... |
elunirn 6413 | Membership in the union of... |
elunirnALT 6414 | Alternate proof of ~ eluni... |
fnunirn 6415 | Membership in a union of s... |
dff13 6416 | A one-to-one function in t... |
dff13f 6417 | A one-to-one function in t... |
f1veqaeq 6418 | If the values of a one-to-... |
f1mpt 6419 | Express injection for a ma... |
f1fveq 6420 | Equality of function value... |
f1elima 6421 | Membership in the image of... |
f1imass 6422 | Taking images under a one-... |
f1imaeq 6423 | Taking images under a one-... |
f1imapss 6424 | Taking images under a one-... |
f1dom3fv3dif 6425 | The function values for a ... |
f1dom3el3dif 6426 | The range of a 1-1 functio... |
dff14a 6427 | A one-to-one function in t... |
dff14b 6428 | A one-to-one function in t... |
f12dfv 6429 | A one-to-one function with... |
f13dfv 6430 | A one-to-one function with... |
dff1o6 6431 | A one-to-one onto function... |
f1ocnvfv1 6432 | The converse value of the ... |
f1ocnvfv2 6433 | The value of the converse ... |
f1ocnvfv 6434 | Relationship between the v... |
f1ocnvfvb 6435 | Relationship between the v... |
nvof1o 6436 | An involution is a bijecti... |
nvocnv 6437 | The converse of an involut... |
fsnex 6438 | Relate a function with a s... |
f1prex 6439 | Relate a one-to-one functi... |
f1ocnvdm 6440 | The value of the converse ... |
f1ocnvfvrneq 6441 | If the values of a one-to-... |
fcof1 6442 | An application is injectiv... |
fcofo 6443 | An application is surjecti... |
cbvfo 6444 | Change bound variable betw... |
cbvexfo 6445 | Change bound variable betw... |
cocan1 6446 | An injection is left-cance... |
cocan2 6447 | A surjection is right-canc... |
fcof1oinvd 6448 | Show that a function is th... |
fcof1od 6449 | A function is bijective if... |
2fcoidinvd 6450 | Show that a function is th... |
fcof1o 6451 | Show that two functions ar... |
2fvcoidd 6452 | Show that the composition ... |
2fvidf1od 6453 | A function is bijective if... |
2fvidinvd 6454 | Show that two functions ar... |
foeqcnvco 6455 | Condition for function equ... |
f1eqcocnv 6456 | Condition for function equ... |
fveqf1o 6457 | Given a bijection ` F ` , ... |
fliftrel 6458 | ` F ` , a function lift, i... |
fliftel 6459 | Elementhood in the relatio... |
fliftel1 6460 | Elementhood in the relatio... |
fliftcnv 6461 | Converse of the relation `... |
fliftfun 6462 | The function ` F ` is the ... |
fliftfund 6463 | The function ` F ` is the ... |
fliftfuns 6464 | The function ` F ` is the ... |
fliftf 6465 | The domain and range of th... |
fliftval 6466 | The value of the function ... |
isoeq1 6467 | Equality theorem for isomo... |
isoeq2 6468 | Equality theorem for isomo... |
isoeq3 6469 | Equality theorem for isomo... |
isoeq4 6470 | Equality theorem for isomo... |
isoeq5 6471 | Equality theorem for isomo... |
nfiso 6472 | Bound-variable hypothesis ... |
isof1o 6473 | An isomorphism is a one-to... |
isof1oidb 6474 | A function is a bijection ... |
isof1oopb 6475 | A function is a bijection ... |
isorel 6476 | An isomorphism connects bi... |
soisores 6477 | Express the condition of i... |
soisoi 6478 | Infer isomorphism from one... |
isoid 6479 | Identity law for isomorphi... |
isocnv 6480 | Converse law for isomorphi... |
isocnv2 6481 | Converse law for isomorphi... |
isocnv3 6482 | Complementation law for is... |
isores2 6483 | An isomorphism from one we... |
isores1 6484 | An isomorphism from one we... |
isores3 6485 | Induced isomorphism on a s... |
isotr 6486 | Composition (transitive) l... |
isomin 6487 | Isomorphisms preserve mini... |
isoini 6488 | Isomorphisms preserve init... |
isoini2 6489 | Isomorphisms are isomorphi... |
isofrlem 6490 | Lemma for ~ isofr . (Cont... |
isoselem 6491 | Lemma for ~ isose . (Cont... |
isofr 6492 | An isomorphism preserves w... |
isose 6493 | An isomorphism preserves s... |
isofr2 6494 | A weak form of ~ isofr tha... |
isopolem 6495 | Lemma for ~ isopo . (Cont... |
isopo 6496 | An isomorphism preserves p... |
isosolem 6497 | Lemma for ~ isoso . (Cont... |
isoso 6498 | An isomorphism preserves s... |
isowe 6499 | An isomorphism preserves w... |
isowe2 6500 | A weak form of ~ isowe tha... |
f1oiso 6501 | Any one-to-one onto functi... |
f1oiso2 6502 | Any one-to-one onto functi... |
f1owe 6503 | Well-ordering of isomorphi... |
weniso 6504 | A set-like well-ordering h... |
weisoeq 6505 | Thus, there is at most one... |
weisoeq2 6506 | Thus, there is at most one... |
knatar 6507 | The Knaster-Tarski theorem... |
canth 6508 | No set ` A ` is equinumero... |
ncanth 6509 | Cantor's theorem fails for... |
riotaeqdv 6512 | Formula-building deduction... |
riotabidv 6513 | Formula-building deduction... |
riotaeqbidv 6514 | Equality deduction for res... |
riotaex 6515 | Restricted iota is a set. ... |
riotav 6516 | An iota restricted to the ... |
riotauni 6517 | Restricted iota in terms o... |
nfriota1 6518 | The abstraction variable i... |
nfriotad 6519 | Deduction version of ~ nfr... |
nfriota 6520 | A variable not free in a w... |
cbvriota 6521 | Change bound variable in a... |
cbvriotav 6522 | Change bound variable in a... |
csbriota 6523 | Interchange class substitu... |
riotacl2 6524 | Membership law for "the un... |
riotacl 6525 | Closure of restricted iota... |
riotasbc 6526 | Substitution law for descr... |
riotabidva 6527 | Equivalent wff's yield equ... |
riotabiia 6528 | Equivalent wff's yield equ... |
riota1 6529 | Property of restricted iot... |
riota1a 6530 | Property of iota. (Contri... |
riota2df 6531 | A deduction version of ~ r... |
riota2f 6532 | This theorem shows a condi... |
riota2 6533 | This theorem shows a condi... |
riotaprop 6534 | Properties of a restricted... |
riota5f 6535 | A method for computing res... |
riota5 6536 | A method for computing res... |
riotass2 6537 | Restriction of a unique el... |
riotass 6538 | Restriction of a unique el... |
moriotass 6539 | Restriction of a unique el... |
snriota 6540 | A restricted class abstrac... |
riotaxfrd 6541 | Change the variable ` x ` ... |
eusvobj2 6542 | Specify the same property ... |
eusvobj1 6543 | Specify the same object in... |
f1ofveu 6544 | There is one domain elemen... |
f1ocnvfv3 6545 | Value of the converse of a... |
riotaund 6546 | Restricted iota equals the... |
riotassuni 6547 | The restricted iota class ... |
riotaclb 6548 | Bidirectional closure of r... |
oveq 6555 | Equality theorem for opera... |
oveq1 6556 | Equality theorem for opera... |
oveq2 6557 | Equality theorem for opera... |
oveq12 6558 | Equality theorem for opera... |
oveq1i 6559 | Equality inference for ope... |
oveq2i 6560 | Equality inference for ope... |
oveq12i 6561 | Equality inference for ope... |
oveqi 6562 | Equality inference for ope... |
oveq123i 6563 | Equality inference for ope... |
oveq1d 6564 | Equality deduction for ope... |
oveq2d 6565 | Equality deduction for ope... |
oveqd 6566 | Equality deduction for ope... |
oveq12d 6567 | Equality deduction for ope... |
oveqan12d 6568 | Equality deduction for ope... |
oveqan12rd 6569 | Equality deduction for ope... |
oveq123d 6570 | Equality deduction for ope... |
ovrspc2v 6571 | If an operation value is e... |
oveqrspc2v 6572 | Restricted specialization ... |
oveqdr 6573 | Equality of two operations... |
nfovd 6574 | Deduction version of bound... |
nfov 6575 | Bound-variable hypothesis ... |
oprabid 6576 | The law of concretion. Sp... |
ovex 6577 | The result of an operation... |
ovexi 6578 | The result of an operation... |
ovssunirn 6579 | The result of an operation... |
0ov 6580 | Operation value of the emp... |
ovprc 6581 | The value of an operation ... |
ovprc1 6582 | The value of an operation ... |
ovprc2 6583 | The value of an operation ... |
ovrcl 6584 | Reverse closure for an ope... |
csbov123 6585 | Move class substitution in... |
csbov 6586 | Move class substitution in... |
csbov12g 6587 | Move class substitution in... |
csbov1g 6588 | Move class substitution in... |
csbov2g 6589 | Move class substitution in... |
rspceov 6590 | A frequently used special ... |
elovimad 6591 | Elementhood of the image s... |
fnotovb 6592 | Equivalence of operation v... |
opabbrex 6593 | A collection of ordered pa... |
fvmptopab1 6594 | The function value of a ma... |
fvmptopab2 6595 | The function value of a ma... |
0neqopab 6596 | The empty set is never an ... |
brabv 6597 | If two classes are in a re... |
brfvopab 6598 | The classes involved in a ... |
dfoprab2 6599 | Class abstraction for oper... |
reloprab 6600 | An operation class abstrac... |
oprabv 6601 | If a pair and a class are ... |
nfoprab1 6602 | The abstraction variables ... |
nfoprab2 6603 | The abstraction variables ... |
nfoprab3 6604 | The abstraction variables ... |
nfoprab 6605 | Bound-variable hypothesis ... |
oprabbid 6606 | Equivalent wff's yield equ... |
oprabbidv 6607 | Equivalent wff's yield equ... |
oprabbii 6608 | Equivalent wff's yield equ... |
ssoprab2 6609 | Equivalence of ordered pai... |
ssoprab2b 6610 | Equivalence of ordered pai... |
eqoprab2b 6611 | Equivalence of ordered pai... |
mpt2eq123 6612 | An equality theorem for th... |
mpt2eq12 6613 | An equality theorem for th... |
mpt2eq123dva 6614 | An equality deduction for ... |
mpt2eq123dv 6615 | An equality deduction for ... |
mpt2eq123i 6616 | An equality inference for ... |
mpt2eq3dva 6617 | Slightly more general equa... |
mpt2eq3ia 6618 | An equality inference for ... |
mpt2eq3dv 6619 | An equality deduction for ... |
nfmpt21 6620 | Bound-variable hypothesis ... |
nfmpt22 6621 | Bound-variable hypothesis ... |
nfmpt2 6622 | Bound-variable hypothesis ... |
mpt20 6623 | A mapping operation with e... |
oprab4 6624 | Two ways to state the doma... |
cbvoprab1 6625 | Rule used to change first ... |
cbvoprab2 6626 | Change the second bound va... |
cbvoprab12 6627 | Rule used to change first ... |
cbvoprab12v 6628 | Rule used to change first ... |
cbvoprab3 6629 | Rule used to change the th... |
cbvoprab3v 6630 | Rule used to change the th... |
cbvmpt2x 6631 | Rule to change the bound v... |
cbvmpt2 6632 | Rule to change the bound v... |
cbvmpt2v 6633 | Rule to change the bound v... |
elimdelov 6634 | Eliminate a hypothesis whi... |
ovif 6635 | Move a conditional outside... |
ovif2 6636 | Move a conditional outside... |
ovif12 6637 | Move a conditional outside... |
ifov 6638 | Move a conditional outside... |
dmoprab 6639 | The domain of an operation... |
dmoprabss 6640 | The domain of an operation... |
rnoprab 6641 | The range of an operation ... |
rnoprab2 6642 | The range of a restricted ... |
reldmoprab 6643 | The domain of an operation... |
oprabss 6644 | Structure of an operation ... |
eloprabga 6645 | The law of concretion for ... |
eloprabg 6646 | The law of concretion for ... |
ssoprab2i 6647 | Inference of operation cla... |
mpt2v 6648 | Operation with universal d... |
mpt2mptx 6649 | Express a two-argument fun... |
mpt2mpt 6650 | Express a two-argument fun... |
mpt2difsnif 6651 | A mapping with two argumen... |
mpt2snif 6652 | A mapping with two argumen... |
fconstmpt2 6653 | Representation of a consta... |
resoprab 6654 | Restriction of an operatio... |
resoprab2 6655 | Restriction of an operator... |
resmpt2 6656 | Restriction of the mapping... |
funoprabg 6657 | "At most one" is a suffici... |
funoprab 6658 | "At most one" is a suffici... |
fnoprabg 6659 | Functionality and domain o... |
mpt2fun 6660 | The maps-to notation for a... |
fnoprab 6661 | Functionality and domain o... |
ffnov 6662 | An operation maps to a cla... |
fovcl 6663 | Closure law for an operati... |
eqfnov 6664 | Equality of two operations... |
eqfnov2 6665 | Two operators with the sam... |
fnov 6666 | Representation of a functi... |
mpt22eqb 6667 | Bidirectional equality the... |
rnmpt2 6668 | The range of an operation ... |
reldmmpt2 6669 | The domain of an operation... |
elrnmpt2g 6670 | Membership in the range of... |
elrnmpt2 6671 | Membership in the range of... |
elrnmpt2res 6672 | Membership in the range of... |
ralrnmpt2 6673 | A restricted quantifier ov... |
rexrnmpt2 6674 | A restricted quantifier ov... |
ovid 6675 | The value of an operation ... |
ovidig 6676 | The value of an operation ... |
ovidi 6677 | The value of an operation ... |
ov 6678 | The value of an operation ... |
ovigg 6679 | The value of an operation ... |
ovig 6680 | The value of an operation ... |
ovmpt4g 6681 | Value of a function given ... |
ovmpt2s 6682 | Value of a function given ... |
ov2gf 6683 | The value of an operation ... |
ovmpt2dxf 6684 | Value of an operation give... |
ovmpt2dx 6685 | Value of an operation give... |
ovmpt2d 6686 | Value of an operation give... |
ovmpt2x 6687 | The value of an operation ... |
ovmpt2ga 6688 | Value of an operation give... |
ovmpt2a 6689 | Value of an operation give... |
ovmpt2df 6690 | Alternate deduction versio... |
ovmpt2dv 6691 | Alternate deduction versio... |
ovmpt2dv2 6692 | Alternate deduction versio... |
ovmpt2g 6693 | Value of an operation give... |
ovmpt2 6694 | Value of an operation give... |
ov3 6695 | The value of an operation ... |
ov6g 6696 | The value of an operation ... |
ovg 6697 | The value of an operation ... |
ovres 6698 | The value of a restricted ... |
ovresd 6699 | Lemma for converting metri... |
oprres 6700 | The restriction of an oper... |
oprssov 6701 | The value of a member of t... |
fovrn 6702 | An operation's value belon... |
fovrnda 6703 | An operation's value belon... |
fovrnd 6704 | An operation's value belon... |
fnrnov 6705 | The range of an operation ... |
foov 6706 | An onto mapping of an oper... |
fnovrn 6707 | An operation's value belon... |
ovelrn 6708 | A member of an operation's... |
funimassov 6709 | Membership relation for th... |
ovelimab 6710 | Operation value in an imag... |
ovima0 6711 | An operation value is a me... |
ovconst2 6712 | The value of a constant op... |
oprssdm 6713 | Domain of closure of an op... |
nssdmovg 6714 | The value of an operation ... |
ndmovg 6715 | The value of an operation ... |
ndmov 6716 | The value of an operation ... |
ndmovcl 6717 | The closure of an operatio... |
ndmovrcl 6718 | Reverse closure law, when ... |
ndmovcom 6719 | Any operation is commutati... |
ndmovass 6720 | Any operation is associati... |
ndmovdistr 6721 | Any operation is distribut... |
ndmovord 6722 | Elimination of redundant a... |
ndmovordi 6723 | Elimination of redundant a... |
caovclg 6724 | Convert an operation closu... |
caovcld 6725 | Convert an operation closu... |
caovcl 6726 | Convert an operation closu... |
caovcomg 6727 | Convert an operation commu... |
caovcomd 6728 | Convert an operation commu... |
caovcom 6729 | Convert an operation commu... |
caovassg 6730 | Convert an operation assoc... |
caovassd 6731 | Convert an operation assoc... |
caovass 6732 | Convert an operation assoc... |
caovcang 6733 | Convert an operation cance... |
caovcand 6734 | Convert an operation cance... |
caovcanrd 6735 | Commute the arguments of a... |
caovcan 6736 | Convert an operation cance... |
caovordig 6737 | Convert an operation order... |
caovordid 6738 | Convert an operation order... |
caovordg 6739 | Convert an operation order... |
caovordd 6740 | Convert an operation order... |
caovord2d 6741 | Operation ordering law wit... |
caovord3d 6742 | Ordering law. (Contribute... |
caovord 6743 | Convert an operation order... |
caovord2 6744 | Operation ordering law wit... |
caovord3 6745 | Ordering law. (Contribute... |
caovdig 6746 | Convert an operation distr... |
caovdid 6747 | Convert an operation distr... |
caovdir2d 6748 | Convert an operation distr... |
caovdirg 6749 | Convert an operation rever... |
caovdird 6750 | Convert an operation distr... |
caovdi 6751 | Convert an operation distr... |
caov32d 6752 | Rearrange arguments in a c... |
caov12d 6753 | Rearrange arguments in a c... |
caov31d 6754 | Rearrange arguments in a c... |
caov13d 6755 | Rearrange arguments in a c... |
caov4d 6756 | Rearrange arguments in a c... |
caov411d 6757 | Rearrange arguments in a c... |
caov42d 6758 | Rearrange arguments in a c... |
caov32 6759 | Rearrange arguments in a c... |
caov12 6760 | Rearrange arguments in a c... |
caov31 6761 | Rearrange arguments in a c... |
caov13 6762 | Rearrange arguments in a c... |
caov4 6763 | Rearrange arguments in a c... |
caov411 6764 | Rearrange arguments in a c... |
caov42 6765 | Rearrange arguments in a c... |
caovdir 6766 | Reverse distributive law. ... |
caovdilem 6767 | Lemma used by real number ... |
caovlem2 6768 | Lemma used in real number ... |
caovmo 6769 | Uniqueness of inverse elem... |
grprinvlem 6770 | Lemma for ~ grprinvd . (C... |
grprinvd 6771 | Deduce right inverse from ... |
grpridd 6772 | Deduce right identity from... |
mpt2ndm0 6773 | The value of an operation ... |
elmpt2cl 6774 | If a two-parameter class i... |
elmpt2cl1 6775 | If a two-parameter class i... |
elmpt2cl2 6776 | If a two-parameter class i... |
elovmpt2 6777 | Utility lemma for two-para... |
elovmpt2rab 6778 | Implications for the value... |
elovmpt2rab1 6779 | Implications for the value... |
2mpt20 6780 | If the operation value of ... |
relmptopab 6781 | Any function to sets of or... |
f1ocnvd 6782 | Describe an implicit one-t... |
f1od 6783 | Describe an implicit one-t... |
f1ocnv2d 6784 | Describe an implicit one-t... |
f1o2d 6785 | Describe an implicit one-t... |
f1opw2 6786 | A one-to-one mapping induc... |
f1opw 6787 | A one-to-one mapping induc... |
elovmpt3imp 6788 | If the value of a function... |
ovmpt3rab1 6789 | The value of an operation ... |
ovmpt3rabdm 6790 | If the value of a function... |
elovmpt3rab1 6791 | Implications for the value... |
elovmpt3rab 6792 | Implications for the value... |
ofeq 6797 | Equality theorem for funct... |
ofreq 6798 | Equality theorem for funct... |
ofexg 6799 | A function operation restr... |
nfof 6800 | Hypothesis builder for fun... |
nfofr 6801 | Hypothesis builder for fun... |
offval 6802 | Value of an operation appl... |
ofrfval 6803 | Value of a relation applie... |
ofval 6804 | Evaluate a function operat... |
ofrval 6805 | Exhibit a function relatio... |
offn 6806 | The function operation pro... |
offval2f 6807 | The function operation exp... |
ofmresval 6808 | Value of a restriction of ... |
fnfvof 6809 | Function value of a pointw... |
off 6810 | The function operation pro... |
ofres 6811 | Restrict the operands of a... |
offval2 6812 | The function operation exp... |
ofrfval2 6813 | The function relation acti... |
ofmpteq 6814 | Value of a pointwise opera... |
ofco 6815 | The composition of a funct... |
offveq 6816 | Convert an identity of the... |
offveqb 6817 | Equivalent expressions for... |
ofc1 6818 | Left operation by a consta... |
ofc2 6819 | Right operation by a const... |
ofc12 6820 | Function operation on two ... |
caofref 6821 | Transfer a reflexive law t... |
caofinvl 6822 | Transfer a left inverse la... |
caofid0l 6823 | Transfer a left identity l... |
caofid0r 6824 | Transfer a right identity ... |
caofid1 6825 | Transfer a right absorptio... |
caofid2 6826 | Transfer a right absorptio... |
caofcom 6827 | Transfer a commutative law... |
caofrss 6828 | Transfer a relation subset... |
caofass 6829 | Transfer an associative la... |
caoftrn 6830 | Transfer a transitivity la... |
caofdi 6831 | Transfer a distributive la... |
caofdir 6832 | Transfer a reverse distrib... |
caonncan 6833 | Transfer ~ nncan -shaped l... |
relrpss 6836 | The proper subset relation... |
brrpssg 6837 | The proper subset relation... |
brrpss 6838 | The proper subset relation... |
porpss 6839 | Every class is partially o... |
sorpss 6840 | Express strict ordering un... |
sorpssi 6841 | Property of a chain of set... |
sorpssun 6842 | A chain of sets is closed ... |
sorpssin 6843 | A chain of sets is closed ... |
sorpssuni 6844 | In a chain of sets, a maxi... |
sorpssint 6845 | In a chain of sets, a mini... |
sorpsscmpl 6846 | The componentwise compleme... |
zfun 6848 | Axiom of Union expressed w... |
axun2 6849 | A variant of the Axiom of ... |
uniex2 6850 | The Axiom of Union using t... |
uniex 6851 | The Axiom of Union in clas... |
vuniex 6852 | The union of a setvar is a... |
uniexg 6853 | The ZF Axiom of Union in c... |
unex 6854 | The union of two sets is a... |
tpex 6855 | An unordered triple of cla... |
unexb 6856 | Existence of union is equi... |
unexg 6857 | A union of two sets is a s... |
xpexg 6858 | The Cartesian product of t... |
3xpexg 6859 | The Cartesian product of t... |
xpex 6860 | The Cartesian product of t... |
sqxpexg 6861 | The Cartesian square of a ... |
snnex 6862 | The class of all singleton... |
difex2 6863 | If the subtrahend of a cla... |
difsnexi 6864 | If the difference of a cla... |
uniuni 6865 | Expression for double unio... |
uniexb 6866 | The Axiom of Union and its... |
pwexb 6867 | The Axiom of Power Sets an... |
eldifpw 6868 | Membership in a power clas... |
elpwun 6869 | Membership in the power cl... |
iunpw 6870 | An indexed union of a powe... |
fr3nr 6871 | A well-founded relation ha... |
epne3 6872 | A set well-founded by epsi... |
dfwe2 6873 | Alternate definition of we... |
ordon 6874 | The class of all ordinal n... |
epweon 6875 | The epsilon relation well-... |
onprc 6876 | No set contains all ordina... |
ssorduni 6877 | The union of a class of or... |
ssonuni 6878 | The union of a set of ordi... |
ssonunii 6879 | The union of a set of ordi... |
ordeleqon 6880 | A way to express the ordin... |
ordsson 6881 | Any ordinal class is a sub... |
onss 6882 | An ordinal number is a sub... |
predon 6883 | For an ordinal, the predec... |
ssonprc 6884 | Two ways of saying a class... |
onuni 6885 | The union of an ordinal nu... |
orduni 6886 | The union of an ordinal cl... |
onint 6887 | The intersection (infimum)... |
onint0 6888 | The intersection of a clas... |
onssmin 6889 | A nonempty class of ordina... |
onminesb 6890 | If a property is true for ... |
onminsb 6891 | If a property is true for ... |
oninton 6892 | The intersection of a none... |
onintrab 6893 | The intersection of a clas... |
onintrab2 6894 | An existence condition equ... |
onnmin 6895 | No member of a set of ordi... |
onnminsb 6896 | An ordinal number smaller ... |
oneqmin 6897 | A way to show that an ordi... |
bm2.5ii 6898 | Problem 2.5(ii) of [BellMa... |
onminex 6899 | If a wff is true for an or... |
sucon 6900 | The class of all ordinal n... |
sucexb 6901 | A successor exists iff its... |
sucexg 6902 | The successor of a set is ... |
sucex 6903 | The successor of a set is ... |
onmindif2 6904 | The minimum of a class of ... |
suceloni 6905 | The successor of an ordina... |
ordsuc 6906 | The successor of an ordina... |
ordpwsuc 6907 | The collection of ordinals... |
onpwsuc 6908 | The collection of ordinal ... |
sucelon 6909 | The successor of an ordina... |
ordsucss 6910 | The successor of an elemen... |
onpsssuc 6911 | An ordinal number is a pro... |
ordelsuc 6912 | A set belongs to an ordina... |
onsucmin 6913 | The successor of an ordina... |
ordsucelsuc 6914 | Membership is inherited by... |
ordsucsssuc 6915 | The subclass relationship ... |
ordsucuniel 6916 | Given an element ` A ` of ... |
ordsucun 6917 | The successor of the maxim... |
ordunpr 6918 | The maximum of two ordinal... |
ordunel 6919 | The maximum of two ordinal... |
onsucuni 6920 | A class of ordinal numbers... |
ordsucuni 6921 | An ordinal class is a subc... |
orduniorsuc 6922 | An ordinal class is either... |
unon 6923 | The class of all ordinal n... |
ordunisuc 6924 | An ordinal class is equal ... |
orduniss2 6925 | The union of the ordinal s... |
onsucuni2 6926 | A successor ordinal is the... |
0elsuc 6927 | The successor of an ordina... |
limon 6928 | The class of ordinal numbe... |
onssi 6929 | An ordinal number is a sub... |
onsuci 6930 | The successor of an ordina... |
onuniorsuci 6931 | An ordinal number is eithe... |
onuninsuci 6932 | A limit ordinal is not a s... |
onsucssi 6933 | A set belongs to an ordina... |
nlimsucg 6934 | A successor is not a limit... |
orduninsuc 6935 | An ordinal equal to its un... |
ordunisuc2 6936 | An ordinal equal to its un... |
ordzsl 6937 | An ordinal is zero, a succ... |
onzsl 6938 | An ordinal number is zero,... |
dflim3 6939 | An alternate definition of... |
dflim4 6940 | An alternate definition of... |
limsuc 6941 | The successor of a member ... |
limsssuc 6942 | A class includes a limit o... |
nlimon 6943 | Two ways to express the cl... |
limuni3 6944 | The union of a nonempty cl... |
tfi 6945 | The Principle of Transfini... |
tfis 6946 | Transfinite Induction Sche... |
tfis2f 6947 | Transfinite Induction Sche... |
tfis2 6948 | Transfinite Induction Sche... |
tfis3 6949 | Transfinite Induction Sche... |
tfisi 6950 | A transfinite induction sc... |
tfinds 6951 | Principle of Transfinite I... |
tfindsg 6952 | Transfinite Induction (inf... |
tfindsg2 6953 | Transfinite Induction (inf... |
tfindes 6954 | Transfinite Induction with... |
tfinds2 6955 | Transfinite Induction (inf... |
tfinds3 6956 | Principle of Transfinite I... |
dfom2 6959 | An alternate definition of... |
elom 6960 | Membership in omega. The ... |
omsson 6961 | Omega is a subset of ` On ... |
limomss 6962 | The class of natural numbe... |
nnon 6963 | A natural number is an ord... |
nnoni 6964 | A natural number is an ord... |
nnord 6965 | A natural number is ordina... |
ordom 6966 | Omega is ordinal. Theorem... |
elnn 6967 | A member of a natural numb... |
omon 6968 | The class of natural numbe... |
omelon2 6969 | Omega is an ordinal number... |
nnlim 6970 | A natural number is not a ... |
omssnlim 6971 | The class of natural numbe... |
limom 6972 | Omega is a limit ordinal. ... |
peano2b 6973 | A class belongs to omega i... |
nnsuc 6974 | A nonzero natural number i... |
ssnlim 6975 | An ordinal subclass of non... |
omsinds 6976 | Strong (or "total") induct... |
peano1 6977 | Zero is a natural number. ... |
peano2 6978 | The successor of any natur... |
peano3 6979 | The successor of any natur... |
peano4 6980 | Two natural numbers are eq... |
peano5 6981 | The induction postulate: a... |
nn0suc 6982 | A natural number is either... |
find 6983 | The Principle of Finite In... |
finds 6984 | Principle of Finite Induct... |
findsg 6985 | Principle of Finite Induct... |
finds2 6986 | Principle of Finite Induct... |
finds1 6987 | Principle of Finite Induct... |
findes 6988 | Finite induction with expl... |
dmexg 6989 | The domain of a set is a s... |
rnexg 6990 | The range of a set is a se... |
dmex 6991 | The domain of a set is a s... |
rnex 6992 | The range of a set is a se... |
iprc 6993 | The identity function is a... |
resiexg 6994 | The existence of a restric... |
imaexg 6995 | The image of a set is a se... |
imaex 6996 | The image of a set is a se... |
opabex2 6997 | Condition for an operation... |
exse2 6998 | Any set relation is set-li... |
xpexr 6999 | If a Cartesian product is ... |
xpexr2 7000 | If a nonempty Cartesian pr... |
xpexcnv 7001 | A condition where the conv... |
soex 7002 | If the relation in a stric... |
elxp4 7003 | Membership in a Cartesian ... |
elxp5 7004 | Membership in a Cartesian ... |
cnvexg 7005 | The converse of a set is a... |
cnvex 7006 | The converse of a set is a... |
relcnvexb 7007 | A relation is a set iff it... |
f1oexrnex 7008 | If the range of a 1-1 onto... |
f1oexbi 7009 | There is a one-to-one onto... |
coexg 7010 | The composition of two set... |
coex 7011 | The composition of two set... |
funcnvuni 7012 | The union of a chain (with... |
fun11uni 7013 | The union of a chain (with... |
fex2 7014 | A function with bounded do... |
fabexg 7015 | Existence of a set of func... |
fabex 7016 | Existence of a set of func... |
dmfex 7017 | If a mapping is a set, its... |
f1oabexg 7018 | The class of all 1-1-onto ... |
fun11iun 7019 | The union of a chain (with... |
ffoss 7020 | Relationship between a map... |
f11o 7021 | Relationship between one-t... |
resfunexgALT 7022 | Alternate proof of ~ resfu... |
cofunexg 7023 | Existence of a composition... |
cofunex2g 7024 | Existence of a composition... |
fnexALT 7025 | Alternate proof of ~ fnex ... |
funrnex 7026 | If the domain of a functio... |
zfrep6 7027 | A version of the Axiom of ... |
fornex 7028 | If the domain of an onto f... |
f1dmex 7029 | If the codomain of a one-t... |
f1ovv 7030 | The range of a 1-1 onto fu... |
fvclex 7031 | Existence of the class of ... |
fvresex 7032 | Existence of the class of ... |
abrexex 7033 | Existence of a class abstr... |
abrexexg 7034 | Existence of a class abstr... |
iunexg 7035 | The existence of an indexe... |
abrexex2g 7036 | Existence of an existentia... |
opabex3d 7037 | Existence of an ordered pa... |
opabex3 7038 | Existence of an ordered pa... |
iunex 7039 | The existence of an indexe... |
abrexex2 7040 | Existence of an existentia... |
abexssex 7041 | Existence of a class abstr... |
abexex 7042 | A condition where a class ... |
f1oweALT 7043 | Alternate proof of ~ f1owe... |
wemoiso 7044 | Thus, there is at most one... |
wemoiso2 7045 | Thus, there is at most one... |
oprabexd 7046 | Existence of an operator a... |
oprabex 7047 | Existence of an operation ... |
oprabex3 7048 | Existence of an operation ... |
oprabrexex2 7049 | Existence of an existentia... |
ab2rexex 7050 | Existence of a class abstr... |
ab2rexex2 7051 | Existence of an existentia... |
xpexgALT 7052 | Alternate proof of ~ xpexg... |
offval3 7053 | General value of ` ( F oF ... |
offres 7054 | Pointwise combination comm... |
ofmres 7055 | Equivalent expressions for... |
ofmresex 7056 | Existence of a restriction... |
1stval 7061 | The value of the function ... |
2ndval 7062 | The value of the function ... |
1stnpr 7063 | Value of the first-member ... |
2ndnpr 7064 | Value of the second-member... |
1st0 7065 | The value of the first-mem... |
2nd0 7066 | The value of the second-me... |
op1st 7067 | Extract the first member o... |
op2nd 7068 | Extract the second member ... |
op1std 7069 | Extract the first member o... |
op2ndd 7070 | Extract the second member ... |
op1stg 7071 | Extract the first member o... |
op2ndg 7072 | Extract the second member ... |
ot1stg 7073 | Extract the first member o... |
ot2ndg 7074 | Extract the second member ... |
ot3rdg 7075 | Extract the third member o... |
1stval2 7076 | Alternate value of the fun... |
2ndval2 7077 | Alternate value of the fun... |
oteqimp 7078 | The components of an order... |
fo1st 7079 | The ` 1st ` function maps ... |
fo2nd 7080 | The ` 2nd ` function maps ... |
f1stres 7081 | Mapping of a restriction o... |
f2ndres 7082 | Mapping of a restriction o... |
fo1stres 7083 | Onto mapping of a restrict... |
fo2ndres 7084 | Onto mapping of a restrict... |
1st2val 7085 | Value of an alternate defi... |
2nd2val 7086 | Value of an alternate defi... |
1stcof 7087 | Composition of the first m... |
2ndcof 7088 | Composition of the second ... |
xp1st 7089 | Location of the first elem... |
xp2nd 7090 | Location of the second ele... |
elxp6 7091 | Membership in a Cartesian ... |
elxp7 7092 | Membership in a Cartesian ... |
eqopi 7093 | Equality with an ordered p... |
xp2 7094 | Representation of Cartesia... |
unielxp 7095 | The membership relation fo... |
1st2nd2 7096 | Reconstruction of a member... |
1st2ndb 7097 | Reconstruction of an order... |
xpopth 7098 | An ordered pair theorem fo... |
eqop 7099 | Two ways to express equali... |
eqop2 7100 | Two ways to express equali... |
op1steq 7101 | Two ways of expressing tha... |
el2xptp 7102 | A member of a nested Carte... |
el2xptp0 7103 | A member of a nested Carte... |
2nd1st 7104 | Swap the members of an ord... |
1st2nd 7105 | Reconstruction of a member... |
1stdm 7106 | The first ordered pair com... |
2ndrn 7107 | The second ordered pair co... |
1st2ndbr 7108 | Express an element of a re... |
releldm2 7109 | Two ways of expressing mem... |
reldm 7110 | An expression for the doma... |
sbcopeq1a 7111 | Equality theorem for subst... |
csbopeq1a 7112 | Equality theorem for subst... |
dfopab2 7113 | A way to define an ordered... |
dfoprab3s 7114 | A way to define an operati... |
dfoprab3 7115 | Operation class abstractio... |
dfoprab4 7116 | Operation class abstractio... |
dfoprab4f 7117 | Operation class abstractio... |
opiota 7118 | The property of a uniquely... |
dfxp3 7119 | Define the Cartesian produ... |
elopabi 7120 | A consequence of membershi... |
eloprabi 7121 | A consequence of membershi... |
mpt2mptsx 7122 | Express a two-argument fun... |
mpt2mpts 7123 | Express a two-argument fun... |
dmmpt2ssx 7124 | The domain of a mapping is... |
fmpt2x 7125 | Functionality, domain and ... |
fmpt2 7126 | Functionality, domain and ... |
fnmpt2 7127 | Functionality and domain o... |
fnmpt2i 7128 | Functionality and domain o... |
dmmpt2 7129 | Domain of a class given by... |
ovmpt2elrn 7130 | An operation's value belon... |
dmmpt2ga 7131 | Domain of an operation giv... |
dmmpt2g 7132 | Domain of an operation giv... |
mpt2exxg 7133 | Existence of an operation ... |
mpt2exg 7134 | Existence of an operation ... |
mpt2exga 7135 | If the domain of a functio... |
mpt2ex 7136 | If the domain of a functio... |
el2mpt2csbcl 7137 | If the operation value of ... |
el2mpt2cl 7138 | If the operation value of ... |
fnmpt2ovd 7139 | A function with a Cartesia... |
offval22 7140 | The function operation exp... |
brovpreldm 7141 | If a binary relation holds... |
bropopvvv 7142 | If a binary relation holds... |
bropfvvvvlem 7143 | Lemma for ~ bropfvvvv . (... |
bropfvvvv 7144 | If a binary relation holds... |
ovmptss 7145 | If all the values of the m... |
relmpt2opab 7146 | Any function to sets of or... |
fmpt2co 7147 | Composition of two functio... |
oprabco 7148 | Composition of a function ... |
oprab2co 7149 | Composition of operator ab... |
df1st2 7150 | An alternate possible defi... |
df2nd2 7151 | An alternate possible defi... |
1stconst 7152 | The mapping of a restricti... |
2ndconst 7153 | The mapping of a restricti... |
dfmpt2 7154 | Alternate definition for t... |
mpt2sn 7155 | An operation (in maps-to n... |
curry1 7156 | Composition with ` ``' ( 2... |
curry1val 7157 | The value of a curried fun... |
curry1f 7158 | Functionality of a curried... |
curry2 7159 | Composition with ` ``' ( 1... |
curry2f 7160 | Functionality of a curried... |
curry2val 7161 | The value of a curried fun... |
cnvf1olem 7162 | Lemma for ~ cnvf1o . (Con... |
cnvf1o 7163 | Describe a function that m... |
fparlem1 7164 | Lemma for ~ fpar . (Contr... |
fparlem2 7165 | Lemma for ~ fpar . (Contr... |
fparlem3 7166 | Lemma for ~ fpar . (Contr... |
fparlem4 7167 | Lemma for ~ fpar . (Contr... |
fpar 7168 | Merge two functions in par... |
fsplit 7169 | A function that can be use... |
f2ndf 7170 | The ` 2nd ` (second member... |
fo2ndf 7171 | The ` 2nd ` (second member... |
f1o2ndf1 7172 | The ` 2nd ` (second member... |
algrflem 7173 | Lemma for ~ algrf and rela... |
frxp 7174 | A lexicographical ordering... |
xporderlem 7175 | Lemma for lexicographical ... |
poxp 7176 | A lexicographical ordering... |
soxp 7177 | A lexicographical ordering... |
wexp 7178 | A lexicographical ordering... |
fnwelem 7179 | Lemma for ~ fnwe . (Contr... |
fnwe 7180 | A variant on lexicographic... |
fnse 7181 | Condition for the well-ord... |
suppval 7184 | The value of the operation... |
supp0prc 7185 | The support of a class is ... |
suppvalbr 7186 | The value of the operation... |
supp0 7187 | The support of the empty s... |
suppval1 7188 | The value of the operation... |
suppvalfn 7189 | The value of the operation... |
elsuppfn 7190 | An element of the support ... |
cnvimadfsn 7191 | The support of functions "... |
suppimacnvss 7192 | The support of functions "... |
suppimacnv 7193 | Support sets of functions ... |
frnsuppeq 7194 | Two ways of writing the su... |
suppssdm 7195 | The support of a function ... |
suppsnop 7196 | The support of a singleton... |
snopsuppss 7197 | The support of a singleton... |
fvn0elsupp 7198 | If the function value for ... |
fvn0elsuppb 7199 | The function value for a g... |
rexsupp 7200 | Existential quantification... |
ressuppss 7201 | The support of the restric... |
suppun 7202 | The support of a class/fun... |
ressuppssdif 7203 | The support of the restric... |
mptsuppdifd 7204 | The support of a function ... |
mptsuppd 7205 | The support of a function ... |
extmptsuppeq 7206 | The support of an extended... |
suppfnss 7207 | The support of a function ... |
funsssuppss 7208 | The support of a function ... |
fnsuppres 7209 | Two ways to express restri... |
fnsuppeq0 7210 | The support of a function ... |
fczsupp0 7211 | The support of a constant ... |
suppss 7212 | Show that the support of a... |
suppssr 7213 | A function is zero outside... |
suppssov1 7214 | Formula building theorem f... |
suppssof1 7215 | Formula building theorem f... |
suppss2 7216 | Show that the support of a... |
suppsssn 7217 | Show that the support of a... |
suppssfv 7218 | Formula building theorem f... |
suppofss1d 7219 | Condition for the support ... |
suppofss2d 7220 | Condition for the support ... |
supp0cosupp0 7221 | The support of the composi... |
imacosupp 7222 | The image of the support o... |
opeliunxp2f 7223 | Membership in a union of C... |
mpt2xeldm 7224 | If there is an element of ... |
mpt2xneldm 7225 | If the first argument of a... |
mpt2xopn0yelv 7226 | If there is an element of ... |
mpt2xopynvov0g 7227 | If the second argument of ... |
mpt2xopxnop0 7228 | If the first argument of a... |
mpt2xopx0ov0 7229 | If the first argument of a... |
mpt2xopxprcov0 7230 | If the components of the f... |
mpt2xopynvov0 7231 | If the second argument of ... |
mpt2xopoveq 7232 | Value of an operation give... |
mpt2xopovel 7233 | Element of the value of an... |
mpt2xopoveqd 7234 | Value of an operation give... |
brovex 7235 | A binary relation of the v... |
brovmpt2ex 7236 | A binary relation of the v... |
sprmpt2d 7237 | The extension of a binary ... |
tposss 7240 | Subset theorem for transpo... |
tposeq 7241 | Equality theorem for trans... |
tposeqd 7242 | Equality theorem for trans... |
tposssxp 7243 | The transposition is a sub... |
reltpos 7244 | The transposition is a rel... |
brtpos2 7245 | Value of the transposition... |
brtpos0 7246 | The behavior of ` tpos ` w... |
reldmtpos 7247 | Necessary and sufficient c... |
brtpos 7248 | The transposition swaps ar... |
ottpos 7249 | The transposition swaps th... |
relbrtpos 7250 | The transposition swaps ar... |
dmtpos 7251 | The domain of ` tpos F ` w... |
rntpos 7252 | The range of ` tpos F ` wh... |
tposexg 7253 | The transposition of a set... |
ovtpos 7254 | The transposition swaps th... |
tposfun 7255 | The transposition of a fun... |
dftpos2 7256 | Alternate definition of ` ... |
dftpos3 7257 | Alternate definition of ` ... |
dftpos4 7258 | Alternate definition of ` ... |
tpostpos 7259 | Value of the double transp... |
tpostpos2 7260 | Value of the double transp... |
tposfn2 7261 | The domain of a transposit... |
tposfo2 7262 | Condition for a surjective... |
tposf2 7263 | The domain and range of a ... |
tposf12 7264 | Condition for an injective... |
tposf1o2 7265 | Condition of a bijective t... |
tposfo 7266 | The domain and range of a ... |
tposf 7267 | The domain and range of a ... |
tposfn 7268 | Functionality of a transpo... |
tpos0 7269 | Transposition of the empty... |
tposco 7270 | Transposition of a composi... |
tpossym 7271 | Two ways to say a function... |
tposeqi 7272 | Equality theorem for trans... |
tposex 7273 | A transposition is a set. ... |
nftpos 7274 | Hypothesis builder for tra... |
tposoprab 7275 | Transposition of a class o... |
tposmpt2 7276 | Transposition of a two-arg... |
tposconst 7277 | The transposition of a con... |
mpt2curryd 7282 | The currying of an operati... |
mpt2curryvald 7283 | The value of a curried ope... |
fvmpt2curryd 7284 | The value of the value of ... |
pwuninel2 7287 | Direct proof of ~ pwuninel... |
pwuninel 7288 | The power set of the union... |
undefval 7289 | Value of the undefined val... |
undefnel2 7290 | The undefined value genera... |
undefnel 7291 | The undefined value genera... |
undefne0 7292 | The undefined value genera... |
wrecseq123 7295 | General equality theorem f... |
nfwrecs 7296 | Bound-variable hypothesis ... |
wrecseq1 7297 | Equality theorem for the w... |
wrecseq2 7298 | Equality theorem for the w... |
wrecseq3 7299 | Equality theorem for the w... |
wfr3g 7300 | Functions defined by well-... |
wfrlem1 7301 | Lemma for well-founded rec... |
wfrlem2 7302 | Lemma for well-founded rec... |
wfrlem3 7303 | Lemma for well-founded rec... |
wfrlem3a 7304 | Lemma for well-founded rec... |
wfrlem4 7305 | Lemma for well-founded rec... |
wfrlem5 7306 | Lemma for well-founded rec... |
wfrrel 7307 | The well-founded recursion... |
wfrdmss 7308 | The domain of the well-fou... |
wfrlem8 7309 | Lemma for well-founded rec... |
wfrdmcl 7310 | Given ` F = wrecs ( R , A ... |
wfrlem10 7311 | Lemma for well-founded rec... |
wfrfun 7312 | The well-founded function ... |
wfrlem12 7313 | Lemma for well-founded rec... |
wfrlem13 7314 | Lemma for well-founded rec... |
wfrlem14 7315 | Lemma for well-founded rec... |
wfrlem15 7316 | Lemma for well-founded rec... |
wfrlem16 7317 | Lemma for well-founded rec... |
wfrlem17 7318 | Without using ~ ax-rep , s... |
wfr2a 7319 | A weak version of ~ wfr2 w... |
wfr1 7320 | The Principle of Well-Foun... |
wfr2 7321 | The Principle of Well-Foun... |
wfr3 7322 | The principle of Well-Foun... |
iunon 7323 | The indexed union of a set... |
iinon 7324 | The nonempty indexed inter... |
onfununi 7325 | A property of functions on... |
onovuni 7326 | A variant of ~ onfununi fo... |
onoviun 7327 | A variant of ~ onovuni wit... |
onnseq 7328 | There are no length ` _om ... |
dfsmo2 7331 | Alternate definition of a ... |
issmo 7332 | Conditions for which ` A `... |
issmo2 7333 | Alternate definition of a ... |
smoeq 7334 | Equality theorem for stric... |
smodm 7335 | The domain of a strictly m... |
smores 7336 | A strictly monotone functi... |
smores3 7337 | A strictly monotone functi... |
smores2 7338 | A strictly monotone ordina... |
smodm2 7339 | The domain of a strictly m... |
smofvon2 7340 | The function values of a s... |
iordsmo 7341 | The identity relation rest... |
smo0 7342 | The null set is a strictly... |
smofvon 7343 | If ` B ` is a strictly mon... |
smoel 7344 | If ` x ` is less than ` y ... |
smoiun 7345 | The value of a strictly mo... |
smoiso 7346 | If ` F ` is an isomorphism... |
smoel2 7347 | A strictly monotone ordina... |
smo11 7348 | A strictly monotone ordina... |
smoord 7349 | A strictly monotone ordina... |
smoword 7350 | A strictly monotone ordina... |
smogt 7351 | A strictly monotone ordina... |
smorndom 7352 | The range of a strictly mo... |
smoiso2 7353 | The strictly monotone ordi... |
dfrecs3 7356 | The old definition of tran... |
recseq 7357 | Equality theorem for ` rec... |
nfrecs 7358 | Bound-variable hypothesis ... |
tfrlem1 7359 | A technical lemma for tran... |
tfrlem3a 7360 | Lemma for transfinite recu... |
tfrlem3 7361 | Lemma for transfinite recu... |
tfrlem4 7362 | Lemma for transfinite recu... |
tfrlem5 7363 | Lemma for transfinite recu... |
recsfval 7364 | Lemma for transfinite recu... |
tfrlem6 7365 | Lemma for transfinite recu... |
tfrlem7 7366 | Lemma for transfinite recu... |
tfrlem8 7367 | Lemma for transfinite recu... |
tfrlem9 7368 | Lemma for transfinite recu... |
tfrlem9a 7369 | Lemma for transfinite recu... |
tfrlem10 7370 | Lemma for transfinite recu... |
tfrlem11 7371 | Lemma for transfinite recu... |
tfrlem12 7372 | Lemma for transfinite recu... |
tfrlem13 7373 | Lemma for transfinite recu... |
tfrlem14 7374 | Lemma for transfinite recu... |
tfrlem15 7375 | Lemma for transfinite recu... |
tfrlem16 7376 | Lemma for finite recursion... |
tfr1a 7377 | A weak version of ~ tfr1 w... |
tfr2a 7378 | A weak version of ~ tfr2 w... |
tfr2b 7379 | Without assuming ~ ax-rep ... |
tfr1 7380 | Principle of Transfinite R... |
tfr2 7381 | Principle of Transfinite R... |
tfr3 7382 | Principle of Transfinite R... |
tfr1ALT 7383 | Alternate proof of ~ tfr1 ... |
tfr2ALT 7384 | Alternate proof of ~ tfr2 ... |
tfr3ALT 7385 | Alternate proof of ~ tfr3 ... |
recsfnon 7386 | Strong transfinite recursi... |
recsval 7387 | Strong transfinite recursi... |
tz7.44lem1 7388 | ` G ` is a function. Lemm... |
tz7.44-1 7389 | The value of ` F ` at ` (/... |
tz7.44-2 7390 | The value of ` F ` at a su... |
tz7.44-3 7391 | The value of ` F ` at a li... |
rdgeq1 7394 | Equality theorem for the r... |
rdgeq2 7395 | Equality theorem for the r... |
rdgeq12 7396 | Equality theorem for the r... |
nfrdg 7397 | Bound-variable hypothesis ... |
rdglem1 7398 | Lemma used with the recurs... |
rdgfun 7399 | The recursive definition g... |
rdgdmlim 7400 | The domain of the recursiv... |
rdgfnon 7401 | The recursive definition g... |
rdgvalg 7402 | Value of the recursive def... |
rdgval 7403 | Value of the recursive def... |
rdg0 7404 | The initial value of the r... |
rdgseg 7405 | The initial segments of th... |
rdgsucg 7406 | The value of the recursive... |
rdgsuc 7407 | The value of the recursive... |
rdglimg 7408 | The value of the recursive... |
rdglim 7409 | The value of the recursive... |
rdg0g 7410 | The initial value of the r... |
rdgsucmptf 7411 | The value of the recursive... |
rdgsucmptnf 7412 | The value of the recursive... |
rdgsucmpt2 7413 | This version of ~ rdgsucmp... |
rdgsucmpt 7414 | The value of the recursive... |
rdglim2 7415 | The value of the recursive... |
rdglim2a 7416 | The value of the recursive... |
frfnom 7417 | The function generated by ... |
fr0g 7418 | The initial value resultin... |
frsuc 7419 | The successor value result... |
frsucmpt 7420 | The successor value result... |
frsucmptn 7421 | The value of the finite re... |
frsucmpt2 7422 | The successor value result... |
tz7.48lem 7423 | A way of showing an ordina... |
tz7.48-2 7424 | Proposition 7.48(2) of [Ta... |
tz7.48-1 7425 | Proposition 7.48(1) of [Ta... |
tz7.48-3 7426 | Proposition 7.48(3) of [Ta... |
tz7.49 7427 | Proposition 7.49 of [Takeu... |
tz7.49c 7428 | Corollary of Proposition 7... |
seqomlem0 7431 | Lemma for ` seqom ` . Cha... |
seqomlem1 7432 | Lemma for ` seqom ` . The... |
seqomlem2 7433 | Lemma for ` seqom ` . (Co... |
seqomlem3 7434 | Lemma for ` seqom ` . (Co... |
seqomlem4 7435 | Lemma for ` seqom ` . (Co... |
seqomeq12 7436 | Equality theorem for ` seq... |
fnseqom 7437 | An index-aware recursive d... |
seqom0g 7438 | Value of an index-aware re... |
seqomsuc 7439 | Value of an index-aware re... |
1on 7454 | Ordinal 1 is an ordinal nu... |
2on 7455 | Ordinal 2 is an ordinal nu... |
2on0 7456 | Ordinal two is not zero. ... |
3on 7457 | Ordinal 3 is an ordinal nu... |
4on 7458 | Ordinal 3 is an ordinal nu... |
df1o2 7459 | Expanded value of the ordi... |
df2o3 7460 | Expanded value of the ordi... |
df2o2 7461 | Expanded value of the ordi... |
1n0 7462 | Ordinal one is not equal t... |
xp01disj 7463 | Cartesian products with th... |
ordgt0ge1 7464 | Two ways to express that a... |
ordge1n0 7465 | An ordinal greater than or... |
el1o 7466 | Membership in ordinal one.... |
dif1o 7467 | Two ways to say that ` A `... |
ondif1 7468 | Two ways to say that ` A `... |
ondif2 7469 | Two ways to say that ` A `... |
2oconcl 7470 | Closure of the pair swappi... |
0lt1o 7471 | Ordinal zero is less than ... |
dif20el 7472 | An ordinal greater than on... |
0we1 7473 | The empty set is a well-or... |
brwitnlem 7474 | Lemma for relations which ... |
fnoa 7475 | Functionality and domain o... |
fnom 7476 | Functionality and domain o... |
fnoe 7477 | Functionality and domain o... |
oav 7478 | Value of ordinal addition.... |
omv 7479 | Value of ordinal multiplic... |
oe0lem 7480 | A helper lemma for ~ oe0 a... |
oev 7481 | Value of ordinal exponenti... |
oevn0 7482 | Value of ordinal exponenti... |
oa0 7483 | Addition with zero. Propo... |
om0 7484 | Ordinal multiplication wit... |
oe0m 7485 | Ordinal exponentiation wit... |
om0x 7486 | Ordinal multiplication wit... |
oe0m0 7487 | Ordinal exponentiation wit... |
oe0m1 7488 | Ordinal exponentiation wit... |
oe0 7489 | Ordinal exponentiation wit... |
oev2 7490 | Alternate value of ordinal... |
oasuc 7491 | Addition with successor. ... |
oesuclem 7492 | Lemma for ~ oesuc . (Cont... |
omsuc 7493 | Multiplication with succes... |
oesuc 7494 | Ordinal exponentiation wit... |
onasuc 7495 | Addition with successor. ... |
onmsuc 7496 | Multiplication with succes... |
onesuc 7497 | Exponentiation with a succ... |
oa1suc 7498 | Addition with 1 is same as... |
oalim 7499 | Ordinal addition with a li... |
omlim 7500 | Ordinal multiplication wit... |
oelim 7501 | Ordinal exponentiation wit... |
oacl 7502 | Closure law for ordinal ad... |
omcl 7503 | Closure law for ordinal mu... |
oecl 7504 | Closure law for ordinal ex... |
oa0r 7505 | Ordinal addition with zero... |
om0r 7506 | Ordinal multiplication wit... |
o1p1e2 7507 | 1 + 1 = 2 for ordinal numb... |
o2p2e4 7508 | 2 + 2 = 4 for ordinal numb... |
om1 7509 | Ordinal multiplication wit... |
om1r 7510 | Ordinal multiplication wit... |
oe1 7511 | Ordinal exponentiation wit... |
oe1m 7512 | Ordinal exponentiation wit... |
oaordi 7513 | Ordering property of ordin... |
oaord 7514 | Ordering property of ordin... |
oacan 7515 | Left cancellation law for ... |
oaword 7516 | Weak ordering property of ... |
oawordri 7517 | Weak ordering property of ... |
oaord1 7518 | An ordinal is less than it... |
oaword1 7519 | An ordinal is less than or... |
oaword2 7520 | An ordinal is less than or... |
oawordeulem 7521 | Lemma for ~ oawordex . (C... |
oawordeu 7522 | Existence theorem for weak... |
oawordexr 7523 | Existence theorem for weak... |
oawordex 7524 | Existence theorem for weak... |
oaordex 7525 | Existence theorem for orde... |
oa00 7526 | An ordinal sum is zero iff... |
oalimcl 7527 | The ordinal sum with a lim... |
oaass 7528 | Ordinal addition is associ... |
oarec 7529 | Recursive definition of or... |
oaf1o 7530 | Left addition by a constan... |
oacomf1olem 7531 | Lemma for ~ oacomf1o . (C... |
oacomf1o 7532 | Define a bijection from ` ... |
omordi 7533 | Ordering property of ordin... |
omord2 7534 | Ordering property of ordin... |
omord 7535 | Ordering property of ordin... |
omcan 7536 | Left cancellation law for ... |
omword 7537 | Weak ordering property of ... |
omwordi 7538 | Weak ordering property of ... |
omwordri 7539 | Weak ordering property of ... |
omword1 7540 | An ordinal is less than or... |
omword2 7541 | An ordinal is less than or... |
om00 7542 | The product of two ordinal... |
om00el 7543 | The product of two nonzero... |
omordlim 7544 | Ordering involving the pro... |
omlimcl 7545 | The product of any nonzero... |
odi 7546 | Distributive law for ordin... |
omass 7547 | Multiplication of ordinal ... |
oneo 7548 | If an ordinal number is ev... |
omeulem1 7549 | Lemma for ~ omeu : existen... |
omeulem2 7550 | Lemma for ~ omeu : uniquen... |
omopth2 7551 | An ordered pair-like theor... |
omeu 7552 | The division algorithm for... |
oen0 7553 | Ordinal exponentiation wit... |
oeordi 7554 | Ordering law for ordinal e... |
oeord 7555 | Ordering property of ordin... |
oecan 7556 | Left cancellation law for ... |
oeword 7557 | Weak ordering property of ... |
oewordi 7558 | Weak ordering property of ... |
oewordri 7559 | Weak ordering property of ... |
oeworde 7560 | Ordinal exponentiation com... |
oeordsuc 7561 | Ordering property of ordin... |
oelim2 7562 | Ordinal exponentiation wit... |
oeoalem 7563 | Lemma for ~ oeoa . (Contr... |
oeoa 7564 | Sum of exponents law for o... |
oeoelem 7565 | Lemma for ~ oeoe . (Contr... |
oeoe 7566 | Product of exponents law f... |
oelimcl 7567 | The ordinal exponential wi... |
oeeulem 7568 | Lemma for ~ oeeu . (Contr... |
oeeui 7569 | The division algorithm for... |
oeeu 7570 | The division algorithm for... |
nna0 7571 | Addition with zero. Theor... |
nnm0 7572 | Multiplication with zero. ... |
nnasuc 7573 | Addition with successor. ... |
nnmsuc 7574 | Multiplication with succes... |
nnesuc 7575 | Exponentiation with a succ... |
nna0r 7576 | Addition to zero. Remark ... |
nnm0r 7577 | Multiplication with zero. ... |
nnacl 7578 | Closure of addition of nat... |
nnmcl 7579 | Closure of multiplication ... |
nnecl 7580 | Closure of exponentiation ... |
nnacli 7581 | ` _om ` is closed under ad... |
nnmcli 7582 | ` _om ` is closed under mu... |
nnarcl 7583 | Reverse closure law for ad... |
nnacom 7584 | Addition of natural number... |
nnaordi 7585 | Ordering property of addit... |
nnaord 7586 | Ordering property of addit... |
nnaordr 7587 | Ordering property of addit... |
nnawordi 7588 | Adding to both sides of an... |
nnaass 7589 | Addition of natural number... |
nndi 7590 | Distributive law for natur... |
nnmass 7591 | Multiplication of natural ... |
nnmsucr 7592 | Multiplication with succes... |
nnmcom 7593 | Multiplication of natural ... |
nnaword 7594 | Weak ordering property of ... |
nnacan 7595 | Cancellation law for addit... |
nnaword1 7596 | Weak ordering property of ... |
nnaword2 7597 | Weak ordering property of ... |
nnmordi 7598 | Ordering property of multi... |
nnmord 7599 | Ordering property of multi... |
nnmword 7600 | Weak ordering property of ... |
nnmcan 7601 | Cancellation law for multi... |
nnmwordi 7602 | Weak ordering property of ... |
nnmwordri 7603 | Weak ordering property of ... |
nnawordex 7604 | Equivalence for weak order... |
nnaordex 7605 | Equivalence for ordering. ... |
1onn 7606 | One is a natural number. ... |
2onn 7607 | The ordinal 2 is a natural... |
3onn 7608 | The ordinal 3 is a natural... |
4onn 7609 | The ordinal 4 is a natural... |
oaabslem 7610 | Lemma for ~ oaabs . (Cont... |
oaabs 7611 | Ordinal addition absorbs a... |
oaabs2 7612 | The absorption law ~ oaabs... |
omabslem 7613 | Lemma for ~ omabs . (Cont... |
omabs 7614 | Ordinal multiplication is ... |
nnm1 7615 | Multiply an element of ` _... |
nnm2 7616 | Multiply an element of ` _... |
nn2m 7617 | Multiply an element of ` _... |
nnneo 7618 | If a natural number is eve... |
nneob 7619 | A natural number is even i... |
omsmolem 7620 | Lemma for ~ omsmo . (Cont... |
omsmo 7621 | A strictly monotonic ordin... |
omopthlem1 7622 | Lemma for ~ omopthi . (Co... |
omopthlem2 7623 | Lemma for ~ omopthi . (Co... |
omopthi 7624 | An ordered pair theorem fo... |
omopth 7625 | An ordered pair theorem fo... |
dfer2 7630 | Alternate definition of eq... |
dfec2 7632 | Alternate definition of ` ... |
ecexg 7633 | An equivalence class modul... |
ecexr 7634 | A nonempty equivalence cla... |
ereq1 7636 | Equality theorem for equiv... |
ereq2 7637 | Equality theorem for equiv... |
errel 7638 | An equivalence relation is... |
erdm 7639 | The domain of an equivalen... |
ercl 7640 | Elementhood in the field o... |
ersym 7641 | An equivalence relation is... |
ercl2 7642 | Elementhood in the field o... |
ersymb 7643 | An equivalence relation is... |
ertr 7644 | An equivalence relation is... |
ertrd 7645 | A transitivity relation fo... |
ertr2d 7646 | A transitivity relation fo... |
ertr3d 7647 | A transitivity relation fo... |
ertr4d 7648 | A transitivity relation fo... |
erref 7649 | An equivalence relation is... |
ercnv 7650 | The converse of an equival... |
errn 7651 | The range and domain of an... |
erssxp 7652 | An equivalence relation is... |
erex 7653 | An equivalence relation is... |
erexb 7654 | An equivalence relation is... |
iserd 7655 | A reflexive, symmetric, tr... |
iseri 7656 | A reflexive, symmetric, tr... |
iseriALT 7657 | Alternate proof of ~ iseri... |
brdifun 7658 | Evaluate the incomparabili... |
swoer 7659 | Incomparability under a st... |
swoord1 7660 | The incomparability equiva... |
swoord2 7661 | The incomparability equiva... |
swoso 7662 | If the incomparability rel... |
eqerlem 7663 | Lemma for ~ eqer . (Contr... |
eqer 7664 | Equivalence relation invol... |
eqerOLD 7665 | Obsolete proof of ~ eqer a... |
ider 7666 | The identity relation is a... |
0er 7667 | The empty set is an equiva... |
0erOLD 7668 | Obsolete proof of ~ 0er as... |
eceq1 7669 | Equality theorem for equiv... |
eceq1d 7670 | Equality theorem for equiv... |
eceq2 7671 | Equality theorem for equiv... |
elecg 7672 | Membership in an equivalen... |
elec 7673 | Membership in an equivalen... |
relelec 7674 | Membership in an equivalen... |
ecss 7675 | An equivalence class is a ... |
ecdmn0 7676 | A representative of a none... |
ereldm 7677 | Equality of equivalence cl... |
erth 7678 | Basic property of equivale... |
erth2 7679 | Basic property of equivale... |
erthi 7680 | Basic property of equivale... |
erdisj 7681 | Equivalence classes do not... |
ecidsn 7682 | An equivalence class modul... |
qseq1 7683 | Equality theorem for quoti... |
qseq2 7684 | Equality theorem for quoti... |
elqsg 7685 | Closed form of ~ elqs . (... |
elqs 7686 | Membership in a quotient s... |
elqsi 7687 | Membership in a quotient s... |
elqsecl 7688 | Membership in a quotient s... |
ecelqsg 7689 | Membership of an equivalen... |
ecelqsi 7690 | Membership of an equivalen... |
ecopqsi 7691 | "Closure" law for equivale... |
qsexg 7692 | A quotient set exists. (C... |
qsex 7693 | A quotient set exists. (C... |
uniqs 7694 | The union of a quotient se... |
qsss 7695 | A quotient set is a set of... |
uniqs2 7696 | The union of a quotient se... |
snec 7697 | The singleton of an equiva... |
ecqs 7698 | Equivalence class in terms... |
ecid 7699 | A set is equal to its conv... |
qsid 7700 | A set is equal to its quot... |
ectocld 7701 | Implicit substitution of c... |
ectocl 7702 | Implicit substitution of c... |
elqsn0 7703 | A quotient set doesn't con... |
ecelqsdm 7704 | Membership of an equivalen... |
xpider 7705 | A square Cartesian product... |
iiner 7706 | The intersection of a none... |
riiner 7707 | The relative intersection ... |
erinxp 7708 | A restricted equivalence r... |
ecinxp 7709 | Restrict the relation in a... |
qsinxp 7710 | Restrict the equivalence r... |
qsdisj 7711 | Members of a quotient set ... |
qsdisj2 7712 | A quotient set is a disjoi... |
qsel 7713 | If an element of a quotien... |
uniinqs 7714 | Class union distributes ov... |
qliftlem 7715 | ` F ` , a function lift, i... |
qliftrel 7716 | ` F ` , a function lift, i... |
qliftel 7717 | Elementhood in the relatio... |
qliftel1 7718 | Elementhood in the relatio... |
qliftfun 7719 | The function ` F ` is the ... |
qliftfund 7720 | The function ` F ` is the ... |
qliftfuns 7721 | The function ` F ` is the ... |
qliftf 7722 | The domain and range of th... |
qliftval 7723 | The value of the function ... |
ecoptocl 7724 | Implicit substitution of c... |
2ecoptocl 7725 | Implicit substitution of c... |
3ecoptocl 7726 | Implicit substitution of c... |
brecop 7727 | Binary relation on a quoti... |
brecop2 7728 | Binary relation on a quoti... |
eroveu 7729 | Lemma for ~ erov and ~ ero... |
erovlem 7730 | Lemma for ~ erov and ~ ero... |
erov 7731 | The value of an operation ... |
eroprf 7732 | Functionality of an operat... |
erov2 7733 | The value of an operation ... |
eroprf2 7734 | Functionality of an operat... |
ecopoveq 7735 | This is the first of sever... |
ecopovsym 7736 | Assuming the operation ` F... |
ecopovtrn 7737 | Assuming that operation ` ... |
ecopover 7738 | Assuming that operation ` ... |
ecopoverOLD 7739 | Obsolete proof of ~ ecopov... |
eceqoveq 7740 | Equality of equivalence re... |
ecovcom 7741 | Lemma used to transfer a c... |
ecovass 7742 | Lemma used to transfer an ... |
ecovdi 7743 | Lemma used to transfer a d... |
mapprc 7748 | When ` A ` is a proper cla... |
pmex 7749 | The class of all partial f... |
mapex 7750 | The class of all functions... |
fnmap 7751 | Set exponentiation has a u... |
fnpm 7752 | Partial function exponenti... |
reldmmap 7753 | Set exponentiation is a we... |
mapvalg 7754 | The value of set exponenti... |
pmvalg 7755 | The value of the partial m... |
mapval 7756 | The value of set exponenti... |
elmapg 7757 | Membership relation for se... |
elmapd 7758 | Deduction form of ~ elmapg... |
elpmg 7759 | The predicate "is a partia... |
elpm2g 7760 | The predicate "is a partia... |
elpm2r 7761 | Sufficient condition for b... |
elpmi 7762 | A partial function is a fu... |
pmfun 7763 | A partial function is a fu... |
elmapex 7764 | Eliminate antecedent for m... |
elmapi 7765 | A mapping is a function, f... |
elmapfn 7766 | A mapping is a function wi... |
elmapfun 7767 | A mapping is always a func... |
elmapssres 7768 | A restricted mapping is a ... |
fpmg 7769 | A total function is a part... |
pmss12g 7770 | Subset relation for the se... |
pmresg 7771 | Elementhood of a restricte... |
elmap 7772 | Membership relation for se... |
mapval2 7773 | Alternate expression for t... |
elpm 7774 | The predicate "is a partia... |
elpm2 7775 | The predicate "is a partia... |
fpm 7776 | A total function is a part... |
mapsspm 7777 | Set exponentiation is a su... |
pmsspw 7778 | Partial maps are a subset ... |
mapsspw 7779 | Set exponentiation is a su... |
fvmptmap 7780 | Special case of ~ fvmpt fo... |
map0e 7781 | Set exponentiation with an... |
map0b 7782 | Set exponentiation with an... |
map0g 7783 | Set exponentiation is empt... |
map0 7784 | Set exponentiation is empt... |
mapsn 7785 | The value of set exponenti... |
mapss 7786 | Subset inheritance for set... |
fdiagfn 7787 | Functionality of the diago... |
fvdiagfn 7788 | Functionality of the diago... |
mapsnconst 7789 | Every singleton map is a c... |
mapsncnv 7790 | Expression for the inverse... |
mapsnf1o2 7791 | Explicit bijection between... |
mapsnf1o3 7792 | Explicit bijection in the ... |
ralxpmap 7793 | Quantification over functi... |
dfixp 7796 | Eliminate the expression `... |
ixpsnval 7797 | The value of an infinite C... |
elixp2 7798 | Membership in an infinite ... |
fvixp 7799 | Projection of a factor of ... |
ixpfn 7800 | A nuple is a function. (C... |
elixp 7801 | Membership in an infinite ... |
elixpconst 7802 | Membership in an infinite ... |
ixpconstg 7803 | Infinite Cartesian product... |
ixpconst 7804 | Infinite Cartesian product... |
ixpeq1 7805 | Equality theorem for infin... |
ixpeq1d 7806 | Equality theorem for infin... |
ss2ixp 7807 | Subclass theorem for infin... |
ixpeq2 7808 | Equality theorem for infin... |
ixpeq2dva 7809 | Equality theorem for infin... |
ixpeq2dv 7810 | Equality theorem for infin... |
cbvixp 7811 | Change bound variable in a... |
cbvixpv 7812 | Change bound variable in a... |
nfixp 7813 | Bound-variable hypothesis ... |
nfixp1 7814 | The index variable in an i... |
ixpprc 7815 | A cartesian product of pro... |
ixpf 7816 | A member of an infinite Ca... |
uniixp 7817 | The union of an infinite C... |
ixpexg 7818 | The existence of an infini... |
ixpin 7819 | The intersection of two in... |
ixpiin 7820 | The indexed intersection o... |
ixpint 7821 | The intersection of a coll... |
ixp0x 7822 | An infinite Cartesian prod... |
ixpssmap2g 7823 | An infinite Cartesian prod... |
ixpssmapg 7824 | An infinite Cartesian prod... |
0elixp 7825 | Membership of the empty se... |
ixpn0 7826 | The infinite Cartesian pro... |
ixp0 7827 | The infinite Cartesian pro... |
ixpssmap 7828 | An infinite Cartesian prod... |
resixp 7829 | Restriction of an element ... |
undifixp 7830 | Union of two projections o... |
mptelixpg 7831 | Condition for an explicit ... |
resixpfo 7832 | Restriction of elements of... |
elixpsn 7833 | Membership in a class of s... |
ixpsnf1o 7834 | A bijection between a clas... |
mapsnf1o 7835 | A bijection between a set ... |
boxriin 7836 | A rectangular subset of a ... |
boxcutc 7837 | The relative complement of... |
relen 7846 | Equinumerosity is a relati... |
reldom 7847 | Dominance is a relation. ... |
relsdom 7848 | Strict dominance is a rela... |
encv 7849 | If two classes are equinum... |
bren 7850 | Equinumerosity relation. ... |
brdomg 7851 | Dominance relation. (Cont... |
brdomi 7852 | Dominance relation. (Cont... |
brdom 7853 | Dominance relation. (Cont... |
domen 7854 | Dominance in terms of equi... |
domeng 7855 | Dominance in terms of equi... |
ctex 7856 | A countable set is a set. ... |
f1oen3g 7857 | The domain and range of a ... |
f1oen2g 7858 | The domain and range of a ... |
f1dom2g 7859 | The domain of a one-to-one... |
f1oeng 7860 | The domain and range of a ... |
f1domg 7861 | The domain of a one-to-one... |
f1oen 7862 | The domain and range of a ... |
f1dom 7863 | The domain of a one-to-one... |
brsdom 7864 | Strict dominance relation,... |
isfi 7865 | Express " ` A ` is finite.... |
enssdom 7866 | Equinumerosity implies dom... |
dfdom2 7867 | Alternate definition of do... |
endom 7868 | Equinumerosity implies dom... |
sdomdom 7869 | Strict dominance implies d... |
sdomnen 7870 | Strict dominance implies n... |
brdom2 7871 | Dominance in terms of stri... |
bren2 7872 | Equinumerosity expressed i... |
enrefg 7873 | Equinumerosity is reflexiv... |
enref 7874 | Equinumerosity is reflexiv... |
eqeng 7875 | Equality implies equinumer... |
domrefg 7876 | Dominance is reflexive. (... |
en2d 7877 | Equinumerosity inference f... |
en3d 7878 | Equinumerosity inference f... |
en2i 7879 | Equinumerosity inference f... |
en3i 7880 | Equinumerosity inference f... |
dom2lem 7881 | A mapping (first hypothesi... |
dom2d 7882 | A mapping (first hypothesi... |
dom3d 7883 | A mapping (first hypothesi... |
dom2 7884 | A mapping (first hypothesi... |
dom3 7885 | A mapping (first hypothesi... |
idssen 7886 | Equality implies equinumer... |
ssdomg 7887 | A set dominates its subset... |
ener 7888 | Equinumerosity is an equiv... |
enerOLD 7889 | Obsolete proof of ~ ener a... |
ensymb 7890 | Symmetry of equinumerosity... |
ensym 7891 | Symmetry of equinumerosity... |
ensymi 7892 | Symmetry of equinumerosity... |
ensymd 7893 | Symmetry of equinumerosity... |
entr 7894 | Transitivity of equinumero... |
domtr 7895 | Transitivity of dominance ... |
entri 7896 | A chained equinumerosity i... |
entr2i 7897 | A chained equinumerosity i... |
entr3i 7898 | A chained equinumerosity i... |
entr4i 7899 | A chained equinumerosity i... |
endomtr 7900 | Transitivity of equinumero... |
domentr 7901 | Transitivity of dominance ... |
f1imaeng 7902 | A one-to-one function's im... |
f1imaen2g 7903 | A one-to-one function's im... |
f1imaen 7904 | A one-to-one function's im... |
en0 7905 | The empty set is equinumer... |
ensn1 7906 | A singleton is equinumerou... |
ensn1g 7907 | A singleton is equinumerou... |
enpr1g 7908 | ` { A , A } ` has only one... |
en1 7909 | A set is equinumerous to o... |
en1b 7910 | A set is equinumerous to o... |
reuen1 7911 | Two ways to express "exact... |
euen1 7912 | Two ways to express "exact... |
euen1b 7913 | Two ways to express " ` A ... |
en1uniel 7914 | A singleton contains its s... |
2dom 7915 | A set that dominates ordin... |
fundmen 7916 | A function is equinumerous... |
fundmeng 7917 | A function is equinumerous... |
cnven 7918 | A relational set is equinu... |
fndmeng 7919 | A function is equinumerate... |
mapsnen 7920 | Set exponentiation to a si... |
map1 7921 | Set exponentiation: ordina... |
en2sn 7922 | Two singletons are equinum... |
snfi 7923 | A singleton is finite. (C... |
fiprc 7924 | The class of finite sets i... |
unen 7925 | Equinumerosity of union of... |
ssct 7926 | Any subset of a countable ... |
difsnen 7927 | All decrements of a set ar... |
domdifsn 7928 | Dominance over a set with ... |
xpsnen 7929 | A set is equinumerous to i... |
xpsneng 7930 | A set is equinumerous to i... |
xp1en 7931 | One times a cardinal numbe... |
endisj 7932 | Any two sets are equinumer... |
undom 7933 | Dominance law for union. ... |
xpcomf1o 7934 | The canonical bijection fr... |
xpcomco 7935 | Composition with the bijec... |
xpcomen 7936 | Commutative law for equinu... |
xpcomeng 7937 | Commutative law for equinu... |
xpsnen2g 7938 | A set is equinumerous to i... |
xpassen 7939 | Associative law for equinu... |
xpdom2 7940 | Dominance law for Cartesia... |
xpdom2g 7941 | Dominance law for Cartesia... |
xpdom1g 7942 | Dominance law for Cartesia... |
xpdom3 7943 | A set is dominated by its ... |
xpdom1 7944 | Dominance law for Cartesia... |
domunsncan 7945 | A singleton cancellation l... |
omxpenlem 7946 | Lemma for ~ omxpen . (Con... |
omxpen 7947 | The cardinal and ordinal p... |
omf1o 7948 | Construct an explicit bije... |
pw2f1olem 7949 | Lemma for ~ pw2f1o . (Con... |
pw2f1o 7950 | The power set of a set is ... |
pw2eng 7951 | The power set of a set is ... |
pw2en 7952 | The power set of a set is ... |
fopwdom 7953 | Covering implies injection... |
enfixsn 7954 | Given two equipollent sets... |
sbthlem1 7955 | Lemma for ~ sbth . (Contr... |
sbthlem2 7956 | Lemma for ~ sbth . (Contr... |
sbthlem3 7957 | Lemma for ~ sbth . (Contr... |
sbthlem4 7958 | Lemma for ~ sbth . (Contr... |
sbthlem5 7959 | Lemma for ~ sbth . (Contr... |
sbthlem6 7960 | Lemma for ~ sbth . (Contr... |
sbthlem7 7961 | Lemma for ~ sbth . (Contr... |
sbthlem8 7962 | Lemma for ~ sbth . (Contr... |
sbthlem9 7963 | Lemma for ~ sbth . (Contr... |
sbthlem10 7964 | Lemma for ~ sbth . (Contr... |
sbth 7965 | Schroeder-Bernstein Theore... |
sbthb 7966 | Schroeder-Bernstein Theore... |
sbthcl 7967 | Schroeder-Bernstein Theore... |
dfsdom2 7968 | Alternate definition of st... |
brsdom2 7969 | Alternate definition of st... |
sdomnsym 7970 | Strict dominance is asymme... |
domnsym 7971 | Theorem 22(i) of [Suppes] ... |
0domg 7972 | Any set dominates the empt... |
dom0 7973 | A set dominated by the emp... |
0sdomg 7974 | A set strictly dominates t... |
0dom 7975 | Any set dominates the empt... |
0sdom 7976 | A set strictly dominates t... |
sdom0 7977 | The empty set does not str... |
sdomdomtr 7978 | Transitivity of strict dom... |
sdomentr 7979 | Transitivity of strict dom... |
domsdomtr 7980 | Transitivity of dominance ... |
ensdomtr 7981 | Transitivity of equinumero... |
sdomirr 7982 | Strict dominance is irrefl... |
sdomtr 7983 | Strict dominance is transi... |
sdomn2lp 7984 | Strict dominance has no 2-... |
enen1 7985 | Equality-like theorem for ... |
enen2 7986 | Equality-like theorem for ... |
domen1 7987 | Equality-like theorem for ... |
domen2 7988 | Equality-like theorem for ... |
sdomen1 7989 | Equality-like theorem for ... |
sdomen2 7990 | Equality-like theorem for ... |
domtriord 7991 | Dominance is trichotomous ... |
sdomel 7992 | Strict dominance implies o... |
sdomdif 7993 | The difference of a set fr... |
onsdominel 7994 | An ordinal with more eleme... |
domunsn 7995 | Dominance over a set with ... |
fodomr 7996 | There exists a mapping fro... |
pwdom 7997 | Injection of sets implies ... |
canth2 7998 | Cantor's Theorem. No set ... |
canth2g 7999 | Cantor's theorem with the ... |
2pwuninel 8000 | The power set of the power... |
2pwne 8001 | No set equals the power se... |
disjen 8002 | A stronger form of ~ pwuni... |
disjenex 8003 | Existence version of ~ dis... |
domss2 8004 | A corollary of ~ disjenex ... |
domssex2 8005 | A corollary of ~ disjenex ... |
domssex 8006 | Weakening of ~ domssex to ... |
xpf1o 8007 | Construct a bijection on a... |
xpen 8008 | Equinumerosity law for Car... |
mapen 8009 | Two set exponentiations ar... |
mapdom1 8010 | Order-preserving property ... |
mapxpen 8011 | Equinumerosity law for dou... |
xpmapenlem 8012 | Lemma for ~ xpmapen . (Co... |
xpmapen 8013 | Equinumerosity law for set... |
mapunen 8014 | Equinumerosity law for set... |
map2xp 8015 | A cardinal power with expo... |
mapdom2 8016 | Order-preserving property ... |
mapdom3 8017 | Set exponentiation dominat... |
pwen 8018 | If two sets are equinumero... |
ssenen 8019 | Equinumerosity of equinume... |
limenpsi 8020 | A limit ordinal is equinum... |
limensuci 8021 | A limit ordinal is equinum... |
limensuc 8022 | A limit ordinal is equinum... |
infensuc 8023 | Any infinite ordinal is eq... |
phplem1 8024 | Lemma for Pigeonhole Princ... |
phplem2 8025 | Lemma for Pigeonhole Princ... |
phplem3 8026 | Lemma for Pigeonhole Princ... |
phplem4 8027 | Lemma for Pigeonhole Princ... |
nneneq 8028 | Two equinumerous natural n... |
php 8029 | Pigeonhole Principle. A n... |
php2 8030 | Corollary of Pigeonhole Pr... |
php3 8031 | Corollary of Pigeonhole Pr... |
php4 8032 | Corollary of the Pigeonhol... |
php5 8033 | Corollary of the Pigeonhol... |
snnen2o 8034 | A singleton ` { A } ` is n... |
onomeneq 8035 | An ordinal number equinume... |
onfin 8036 | An ordinal number is finit... |
onfin2 8037 | A set is a natural number ... |
nnfi 8038 | Natural numbers are finite... |
nndomo 8039 | Cardinal ordering agrees w... |
nnsdomo 8040 | Cardinal ordering agrees w... |
sucdom2 8041 | Strict dominance of a set ... |
sucdom 8042 | Strict dominance of a set ... |
0sdom1dom 8043 | Strict dominance over zero... |
1sdom2 8044 | Ordinal 1 is strictly domi... |
sdom1 8045 | A set has less than one me... |
modom 8046 | Two ways to express "at mo... |
modom2 8047 | Two ways to express "at mo... |
1sdom 8048 | A set that strictly domina... |
unxpdomlem1 8049 | Lemma for ~ unxpdom . (Tr... |
unxpdomlem2 8050 | Lemma for ~ unxpdom . (Co... |
unxpdomlem3 8051 | Lemma for ~ unxpdom . (Co... |
unxpdom 8052 | Cartesian product dominate... |
unxpdom2 8053 | Corollary of ~ unxpdom . ... |
sucxpdom 8054 | Cartesian product dominate... |
pssinf 8055 | A set equinumerous to a pr... |
fisseneq 8056 | A finite set is equal to i... |
ominf 8057 | The set of natural numbers... |
isinf 8058 | Any set that is not finite... |
fineqvlem 8059 | Lemma for ~ fineqv . (Con... |
fineqv 8060 | If the Axiom of Infinity i... |
enfi 8061 | Equinumerous sets have the... |
enfii 8062 | A set equinumerous to a fi... |
pssnn 8063 | A proper subset of a natur... |
ssnnfi 8064 | A subset of a natural numb... |
ssfi 8065 | A subset of a finite set i... |
domfi 8066 | A set dominated by a finit... |
xpfir 8067 | The components of a nonemp... |
ssfid 8068 | A subset of a finite set i... |
infi 8069 | The intersection of two se... |
rabfi 8070 | A restricted class built f... |
finresfin 8071 | The restriction of a finit... |
f1finf1o 8072 | Any injection from one fin... |
0fin 8073 | The empty set is finite. ... |
nfielex 8074 | If a class is not finite, ... |
en1eqsn 8075 | A set with one element is ... |
en1eqsnbi 8076 | A set containing an elemen... |
diffi 8077 | If ` A ` is finite, ` ( A ... |
dif1en 8078 | If a set ` A ` is equinume... |
enp1ilem 8079 | Lemma for uses of ~ enp1i ... |
enp1i 8080 | Proof induction for ~ en2i... |
en2 8081 | A set equinumerous to ordi... |
en3 8082 | A set equinumerous to ordi... |
en4 8083 | A set equinumerous to ordi... |
findcard 8084 | Schema for induction on th... |
findcard2 8085 | Schema for induction on th... |
findcard2s 8086 | Variation of ~ findcard2 r... |
findcard2d 8087 | Deduction version of ~ fin... |
findcard3 8088 | Schema for strong inductio... |
ac6sfi 8089 | A version of ~ ac6s for fi... |
frfi 8090 | A partial order is well-fo... |
fimax2g 8091 | A finite set has a maximum... |
fimaxg 8092 | A finite set has a maximum... |
fisupg 8093 | Lemma showing existence an... |
wofi 8094 | A total order on a finite ... |
ordunifi 8095 | The maximum of a finite co... |
nnunifi 8096 | The union (supremum) of a ... |
unblem1 8097 | Lemma for ~ unbnn . After... |
unblem2 8098 | Lemma for ~ unbnn . The v... |
unblem3 8099 | Lemma for ~ unbnn . The v... |
unblem4 8100 | Lemma for ~ unbnn . The f... |
unbnn 8101 | Any unbounded subset of na... |
unbnn2 8102 | Version of ~ unbnn that do... |
isfinite2 8103 | Any set strictly dominated... |
nnsdomg 8104 | Omega strictly dominates a... |
isfiniteg 8105 | A set is finite iff it is ... |
infsdomnn 8106 | An infinite set strictly d... |
infn0 8107 | An infinite set is not emp... |
fin2inf 8108 | This (useless) theorem, wh... |
unfilem1 8109 | Lemma for proving that the... |
unfilem2 8110 | Lemma for proving that the... |
unfilem3 8111 | Lemma for proving that the... |
unfi 8112 | The union of two finite se... |
unfir 8113 | If a union is finite, the ... |
unfi2 8114 | The union of two finite se... |
difinf 8115 | An infinite set ` A ` minu... |
xpfi 8116 | The Cartesian product of t... |
3xpfi 8117 | The Cartesian product of t... |
domunfican 8118 | A finite set union cancell... |
infcntss 8119 | Every infinite set has a d... |
prfi 8120 | An unordered pair is finit... |
tpfi 8121 | An unordered triple is fin... |
fiint 8122 | Equivalent ways of stating... |
fnfi 8123 | A version of ~ fnex for fi... |
fodomfi 8124 | An onto function implies d... |
fodomfib 8125 | Equivalence of an onto map... |
fofinf1o 8126 | Any surjection from one fi... |
rneqdmfinf1o 8127 | Any function from a finite... |
fidomdm 8128 | Any finite set dominates i... |
dmfi 8129 | The domain of a finite set... |
fundmfibi 8130 | A function (set) is finite... |
cnvfi 8131 | If a set is finite, its co... |
rnfi 8132 | The range of a finite set ... |
f1dmvrnfibi 8133 | A 1-1 function (class) wit... |
f1vrnfibi 8134 | A 1-1 function (set) is fi... |
fofi 8135 | If a function has a finite... |
f1fi 8136 | If a 1-to-1 function has a... |
iunfi 8137 | The finite union of finite... |
unifi 8138 | The finite union of finite... |
unifi2 8139 | The finite union of finite... |
infssuni 8140 | If an infinite set ` A ` i... |
unirnffid 8141 | The union of the range of ... |
imafi 8142 | Images of finite sets are ... |
pwfilem 8143 | Lemma for ~ pwfi . (Contr... |
pwfi 8144 | The power set of a finite ... |
mapfi 8145 | Set exponentiation of fini... |
ixpfi 8146 | A Cartesian product of fin... |
ixpfi2 8147 | A Cartesian product of fin... |
mptfi 8148 | A finite mapping set is fi... |
abrexfi 8149 | An image set from a finite... |
cnvimamptfin 8150 | A preimage of a mapping wi... |
elfpw 8151 | Membership in a class of f... |
unifpw 8152 | A set is the union of its ... |
f1opwfi 8153 | A one-to-one mapping induc... |
fissuni 8154 | A finite subset of a union... |
fipreima 8155 | Given a finite subset ` A ... |
finsschain 8156 | A finite subset of the uni... |
indexfi 8157 | If for every element of a ... |
relfsupp 8160 | The property of a function... |
relprcnfsupp 8161 | A proper class is never fi... |
isfsupp 8162 | The property of a class to... |
funisfsupp 8163 | The property of a function... |
fsuppimp 8164 | Implications of a class be... |
fsuppimpd 8165 | A finitely supported funct... |
fisuppfi 8166 | A function on a finite set... |
fdmfisuppfi 8167 | The support of a function ... |
fdmfifsupp 8168 | A function with a finite d... |
fsuppmptdm 8169 | A mapping with a finite do... |
fndmfisuppfi 8170 | The support of a function ... |
fndmfifsupp 8171 | A function with a finite d... |
suppeqfsuppbi 8172 | If two functions have the ... |
suppssfifsupp 8173 | If the support of a functi... |
fsuppsssupp 8174 | If the support of a functi... |
fsuppxpfi 8175 | The cartesian product of t... |
fczfsuppd 8176 | A constant function with v... |
fsuppun 8177 | The union of two finitely ... |
fsuppunfi 8178 | The union of the support o... |
fsuppunbi 8179 | If the union of two classe... |
0fsupp 8180 | The empty set is a finitel... |
snopfsupp 8181 | A singleton containing an ... |
funsnfsupp 8182 | Finite support for a funct... |
fsuppres 8183 | The restriction of a finit... |
ressuppfi 8184 | If the support of the rest... |
resfsupp 8185 | If the restriction of a fu... |
resfifsupp 8186 | The restriction of a funct... |
frnfsuppbi 8187 | Two ways of saying that a ... |
fsuppmptif 8188 | A function mapping an argu... |
fsuppcolem 8189 | Lemma for ~ fsuppco . For... |
fsuppco 8190 | The composition of a 1-1 f... |
fsuppco2 8191 | The composition of a funct... |
fsuppcor 8192 | The composition of a funct... |
mapfienlem1 8193 | Lemma 1 for ~ mapfien . (... |
mapfienlem2 8194 | Lemma 2 for ~ mapfien . (... |
mapfienlem3 8195 | Lemma 3 for ~ mapfien . (... |
mapfien 8196 | A bijection of the base se... |
mapfien2 8197 | Equinumerousity relation f... |
sniffsupp 8198 | A function mapping all but... |
fival 8201 | The set of all the finite ... |
elfi 8202 | Specific properties of an ... |
elfi2 8203 | The empty intersection nee... |
elfir 8204 | Sufficient condition for a... |
intrnfi 8205 | Sufficient condition for t... |
iinfi 8206 | An indexed intersection of... |
inelfi 8207 | The intersection of two se... |
ssfii 8208 | Any element of a set ` A `... |
fi0 8209 | The set of finite intersec... |
fieq0 8210 | If ` A ` is not empty, the... |
fiin 8211 | The elements of ` ( fi `` ... |
dffi2 8212 | The set of finite intersec... |
fiss 8213 | Subset relationship for fu... |
inficl 8214 | A set which is closed unde... |
fipwuni 8215 | The set of finite intersec... |
fisn 8216 | A singleton is closed unde... |
fiuni 8217 | The union of the finite in... |
fipwss 8218 | If a set is a family of su... |
elfiun 8219 | A finite intersection of e... |
dffi3 8220 | The set of finite intersec... |
fifo 8221 | Describe a surjection from... |
marypha1lem 8222 | Core induction for Philip ... |
marypha1 8223 | (Philip) Hall's marriage t... |
marypha2lem1 8224 | Lemma for ~ marypha2 . Pr... |
marypha2lem2 8225 | Lemma for ~ marypha2 . Pr... |
marypha2lem3 8226 | Lemma for ~ marypha2 . Pr... |
marypha2lem4 8227 | Lemma for ~ marypha2 . Pr... |
marypha2 8228 | Version of ~ marypha1 usin... |
dfsup2 8233 | Quantifier free definition... |
supeq1 8234 | Equality theorem for supre... |
supeq1d 8235 | Equality deduction for sup... |
supeq1i 8236 | Equality inference for sup... |
supeq2 8237 | Equality theorem for supre... |
supeq3 8238 | Equality theorem for supre... |
supeq123d 8239 | Equality deduction for sup... |
nfsup 8240 | Hypothesis builder for sup... |
supmo 8241 | Any class ` B ` has at mos... |
supexd 8242 | A supremum is a set. (Con... |
supeu 8243 | A supremum is unique. Sim... |
supval2 8244 | Alternate expression for t... |
eqsup 8245 | Sufficient condition for a... |
eqsupd 8246 | Sufficient condition for a... |
supcl 8247 | A supremum belongs to its ... |
supub 8248 | A supremum is an upper bou... |
suplub 8249 | A supremum is the least up... |
suplub2 8250 | Bidirectional form of ~ su... |
supnub 8251 | An upper bound is not less... |
supex 8252 | A supremum is a set. (Con... |
sup00 8253 | The supremum under an empt... |
sup0riota 8254 | The supremum of an empty s... |
sup0 8255 | The supremum of an empty s... |
supmax 8256 | The greatest element of a ... |
fisup2g 8257 | A finite set satisfies the... |
fisupcl 8258 | A nonempty finite set cont... |
supgtoreq 8259 | The supremum of a finite s... |
suppr 8260 | The supremum of a pair. (... |
supsn 8261 | The supremum of a singleto... |
supisolem 8262 | Lemma for ~ supiso . (Con... |
supisoex 8263 | Lemma for ~ supiso . (Con... |
supiso 8264 | Image of a supremum under ... |
infeq1 8265 | Equality theorem for infim... |
infeq1d 8266 | Equality deduction for inf... |
infeq1i 8267 | Equality inference for inf... |
infeq2 8268 | Equality theorem for infim... |
infeq3 8269 | Equality theorem for infim... |
infeq123d 8270 | Equality deduction for inf... |
nfinf 8271 | Hypothesis builder for inf... |
infexd 8272 | An infimum is a set. (Con... |
eqinf 8273 | Sufficient condition for a... |
eqinfd 8274 | Sufficient condition for a... |
infval 8275 | Alternate expression for t... |
infcllem 8276 | Lemma for ~ infcl , ~ infl... |
infcl 8277 | An infimum belongs to its ... |
inflb 8278 | An infimum is a lower boun... |
infglb 8279 | An infimum is the greatest... |
infglbb 8280 | Bidirectional form of ~ in... |
infnlb 8281 | A lower bound is not great... |
infex 8282 | An infimum is a set. (Con... |
infmin 8283 | The smallest element of a ... |
infmo 8284 | Any class ` B ` has at mos... |
infeu 8285 | An infimum is unique. (Co... |
fimin2g 8286 | A finite set has a minimum... |
fiming 8287 | A finite set has a minimum... |
fiinfg 8288 | Lemma showing existence an... |
fiinf2g 8289 | A finite set satisfies the... |
fiinfcl 8290 | A nonempty finite set cont... |
infltoreq 8291 | The infimum of a finite se... |
infpr 8292 | The infimum of a pair. (C... |
infsn 8293 | The infimum of a singleton... |
inf00 8294 | The infimum regarding an e... |
infempty 8295 | The infimum of an empty se... |
infiso 8296 | Image of an infimum under ... |
dfoi 8299 | Rewrite ~ df-oi with abbre... |
oieq1 8300 | Equality theorem for ordin... |
oieq2 8301 | Equality theorem for ordin... |
nfoi 8302 | Hypothesis builder for ord... |
ordiso2 8303 | Generalize ~ ordiso to pro... |
ordiso 8304 | Order-isomorphic ordinal n... |
ordtypecbv 8305 | Lemma for ~ ordtype . (Co... |
ordtypelem1 8306 | Lemma for ~ ordtype . (Co... |
ordtypelem2 8307 | Lemma for ~ ordtype . (Co... |
ordtypelem3 8308 | Lemma for ~ ordtype . (Co... |
ordtypelem4 8309 | Lemma for ~ ordtype . (Co... |
ordtypelem5 8310 | Lemma for ~ ordtype . (Co... |
ordtypelem6 8311 | Lemma for ~ ordtype . (Co... |
ordtypelem7 8312 | Lemma for ~ ordtype . ` ra... |
ordtypelem8 8313 | Lemma for ~ ordtype . (Co... |
ordtypelem9 8314 | Lemma for ~ ordtype . Eit... |
ordtypelem10 8315 | Lemma for ~ ordtype . Usi... |
oi0 8316 | Definition of the ordinal ... |
oicl 8317 | The order type of the well... |
oif 8318 | The order isomorphism of t... |
oiiso2 8319 | The order isomorphism of t... |
ordtype 8320 | For any set-like well-orde... |
oiiniseg 8321 | ` ran F ` is an initial se... |
ordtype2 8322 | For any set-like well-orde... |
oiexg 8323 | The order isomorphism on a... |
oion 8324 | The order type of the well... |
oiiso 8325 | The order isomorphism of t... |
oien 8326 | The order type of a well-o... |
oieu 8327 | Uniqueness of the unique o... |
oismo 8328 | When ` A ` is a subclass o... |
oiid 8329 | The order type of an ordin... |
hartogslem1 8330 | Lemma for ~ hartogs . (Co... |
hartogslem2 8331 | Lemma for ~ hartogs . (Co... |
hartogs 8332 | Given any set, the Hartogs... |
wofib 8333 | The only sets which are we... |
wemaplem1 8334 | Value of the lexicographic... |
wemaplem2 8335 | Lemma for ~ wemapso . Tra... |
wemaplem3 8336 | Lemma for ~ wemapso . Tra... |
wemappo 8337 | Construct lexicographic or... |
wemapsolem 8338 | Lemma for ~ wemapso . (Co... |
wemapso 8339 | Construct lexicographic or... |
wemapso2lem 8340 | Lemma for ~ wemapso2 . (C... |
wemapso2 8341 | An alternative to having a... |
card2on 8342 | Proof that the alternate d... |
card2inf 8343 | The definition ~ cardval2 ... |
harf 8348 | Functionality of the Harto... |
harcl 8349 | Closure of the Hartogs fun... |
harval 8350 | Function value of the Hart... |
elharval 8351 | The Hartogs number of a se... |
harndom 8352 | The Hartogs number of a se... |
harword 8353 | Weak ordering property of ... |
relwdom 8354 | Weak dominance is a relati... |
brwdom 8355 | Property of weak dominance... |
brwdomi 8356 | Property of weak dominance... |
brwdomn0 8357 | Weak dominance over nonemp... |
0wdom 8358 | Any set weakly dominates t... |
fowdom 8359 | An onto function implies w... |
wdomref 8360 | Reflexivity of weak domina... |
brwdom2 8361 | Alternate characterization... |
domwdom 8362 | Weak dominance is implied ... |
wdomtr 8363 | Transitivity of weak domin... |
wdomen1 8364 | Equality-like theorem for ... |
wdomen2 8365 | Equality-like theorem for ... |
wdompwdom 8366 | Weak dominance strengthens... |
canthwdom 8367 | Cantor's Theorem, stated u... |
wdom2d 8368 | Deduce weak dominance from... |
wdomd 8369 | Deduce weak dominance from... |
brwdom3 8370 | Condition for weak dominan... |
brwdom3i 8371 | Weak dominance implies exi... |
unwdomg 8372 | Weak dominance of a (disjo... |
xpwdomg 8373 | Weak dominance of a Cartes... |
wdomima2g 8374 | A set is weakly dominant o... |
wdomimag 8375 | A set is weakly dominant o... |
unxpwdom2 8376 | Lemma for ~ unxpwdom . (C... |
unxpwdom 8377 | If a Cartesian product is ... |
harwdom 8378 | The Hartogs function is we... |
ixpiunwdom 8379 | Describe an onto function ... |
axreg2 8381 | Axiom of Regularity expres... |
zfregcl 8382 | The Axiom of Regularity wi... |
zfreg 8383 | The Axiom of Regularity us... |
zfregclOLD 8384 | Obsolete version of ~ zfre... |
zfregOLD 8385 | Obsolete version of ~ zfre... |
zfreg2OLD 8386 | Alternate version of ~ zfr... |
elirrv 8387 | The membership relation is... |
elirr 8388 | No class is a member of it... |
sucprcreg 8389 | A class is equal to its su... |
ruv 8390 | The Russell class is equal... |
ruALT 8391 | Alternate proof of ~ ru , ... |
zfregfr 8392 | The epsilon relation is we... |
en2lp 8393 | No class has 2-cycle membe... |
en3lplem1 8394 | Lemma for ~ en3lp . (Cont... |
en3lplem2 8395 | Lemma for ~ en3lp . (Cont... |
en3lp 8396 | No class has 3-cycle membe... |
preleq 8397 | Equality of two unordered ... |
opthreg 8398 | Theorem for alternate repr... |
suc11reg 8399 | The successor operation be... |
dford2 8400 | Assuming ~ ax-reg , an ord... |
inf0 8401 | Our Axiom of Infinity deri... |
inf1 8402 | Variation of Axiom of Infi... |
inf2 8403 | Variation of Axiom of Infi... |
inf3lema 8404 | Lemma for our Axiom of Inf... |
inf3lemb 8405 | Lemma for our Axiom of Inf... |
inf3lemc 8406 | Lemma for our Axiom of Inf... |
inf3lemd 8407 | Lemma for our Axiom of Inf... |
inf3lem1 8408 | Lemma for our Axiom of Inf... |
inf3lem2 8409 | Lemma for our Axiom of Inf... |
inf3lem3 8410 | Lemma for our Axiom of Inf... |
inf3lem4 8411 | Lemma for our Axiom of Inf... |
inf3lem5 8412 | Lemma for our Axiom of Inf... |
inf3lem6 8413 | Lemma for our Axiom of Inf... |
inf3lem7 8414 | Lemma for our Axiom of Inf... |
inf3 8415 | Our Axiom of Infinity ~ ax... |
infeq5i 8416 | Half of ~ infeq5 . (Contr... |
infeq5 8417 | The statement "there exist... |
zfinf 8419 | Axiom of Infinity expresse... |
axinf2 8420 | A standard version of Axio... |
zfinf2 8422 | A standard version of the ... |
omex 8423 | The existence of omega (th... |
axinf 8424 | The first version of the A... |
inf5 8425 | The statement "there exist... |
omelon 8426 | Omega is an ordinal number... |
dfom3 8427 | The class of natural numbe... |
elom3 8428 | A simplification of ~ elom... |
dfom4 8429 | A simplification of ~ df-o... |
dfom5 8430 | ` _om ` is the smallest li... |
oancom 8431 | Ordinal addition is not co... |
isfinite 8432 | A set is finite iff it is ... |
fict 8433 | A finite set is countable ... |
nnsdom 8434 | A natural number is strict... |
omenps 8435 | Omega is equinumerous to a... |
omensuc 8436 | The set of natural numbers... |
infdifsn 8437 | Removing a singleton from ... |
infdiffi 8438 | Removing a finite set from... |
unbnn3 8439 | Any unbounded subset of na... |
noinfep 8440 | Using the Axiom of Regular... |
cantnffval 8443 | The value of the Cantor no... |
cantnfdm 8444 | The domain of the Cantor n... |
cantnfvalf 8445 | Lemma for ~ cantnf . The ... |
cantnfs 8446 | Elementhood in the set of ... |
cantnfcl 8447 | Basic properties of the or... |
cantnfval 8448 | The value of the Cantor no... |
cantnfval2 8449 | Alternate expression for t... |
cantnfsuc 8450 | The value of the recursive... |
cantnfle 8451 | A lower bound on the ` CNF... |
cantnflt 8452 | An upper bound on the part... |
cantnflt2 8453 | An upper bound on the ` CN... |
cantnff 8454 | The ` CNF ` function is a ... |
cantnf0 8455 | The value of the zero func... |
cantnfrescl 8456 | A function is finitely sup... |
cantnfres 8457 | The ` CNF ` function respe... |
cantnfp1lem1 8458 | Lemma for ~ cantnfp1 . (C... |
cantnfp1lem2 8459 | Lemma for ~ cantnfp1 . (C... |
cantnfp1lem3 8460 | Lemma for ~ cantnfp1 . (C... |
cantnfp1 8461 | If ` F ` is created by add... |
oemapso 8462 | The relation ` T ` is a st... |
oemapval 8463 | Value of the relation ` T ... |
oemapvali 8464 | If ` F < G ` , then there ... |
cantnflem1a 8465 | Lemma for ~ cantnf . (Con... |
cantnflem1b 8466 | Lemma for ~ cantnf . (Con... |
cantnflem1c 8467 | Lemma for ~ cantnf . (Con... |
cantnflem1d 8468 | Lemma for ~ cantnf . (Con... |
cantnflem1 8469 | Lemma for ~ cantnf . This... |
cantnflem2 8470 | Lemma for ~ cantnf . (Con... |
cantnflem3 8471 | Lemma for ~ cantnf . Here... |
cantnflem4 8472 | Lemma for ~ cantnf . Comp... |
cantnf 8473 | The Cantor Normal Form the... |
oemapwe 8474 | The lexicographic order on... |
cantnffval2 8475 | An alternate definition of... |
cantnff1o 8476 | Simplify the isomorphism o... |
wemapwe 8477 | Construct lexicographic or... |
oef1o 8478 | A bijection of the base se... |
cnfcomlem 8479 | Lemma for ~ cnfcom . (Con... |
cnfcom 8480 | Any ordinal ` B ` is equin... |
cnfcom2lem 8481 | Lemma for ~ cnfcom2 . (Co... |
cnfcom2 8482 | Any nonzero ordinal ` B ` ... |
cnfcom3lem 8483 | Lemma for ~ cnfcom3 . (Co... |
cnfcom3 8484 | Any infinite ordinal ` B `... |
cnfcom3clem 8485 | Lemma for ~ cnfcom3c . (C... |
cnfcom3c 8486 | Wrap the construction of ~... |
trcl 8487 | For any set ` A ` , show t... |
tz9.1 8488 | Every set has a transitive... |
tz9.1c 8489 | Alternate expression for t... |
epfrs 8490 | The strong form of the Axi... |
zfregs 8491 | The strong form of the Axi... |
zfregs2 8492 | Alternate strong form of t... |
setind 8493 | Set (epsilon) induction. ... |
setind2 8494 | Set (epsilon) induction, s... |
tcvalg 8497 | Value of the transitive cl... |
tcid 8498 | Defining property of the t... |
tctr 8499 | Defining property of the t... |
tcmin 8500 | Defining property of the t... |
tc2 8501 | A variant of the definitio... |
tcsni 8502 | The transitive closure of ... |
tcss 8503 | The transitive closure fun... |
tcel 8504 | The transitive closure fun... |
tcidm 8505 | The transitive closure fun... |
tc0 8506 | The transitive closure of ... |
tc00 8507 | The transitive closure is ... |
r1funlim 8512 | The cumulative hierarchy o... |
r1fnon 8513 | The cumulative hierarchy o... |
r10 8514 | Value of the cumulative hi... |
r1sucg 8515 | Value of the cumulative hi... |
r1suc 8516 | Value of the cumulative hi... |
r1limg 8517 | Value of the cumulative hi... |
r1lim 8518 | Value of the cumulative hi... |
r1fin 8519 | The first ` _om ` levels o... |
r1sdom 8520 | Each stage in the cumulati... |
r111 8521 | The cumulative hierarchy i... |
r1tr 8522 | The cumulative hierarchy o... |
r1tr2 8523 | The union of a cumulative ... |
r1ordg 8524 | Ordering relation for the ... |
r1ord3g 8525 | Ordering relation for the ... |
r1ord 8526 | Ordering relation for the ... |
r1ord2 8527 | Ordering relation for the ... |
r1ord3 8528 | Ordering relation for the ... |
r1sssuc 8529 | The value of the cumulativ... |
r1pwss 8530 | Each set of the cumulative... |
r1sscl 8531 | Each set of the cumulative... |
r1val1 8532 | The value of the cumulativ... |
tz9.12lem1 8533 | Lemma for ~ tz9.12 . (Con... |
tz9.12lem2 8534 | Lemma for ~ tz9.12 . (Con... |
tz9.12lem3 8535 | Lemma for ~ tz9.12 . (Con... |
tz9.12 8536 | A set is well-founded if a... |
tz9.13 8537 | Every set is well-founded,... |
tz9.13g 8538 | Every set is well-founded,... |
rankwflemb 8539 | Two ways of saying a set i... |
rankf 8540 | The domain and range of th... |
rankon 8541 | The rank of a set is an or... |
r1elwf 8542 | Any member of the cumulati... |
rankvalb 8543 | Value of the rank function... |
rankr1ai 8544 | One direction of ~ rankr1a... |
rankvaln 8545 | Value of the rank function... |
rankidb 8546 | Identity law for the rank ... |
rankdmr1 8547 | A rank is a member of the ... |
rankr1ag 8548 | A version of ~ rankr1a tha... |
rankr1bg 8549 | A relationship between ran... |
r1rankidb 8550 | Any set is a subset of the... |
r1elssi 8551 | The range of the ` R1 ` fu... |
r1elss 8552 | The range of the ` R1 ` fu... |
pwwf 8553 | A power set is well-founde... |
sswf 8554 | A subset of a well-founded... |
snwf 8555 | A singleton is well-founde... |
unwf 8556 | A binary union is well-fou... |
prwf 8557 | An unordered pair is well-... |
opwf 8558 | An ordered pair is well-fo... |
unir1 8559 | The cumulative hierarchy o... |
jech9.3 8560 | Every set belongs to some ... |
rankwflem 8561 | Every set is well-founded,... |
rankval 8562 | Value of the rank function... |
rankvalg 8563 | Value of the rank function... |
rankval2 8564 | Value of an alternate defi... |
uniwf 8565 | A union is well-founded if... |
rankr1clem 8566 | Lemma for ~ rankr1c . (Co... |
rankr1c 8567 | A relationship between the... |
rankidn 8568 | A relationship between the... |
rankpwi 8569 | The rank of a power set. ... |
rankelb 8570 | The membership relation is... |
wfelirr 8571 | A well-founded set is not ... |
rankval3b 8572 | The value of the rank func... |
ranksnb 8573 | The rank of a singleton. ... |
rankonidlem 8574 | Lemma for ~ rankonid . (C... |
rankonid 8575 | The rank of an ordinal num... |
onwf 8576 | The ordinals are all well-... |
onssr1 8577 | Initial segments of the or... |
rankr1g 8578 | A relationship between the... |
rankid 8579 | Identity law for the rank ... |
rankr1 8580 | A relationship between the... |
ssrankr1 8581 | A relationship between an ... |
rankr1a 8582 | A relationship between ran... |
r1val2 8583 | The value of the cumulativ... |
r1val3 8584 | The value of the cumulativ... |
rankel 8585 | The membership relation is... |
rankval3 8586 | The value of the rank func... |
bndrank 8587 | Any class whose elements h... |
unbndrank 8588 | The elements of a proper c... |
rankpw 8589 | The rank of a power set. ... |
ranklim 8590 | The rank of a set belongs ... |
r1pw 8591 | A stronger property of ` R... |
r1pwALT 8592 | Alternate shorter proof of... |
r1pwcl 8593 | The cumulative hierarchy o... |
rankssb 8594 | The subset relation is inh... |
rankss 8595 | The subset relation is inh... |
rankunb 8596 | The rank of the union of t... |
rankprb 8597 | The rank of an unordered p... |
rankopb 8598 | The rank of an ordered pai... |
rankuni2b 8599 | The value of the rank func... |
ranksn 8600 | The rank of a singleton. ... |
rankuni2 8601 | The rank of a union. Part... |
rankun 8602 | The rank of the union of t... |
rankpr 8603 | The rank of an unordered p... |
rankop 8604 | The rank of an ordered pai... |
r1rankid 8605 | Any set is a subset of the... |
rankeq0b 8606 | A set is empty iff its ran... |
rankeq0 8607 | A set is empty iff its ran... |
rankr1id 8608 | The rank of the hierarchy ... |
rankuni 8609 | The rank of a union. Part... |
rankr1b 8610 | A relationship between ran... |
ranksuc 8611 | The rank of a successor. ... |
rankuniss 8612 | Upper bound of the rank of... |
rankval4 8613 | The rank of a set is the s... |
rankbnd 8614 | The rank of a set is bound... |
rankbnd2 8615 | The rank of a set is bound... |
rankc1 8616 | A relationship that can be... |
rankc2 8617 | A relationship that can be... |
rankelun 8618 | Rank membership is inherit... |
rankelpr 8619 | Rank membership is inherit... |
rankelop 8620 | Rank membership is inherit... |
rankxpl 8621 | A lower bound on the rank ... |
rankxpu 8622 | An upper bound on the rank... |
rankfu 8623 | An upper bound on the rank... |
rankmapu 8624 | An upper bound on the rank... |
rankxplim 8625 | The rank of a Cartesian pr... |
rankxplim2 8626 | If the rank of a Cartesian... |
rankxplim3 8627 | The rank of a Cartesian pr... |
rankxpsuc 8628 | The rank of a Cartesian pr... |
tcwf 8629 | The transitive closure fun... |
tcrank 8630 | This theorem expresses two... |
scottex 8631 | Scott's trick collects all... |
scott0 8632 | Scott's trick collects all... |
scottexs 8633 | Theorem scheme version of ... |
scott0s 8634 | Theorem scheme version of ... |
cplem1 8635 | Lemma for the Collection P... |
cplem2 8636 | -Lemma for the Collection ... |
cp 8637 | Collection Principle. Thi... |
bnd 8638 | A very strong generalizati... |
bnd2 8639 | A variant of the Boundedne... |
kardex 8640 | The collection of all sets... |
karden 8641 | If we allow the Axiom of R... |
htalem 8642 | Lemma for defining an emul... |
hta 8643 | A ZFC emulation of Hilbert... |
cardf2 8652 | The cardinality function i... |
cardon 8653 | The cardinal number of a s... |
isnum2 8654 | A way to express well-orde... |
isnumi 8655 | A set equinumerous to an o... |
ennum 8656 | Equinumerous sets are equi... |
finnum 8657 | Every finite set is numera... |
onenon 8658 | Every ordinal number is nu... |
tskwe 8659 | A Tarski set is well-order... |
xpnum 8660 | The cartesian product of n... |
cardval3 8661 | An alternate definition of... |
cardid2 8662 | Any numerable set is equin... |
isnum3 8663 | A set is numerable iff it ... |
oncardval 8664 | The value of the cardinal ... |
oncardid 8665 | Any ordinal number is equi... |
cardonle 8666 | The cardinal of an ordinal... |
card0 8667 | The cardinality of the emp... |
cardidm 8668 | The cardinality function i... |
oncard 8669 | A set is a cardinal number... |
ficardom 8670 | The cardinal number of a f... |
ficardid 8671 | A finite set is equinumero... |
cardnn 8672 | The cardinality of a natur... |
cardnueq0 8673 | The empty set is the only ... |
cardne 8674 | No member of a cardinal nu... |
carden2a 8675 | If two sets have equal non... |
carden2b 8676 | If two sets are equinumero... |
card1 8677 | A set has cardinality one ... |
cardsn 8678 | A singleton has cardinalit... |
carddomi2 8679 | Two sets have the dominanc... |
sdomsdomcardi 8680 | A set strictly dominates i... |
cardlim 8681 | An infinite cardinal is a ... |
cardsdomelir 8682 | A cardinal strictly domina... |
cardsdomel 8683 | A cardinal strictly domina... |
iscard 8684 | Two ways to express the pr... |
iscard2 8685 | Two ways to express the pr... |
carddom2 8686 | Two numerable sets have th... |
harcard 8687 | The class of ordinal numbe... |
cardprclem 8688 | Lemma for ~ cardprc . (Co... |
cardprc 8689 | The class of all cardinal ... |
carduni 8690 | The union of a set of card... |
cardiun 8691 | The indexed union of a set... |
cardennn 8692 | If ` A ` is equinumerous t... |
cardsucinf 8693 | The cardinality of the suc... |
cardsucnn 8694 | The cardinality of the suc... |
cardom 8695 | The set of natural numbers... |
carden2 8696 | Two numerable sets are equ... |
cardsdom2 8697 | A numerable set is strictl... |
domtri2 8698 | Trichotomy of dominance fo... |
nnsdomel 8699 | Strict dominance and eleme... |
cardval2 8700 | An alternate version of th... |
isinffi 8701 | An infinite set contains s... |
fidomtri 8702 | Trichotomy of dominance wi... |
fidomtri2 8703 | Trichotomy of dominance wi... |
harsdom 8704 | The Hartogs number of a we... |
onsdom 8705 | Any well-orderable set is ... |
harval2 8706 | An alternate expression fo... |
cardmin2 8707 | The smallest ordinal that ... |
pm54.43lem 8708 | In Theorem *54.43 of [Whit... |
pm54.43 8709 | Theorem *54.43 of [Whitehe... |
pr2nelem 8710 | Lemma for ~ pr2ne . (Cont... |
pr2ne 8711 | If an unordered pair has t... |
prdom2 8712 | An unordered pair has at m... |
en2eqpr 8713 | Building a set with two el... |
en2eleq 8714 | Express a set of pair card... |
en2other2 8715 | Taking the other element t... |
dif1card 8716 | The cardinality of a nonem... |
leweon 8717 | Lexicographical order is a... |
r0weon 8718 | A set-like well-ordering o... |
infxpenlem 8719 | Lemma for ~ infxpen . (Co... |
infxpen 8720 | Every infinite ordinal is ... |
xpomen 8721 | The Cartesian product of o... |
xpct 8722 | The cartesian product of t... |
infxpidm2 8723 | The Cartesian product of a... |
infxpenc 8724 | A canonical version of ~ i... |
infxpenc2lem1 8725 | Lemma for ~ infxpenc2 . (... |
infxpenc2lem2 8726 | Lemma for ~ infxpenc2 . (... |
infxpenc2lem3 8727 | Lemma for ~ infxpenc2 . (... |
infxpenc2 8728 | Existence form of ~ infxpe... |
iunmapdisj 8729 | The union ` U_ n e. C ( A ... |
fseqenlem1 8730 | Lemma for ~ fseqen . (Con... |
fseqenlem2 8731 | Lemma for ~ fseqen . (Con... |
fseqdom 8732 | One half of ~ fseqen . (C... |
fseqen 8733 | A set that is equinumerous... |
infpwfidom 8734 | The collection of finite s... |
dfac8alem 8735 | Lemma for ~ dfac8a . If t... |
dfac8a 8736 | Numeration theorem: every ... |
dfac8b 8737 | The well-ordering theorem:... |
dfac8clem 8738 | Lemma for ~ dfac8c . (Con... |
dfac8c 8739 | If the union of a set is w... |
ac10ct 8740 | A proof of the Well orderi... |
ween 8741 | A set is numerable iff it ... |
ac5num 8742 | A version of ~ ac5b with t... |
ondomen 8743 | If a set is dominated by a... |
numdom 8744 | A set dominated by a numer... |
ssnum 8745 | A subset of a numerable se... |
onssnum 8746 | All subsets of the ordinal... |
indcardi 8747 | Indirect strong induction ... |
acnrcl 8748 | Reverse closure for the ch... |
acneq 8749 | Equality theorem for the c... |
isacn 8750 | The property of being a ch... |
acni 8751 | The property of being a ch... |
acni2 8752 | The property of being a ch... |
acni3 8753 | The property of being a ch... |
acnlem 8754 | Construct a mapping satisf... |
numacn 8755 | A well-orderable set has c... |
finacn 8756 | Every set has finite choic... |
acndom 8757 | A set with long choice seq... |
acnnum 8758 | A set ` X ` which has choi... |
acnen 8759 | The class of choice sets o... |
acndom2 8760 | A set smaller than one wit... |
acnen2 8761 | The class of sets with cho... |
fodomacn 8762 | A version of ~ fodom that ... |
fodomnum 8763 | A version of ~ fodom that ... |
fonum 8764 | A surjection maps numerabl... |
numwdom 8765 | A surjection maps numerabl... |
fodomfi2 8766 | Onto functions define domi... |
wdomfil 8767 | Weak dominance agrees with... |
infpwfien 8768 | Any infinite well-orderabl... |
inffien 8769 | The set of finite intersec... |
wdomnumr 8770 | Weak dominance agrees with... |
alephfnon 8771 | The aleph function is a fu... |
aleph0 8772 | The first infinite cardina... |
alephlim 8773 | Value of the aleph functio... |
alephsuc 8774 | Value of the aleph functio... |
alephon 8775 | An aleph is an ordinal num... |
alephcard 8776 | Every aleph is a cardinal ... |
alephnbtwn 8777 | No cardinal can be sandwic... |
alephnbtwn2 8778 | No set has equinumerosity ... |
alephordilem1 8779 | Lemma for ~ alephordi . (... |
alephordi 8780 | Strict ordering property o... |
alephord 8781 | Ordering property of the a... |
alephord2 8782 | Ordering property of the a... |
alephord2i 8783 | Ordering property of the a... |
alephord3 8784 | Ordering property of the a... |
alephsucdom 8785 | A set dominated by an alep... |
alephsuc2 8786 | An alternate representatio... |
alephdom 8787 | Relationship between inclu... |
alephgeom 8788 | Every aleph is greater tha... |
alephislim 8789 | Every aleph is a limit ord... |
aleph11 8790 | The aleph function is one-... |
alephf1 8791 | The aleph function is a on... |
alephsdom 8792 | If an ordinal is smaller t... |
alephdom2 8793 | A dominated initial ordina... |
alephle 8794 | The argument of the aleph ... |
cardaleph 8795 | Given any transfinite card... |
cardalephex 8796 | Every transfinite cardinal... |
infenaleph 8797 | An infinite numerable set ... |
isinfcard 8798 | Two ways to express the pr... |
iscard3 8799 | Two ways to express the pr... |
cardnum 8800 | Two ways to express the cl... |
alephinit 8801 | An infinite initial ordina... |
carduniima 8802 | The union of the image of ... |
cardinfima 8803 | If a mapping to cardinals ... |
alephiso 8804 | Aleph is an order isomorph... |
alephprc 8805 | The class of all transfini... |
alephsson 8806 | The class of transfinite c... |
unialeph 8807 | The union of the class of ... |
alephsmo 8808 | The aleph function is stri... |
alephf1ALT 8809 | Alternate proof of ~ aleph... |
alephfplem1 8810 | Lemma for ~ alephfp . (Co... |
alephfplem2 8811 | Lemma for ~ alephfp . (Co... |
alephfplem3 8812 | Lemma for ~ alephfp . (Co... |
alephfplem4 8813 | Lemma for ~ alephfp . (Co... |
alephfp 8814 | The aleph function has a f... |
alephfp2 8815 | The aleph function has at ... |
alephval3 8816 | An alternate way to expres... |
alephsucpw2 8817 | The power set of an aleph ... |
mappwen 8818 | Power rule for cardinal ar... |
finnisoeu 8819 | A finite totally ordered s... |
iunfictbso 8820 | Countability of a countabl... |
aceq1 8823 | Equivalence of two version... |
aceq0 8824 | Equivalence of two version... |
aceq2 8825 | Equivalence of two version... |
aceq3lem 8826 | Lemma for ~ dfac3 . (Cont... |
dfac3 8827 | Equivalence of two version... |
dfac4 8828 | Equivalence of two version... |
dfac5lem1 8829 | Lemma for ~ dfac5 . (Cont... |
dfac5lem2 8830 | Lemma for ~ dfac5 . (Cont... |
dfac5lem3 8831 | Lemma for ~ dfac5 . (Cont... |
dfac5lem4 8832 | Lemma for ~ dfac5 . (Cont... |
dfac5lem5 8833 | Lemma for ~ dfac5 . (Cont... |
dfac5 8834 | Equivalence of two version... |
dfac2a 8835 | Our Axiom of Choice (in th... |
dfac2 8836 | Axiom of Choice (first for... |
dfac7 8837 | Equivalence of the Axiom o... |
dfac0 8838 | Equivalence of two version... |
dfac1 8839 | Equivalence of two version... |
dfac8 8840 | A proof of the equivalency... |
dfac9 8841 | Equivalence of the axiom o... |
dfac10 8842 | Axiom of Choice equivalent... |
dfac10c 8843 | Axiom of Choice equivalent... |
dfac10b 8844 | Axiom of Choice equivalent... |
acacni 8845 | A choice equivalent: every... |
dfacacn 8846 | A choice equivalent: every... |
dfac13 8847 | The axiom of choice holds ... |
dfac12lem1 8848 | Lemma for ~ dfac12 . (Con... |
dfac12lem2 8849 | Lemma for ~ dfac12 . (Con... |
dfac12lem3 8850 | Lemma for ~ dfac12 . (Con... |
dfac12r 8851 | The axiom of choice holds ... |
dfac12k 8852 | Equivalence of ~ dfac12 an... |
dfac12a 8853 | The axiom of choice holds ... |
dfac12 8854 | The axiom of choice holds ... |
kmlem1 8855 | Lemma for 5-quantifier AC ... |
kmlem2 8856 | Lemma for 5-quantifier AC ... |
kmlem3 8857 | Lemma for 5-quantifier AC ... |
kmlem4 8858 | Lemma for 5-quantifier AC ... |
kmlem5 8859 | Lemma for 5-quantifier AC ... |
kmlem6 8860 | Lemma for 5-quantifier AC ... |
kmlem7 8861 | Lemma for 5-quantifier AC ... |
kmlem8 8862 | Lemma for 5-quantifier AC ... |
kmlem9 8863 | Lemma for 5-quantifier AC ... |
kmlem10 8864 | Lemma for 5-quantifier AC ... |
kmlem11 8865 | Lemma for 5-quantifier AC ... |
kmlem12 8866 | Lemma for 5-quantifier AC ... |
kmlem13 8867 | Lemma for 5-quantifier AC ... |
kmlem14 8868 | Lemma for 5-quantifier AC ... |
kmlem15 8869 | Lemma for 5-quantifier AC ... |
kmlem16 8870 | Lemma for 5-quantifier AC ... |
dfackm 8871 | Equivalence of the Axiom o... |
cdafn 8874 | Cardinal number addition i... |
cdaval 8875 | Value of cardinal addition... |
uncdadom 8876 | Cardinal addition dominate... |
cdaun 8877 | Cardinal addition is equin... |
cdaen 8878 | Cardinal addition of equin... |
cdaenun 8879 | Cardinal addition is equin... |
cda1en 8880 | Cardinal addition with car... |
cda1dif 8881 | Adding and subtracting one... |
pm110.643 8882 | 1+1=2 for cardinal number ... |
pm110.643ALT 8883 | Alternate proof of ~ pm110... |
cda0en 8884 | Cardinal addition with car... |
xp2cda 8885 | Two times a cardinal numbe... |
cdacomen 8886 | Commutative law for cardin... |
cdaassen 8887 | Associative law for cardin... |
xpcdaen 8888 | Cardinal multiplication di... |
mapcdaen 8889 | Sum of exponents law for c... |
pwcdaen 8890 | Sum of exponents law for c... |
cdadom1 8891 | Ordering law for cardinal ... |
cdadom2 8892 | Ordering law for cardinal ... |
cdadom3 8893 | A set is dominated by its ... |
cdaxpdom 8894 | Cartesian product dominate... |
cdafi 8895 | The cardinal sum of two fi... |
cdainflem 8896 | Any partition of omega int... |
cdainf 8897 | A set is infinite iff the ... |
infcda1 8898 | An infinite set is equinum... |
pwcda1 8899 | The sum of a powerset with... |
pwcdaidm 8900 | If the natural numbers inj... |
cdalepw 8901 | If ` A ` is idempotent und... |
onacda 8902 | The cardinal and ordinal s... |
cardacda 8903 | The cardinal sum is equinu... |
cdanum 8904 | The cardinal sum of two nu... |
unnum 8905 | The union of two numerable... |
nnacda 8906 | The cardinal and ordinal s... |
ficardun 8907 | The cardinality of the uni... |
ficardun2 8908 | The cardinality of the uni... |
pwsdompw 8909 | Lemma for ~ domtriom . Th... |
unctb 8910 | The union of two countable... |
infcdaabs 8911 | Absorption law for additio... |
infunabs 8912 | An infinite set is equinum... |
infcda 8913 | The sum of two cardinal nu... |
infdif 8914 | The cardinality of an infi... |
infdif2 8915 | Cardinality ordering for a... |
infxpdom 8916 | Dominance law for multipli... |
infxpabs 8917 | Absorption law for multipl... |
infunsdom1 8918 | The union of two sets that... |
infunsdom 8919 | The union of two sets that... |
infxp 8920 | Absorption law for multipl... |
pwcdadom 8921 | A property of dominance ov... |
infpss 8922 | Every infinite set has an ... |
infmap2 8923 | An exponentiation law for ... |
ackbij2lem1 8924 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem1 8925 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem2 8926 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem3 8927 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem4 8928 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem5 8929 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem6 8930 | Lemma for ~ ackbij2 . (Co... |
ackbij1lem7 8931 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem8 8932 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem9 8933 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem10 8934 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem11 8935 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem12 8936 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem13 8937 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem14 8938 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem15 8939 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem16 8940 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem17 8941 | Lemma for ~ ackbij1 . (Co... |
ackbij1lem18 8942 | Lemma for ~ ackbij1 . (Co... |
ackbij1 8943 | The Ackermann bijection, p... |
ackbij1b 8944 | The Ackermann bijection, p... |
ackbij2lem2 8945 | Lemma for ~ ackbij2 . (Co... |
ackbij2lem3 8946 | Lemma for ~ ackbij2 . (Co... |
ackbij2lem4 8947 | Lemma for ~ ackbij2 . (Co... |
ackbij2 8948 | The Ackermann bijection, p... |
r1om 8949 | The set of hereditarily fi... |
fictb 8950 | A set is countable iff its... |
cflem 8951 | A lemma used to simplify c... |
cfval 8952 | Value of the cofinality fu... |
cff 8953 | Cofinality is a function o... |
cfub 8954 | An upper bound on cofinali... |
cflm 8955 | Value of the cofinality fu... |
cf0 8956 | Value of the cofinality fu... |
cardcf 8957 | Cofinality is a cardinal n... |
cflecard 8958 | Cofinality is bounded by t... |
cfle 8959 | Cofinality is bounded by i... |
cfon 8960 | The cofinality of any set ... |
cfeq0 8961 | Only the ordinal zero has ... |
cfsuc 8962 | Value of the cofinality fu... |
cff1 8963 | There is always a map from... |
cfflb 8964 | If there is a cofinal map ... |
cfval2 8965 | Another expression for the... |
coflim 8966 | A simpler expression for t... |
cflim3 8967 | Another expression for the... |
cflim2 8968 | The cofinality function is... |
cfom 8969 | Value of the cofinality fu... |
cfss 8970 | There is a cofinal subset ... |
cfslb 8971 | Any cofinal subset of ` A ... |
cfslbn 8972 | Any subset of ` A ` smalle... |
cfslb2n 8973 | Any small collection of sm... |
cofsmo 8974 | Any cofinal map implies th... |
cfsmolem 8975 | Lemma for ~ cfsmo . (Cont... |
cfsmo 8976 | The map in ~ cff1 can be a... |
cfcoflem 8977 | Lemma for ~ cfcof , showin... |
coftr 8978 | If there is a cofinal map ... |
cfcof 8979 | If there is a cofinal map ... |
cfidm 8980 | The cofinality function is... |
alephsing 8981 | The cofinality of a limit ... |
sornom 8982 | The range of a single-step... |
isfin1a 8997 | Definition of a Ia-finite ... |
fin1ai 8998 | Property of a Ia-finite se... |
isfin2 8999 | Definition of a II-finite ... |
fin2i 9000 | Property of a II-finite se... |
isfin3 9001 | Definition of a III-finite... |
isfin4 9002 | Definition of a IV-finite ... |
fin4i 9003 | Infer that a set is IV-inf... |
isfin5 9004 | Definition of a V-finite s... |
isfin6 9005 | Definition of a VI-finite ... |
isfin7 9006 | Definition of a VII-finite... |
sdom2en01 9007 | A set with less than two e... |
infpssrlem1 9008 | Lemma for ~ infpssr . (Co... |
infpssrlem2 9009 | Lemma for ~ infpssr . (Co... |
infpssrlem3 9010 | Lemma for ~ infpssr . (Co... |
infpssrlem4 9011 | Lemma for ~ infpssr . (Co... |
infpssrlem5 9012 | Lemma for ~ infpssr . (Co... |
infpssr 9013 | Dedekind infinity implies ... |
fin4en1 9014 | Dedekind finite is a cardi... |
ssfin4 9015 | Dedekind finite sets have ... |
domfin4 9016 | A set dominated by a Dedek... |
ominf4 9017 | ` _om ` is Dedekind infini... |
infpssALT 9018 | Alternate proof of ~ infps... |
isfin4-2 9019 | Alternate definition of IV... |
isfin4-3 9020 | Alternate definition of IV... |
fin23lem7 9021 | Lemma for ~ isfin2-2 . Th... |
fin23lem11 9022 | Lemma for ~ isfin2-2 . (C... |
fin2i2 9023 | A II-finite set contains m... |
isfin2-2 9024 | ` Fin2 ` expressed in term... |
ssfin2 9025 | A subset of a II-finite se... |
enfin2i 9026 | II-finiteness is a cardina... |
fin23lem24 9027 | Lemma for ~ fin23 . In a ... |
fincssdom 9028 | In a chain of finite sets,... |
fin23lem25 9029 | Lemma for ~ fin23 . In a ... |
fin23lem26 9030 | Lemma for ~ fin23lem22 . ... |
fin23lem23 9031 | Lemma for ~ fin23lem22 . ... |
fin23lem22 9032 | Lemma for ~ fin23 but coul... |
fin23lem27 9033 | The mapping constructed in... |
isfin3ds 9034 | Property of a III-finite s... |
ssfin3ds 9035 | A subset of a III-finite s... |
fin23lem12 9036 | The beginning of the proof... |
fin23lem13 9037 | Lemma for ~ fin23 . Each ... |
fin23lem14 9038 | Lemma for ~ fin23 . ` U ` ... |
fin23lem15 9039 | Lemma for ~ fin23 . ` U ` ... |
fin23lem16 9040 | Lemma for ~ fin23 . ` U ` ... |
fin23lem19 9041 | Lemma for ~ fin23 . The f... |
fin23lem20 9042 | Lemma for ~ fin23 . ` X ` ... |
fin23lem17 9043 | Lemma for ~ fin23 . By ? ... |
fin23lem21 9044 | Lemma for ~ fin23 . ` X ` ... |
fin23lem28 9045 | Lemma for ~ fin23 . The r... |
fin23lem29 9046 | Lemma for ~ fin23 . The r... |
fin23lem30 9047 | Lemma for ~ fin23 . The r... |
fin23lem31 9048 | Lemma for ~ fin23 . The r... |
fin23lem32 9049 | Lemma for ~ fin23 . Wrap ... |
fin23lem33 9050 | Lemma for ~ fin23 . Disch... |
fin23lem34 9051 | Lemma for ~ fin23 . Estab... |
fin23lem35 9052 | Lemma for ~ fin23 . Stric... |
fin23lem36 9053 | Lemma for ~ fin23 . Weak ... |
fin23lem38 9054 | Lemma for ~ fin23 . The c... |
fin23lem39 9055 | Lemma for ~ fin23 . Thus,... |
fin23lem40 9056 | Lemma for ~ fin23 . ` Fin2... |
fin23lem41 9057 | Lemma for ~ fin23 . A set... |
isf32lem1 9058 | Lemma for ~ isfin3-2 . De... |
isf32lem2 9059 | Lemma for ~ isfin3-2 . No... |
isf32lem3 9060 | Lemma for ~ isfin3-2 . Be... |
isf32lem4 9061 | Lemma for ~ isfin3-2 . Be... |
isf32lem5 9062 | Lemma for ~ isfin3-2 . Th... |
isf32lem6 9063 | Lemma for ~ isfin3-2 . Ea... |
isf32lem7 9064 | Lemma for ~ isfin3-2 . Di... |
isf32lem8 9065 | Lemma for ~ isfin3-2 . K ... |
isf32lem9 9066 | Lemma for ~ isfin3-2 . Co... |
isf32lem10 9067 | Lemma for isfin3-2 . Writ... |
isf32lem11 9068 | Lemma for ~ isfin3-2 . Re... |
isf32lem12 9069 | Lemma for ~ isfin3-2 . (C... |
isfin32i 9070 | One half of ~ isfin3-2 . ... |
isf33lem 9071 | Lemma for ~ isfin3-3 . (C... |
isfin3-2 9072 | Weakly Dedekind-infinite s... |
isfin3-3 9073 | Weakly Dedekind-infinite s... |
fin33i 9074 | Inference from ~ isfin3-3 ... |
compsscnvlem 9075 | Lemma for ~ compsscnv . (... |
compsscnv 9076 | Complementation on a power... |
isf34lem1 9077 | Lemma for ~ isfin3-4 . (C... |
isf34lem2 9078 | Lemma for ~ isfin3-4 . (C... |
compssiso 9079 | Complementation is an anti... |
isf34lem3 9080 | Lemma for ~ isfin3-4 . (C... |
compss 9081 | Express image under of the... |
isf34lem4 9082 | Lemma for ~ isfin3-4 . (C... |
isf34lem5 9083 | Lemma for ~ isfin3-4 . (C... |
isf34lem7 9084 | Lemma for ~ isfin3-4 . (C... |
isf34lem6 9085 | Lemma for ~ isfin3-4 . (C... |
fin34i 9086 | Inference from ~ isfin3-4 ... |
isfin3-4 9087 | Weakly Dedekind-infinite s... |
fin11a 9088 | Every I-finite set is Ia-f... |
enfin1ai 9089 | Ia-finiteness is a cardina... |
isfin1-2 9090 | A set is finite in the usu... |
isfin1-3 9091 | A set is I-finite iff ever... |
isfin1-4 9092 | A set is I-finite iff ever... |
dffin1-5 9093 | Compact quantifier-free ve... |
fin23 9094 | Every II-finite set (every... |
fin34 9095 | Every III-finite set is IV... |
isfin5-2 9096 | Alternate definition of V-... |
fin45 9097 | Every IV-finite set is V-f... |
fin56 9098 | Every V-finite set is VI-f... |
fin17 9099 | Every I-finite set is VII-... |
fin67 9100 | Every VI-finite set is VII... |
isfin7-2 9101 | A set is VII-finite iff it... |
fin71num 9102 | A well-orderable set is VI... |
dffin7-2 9103 | Class form of ~ isfin7-2 .... |
dfacfin7 9104 | Axiom of Choice equivalent... |
fin1a2lem1 9105 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem2 9106 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem3 9107 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem4 9108 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem5 9109 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem6 9110 | Lemma for ~ fin1a2 . Esta... |
fin1a2lem7 9111 | Lemma for ~ fin1a2 . Spli... |
fin1a2lem8 9112 | Lemma for ~ fin1a2 . Spli... |
fin1a2lem9 9113 | Lemma for ~ fin1a2 . In a... |
fin1a2lem10 9114 | Lemma for ~ fin1a2 . A no... |
fin1a2lem11 9115 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem12 9116 | Lemma for ~ fin1a2 . (Con... |
fin1a2lem13 9117 | Lemma for ~ fin1a2 . (Con... |
fin12 9118 | Weak theorem which skips I... |
fin1a2s 9119 | An II-infinite set can hav... |
fin1a2 9120 | Every Ia-finite set is II-... |
itunifval 9121 | Function value of iterated... |
itunifn 9122 | Functionality of the itera... |
ituni0 9123 | A zero-fold iterated union... |
itunisuc 9124 | Successor iterated union. ... |
itunitc1 9125 | Each union iterate is a me... |
itunitc 9126 | The union of all union ite... |
ituniiun 9127 | Unwrap an iterated union f... |
hsmexlem7 9128 | Lemma for ~ hsmex . Prope... |
hsmexlem8 9129 | Lemma for ~ hsmex . Prope... |
hsmexlem9 9130 | Lemma for ~ hsmex . Prope... |
hsmexlem1 9131 | Lemma for ~ hsmex . Bound... |
hsmexlem2 9132 | Lemma for ~ hsmex . Bound... |
hsmexlem3 9133 | Lemma for ~ hsmex . Clear... |
hsmexlem4 9134 | Lemma for ~ hsmex . The c... |
hsmexlem5 9135 | Lemma for ~ hsmex . Combi... |
hsmexlem6 9136 | Lemma for ~ hsmex . (Cont... |
hsmex 9137 | The collection of heredita... |
hsmex2 9138 | The set of hereditary size... |
hsmex3 9139 | The set of hereditary size... |
axcc2lem 9141 | Lemma for ~ axcc2 . (Cont... |
axcc2 9142 | A possibly more useful ver... |
axcc3 9143 | A possibly more useful ver... |
axcc4 9144 | A version of ~ axcc3 that ... |
acncc 9145 | An ~ ax-cc equivalent: eve... |
axcc4dom 9146 | Relax the constraint on ~ ... |
domtriomlem 9147 | Lemma for ~ domtriom . (C... |
domtriom 9148 | Trichotomy of equinumerosi... |
fin41 9149 | Under countable choice, th... |
dominf 9150 | A nonempty set that is a s... |
dcomex 9152 | The Axiom of Dependent Cho... |
axdc2lem 9153 | Lemma for ~ axdc2 . We co... |
axdc2 9154 | An apparent strengthening ... |
axdc3lem 9155 | The class ` S ` of finite ... |
axdc3lem2 9156 | Lemma for ~ axdc3 . We ha... |
axdc3lem3 9157 | Simple substitution lemma ... |
axdc3lem4 9158 | Lemma for ~ axdc3 . We ha... |
axdc3 9159 | Dependent Choice. Axiom D... |
axdc4lem 9160 | Lemma for ~ axdc4 . (Cont... |
axdc4 9161 | A more general version of ... |
axcclem 9162 | Lemma for ~ axcc . (Contr... |
axcc 9163 | Although CC can be proven ... |
zfac 9165 | Axiom of Choice expressed ... |
ac2 9166 | Axiom of Choice equivalent... |
ac3 9167 | Axiom of Choice using abbr... |
axac3 9169 | This theorem asserts that ... |
ackm 9170 | A remarkable equivalent to... |
axac2 9171 | Derive ~ ax-ac2 from ~ ax-... |
axac 9172 | Derive ~ ax-ac from ~ ax-a... |
axaci 9173 | Apply a choice equivalent.... |
cardeqv 9174 | All sets are well-orderabl... |
numth3 9175 | All sets are well-orderabl... |
numth2 9176 | Numeration theorem: any se... |
numth 9177 | Numeration theorem: every ... |
ac7 9178 | An Axiom of Choice equival... |
ac7g 9179 | An Axiom of Choice equival... |
ac4 9180 | Equivalent of Axiom of Cho... |
ac4c 9181 | Equivalent of Axiom of Cho... |
ac5 9182 | An Axiom of Choice equival... |
ac5b 9183 | Equivalent of Axiom of Cho... |
ac6num 9184 | A version of ~ ac6 which t... |
ac6 9185 | Equivalent of Axiom of Cho... |
ac6c4 9186 | Equivalent of Axiom of Cho... |
ac6c5 9187 | Equivalent of Axiom of Cho... |
ac9 9188 | An Axiom of Choice equival... |
ac6s 9189 | Equivalent of Axiom of Cho... |
ac6n 9190 | Equivalent of Axiom of Cho... |
ac6s2 9191 | Generalization of the Axio... |
ac6s3 9192 | Generalization of the Axio... |
ac6sg 9193 | ~ ac6s with sethood as ant... |
ac6sf 9194 | Version of ~ ac6 with boun... |
ac6s4 9195 | Generalization of the Axio... |
ac6s5 9196 | Generalization of the Axio... |
ac8 9197 | An Axiom of Choice equival... |
ac9s 9198 | An Axiom of Choice equival... |
numthcor 9199 | Any set is strictly domina... |
weth 9200 | Well-ordering theorem: any... |
zorn2lem1 9201 | Lemma for ~ zorn2 . (Cont... |
zorn2lem2 9202 | Lemma for ~ zorn2 . (Cont... |
zorn2lem3 9203 | Lemma for ~ zorn2 . (Cont... |
zorn2lem4 9204 | Lemma for ~ zorn2 . (Cont... |
zorn2lem5 9205 | Lemma for ~ zorn2 . (Cont... |
zorn2lem6 9206 | Lemma for ~ zorn2 . (Cont... |
zorn2lem7 9207 | Lemma for ~ zorn2 . (Cont... |
zorn2g 9208 | Zorn's Lemma of [Monk1] p.... |
zorng 9209 | Zorn's Lemma. If the unio... |
zornn0g 9210 | Variant of Zorn's lemma ~ ... |
zorn2 9211 | Zorn's Lemma of [Monk1] p.... |
zorn 9212 | Zorn's Lemma. If the unio... |
zornn0 9213 | Variant of Zorn's lemma ~ ... |
ttukeylem1 9214 | Lemma for ~ ttukey . Expa... |
ttukeylem2 9215 | Lemma for ~ ttukey . A pr... |
ttukeylem3 9216 | Lemma for ~ ttukey . (Con... |
ttukeylem4 9217 | Lemma for ~ ttukey . (Con... |
ttukeylem5 9218 | Lemma for ~ ttukey . The ... |
ttukeylem6 9219 | Lemma for ~ ttukey . (Con... |
ttukeylem7 9220 | Lemma for ~ ttukey . (Con... |
ttukey2g 9221 | The Teichmüller-Tukey... |
ttukeyg 9222 | The Teichmüller-Tukey... |
ttukey 9223 | The Teichmüller-Tukey... |
axdclem 9224 | Lemma for ~ axdc . (Contr... |
axdclem2 9225 | Lemma for ~ axdc . Using ... |
axdc 9226 | This theorem derives ~ ax-... |
fodom 9227 | An onto function implies d... |
fodomg 9228 | An onto function implies d... |
fodomb 9229 | Equivalence of an onto map... |
wdomac 9230 | When assuming AC, weak and... |
brdom3 9231 | Equivalence to a dominance... |
brdom5 9232 | An equivalence to a domina... |
brdom4 9233 | An equivalence to a domina... |
brdom7disj 9234 | An equivalence to a domina... |
brdom6disj 9235 | An equivalence to a domina... |
fin71ac 9236 | Once we allow AC, the "str... |
imadomg 9237 | An image of a function und... |
fimact 9238 | The image by a function of... |
fnrndomg 9239 | The range of a function is... |
iunfo 9240 | Existence of an onto funct... |
iundom2g 9241 | An upper bound for the car... |
iundomg 9242 | An upper bound for the car... |
iundom 9243 | An upper bound for the car... |
unidom 9244 | An upper bound for the car... |
uniimadom 9245 | An upper bound for the car... |
uniimadomf 9246 | An upper bound for the car... |
cardval 9247 | The value of the cardinal ... |
cardid 9248 | Any set is equinumerous to... |
cardidg 9249 | Any set is equinumerous to... |
cardidd 9250 | Any set is equinumerous to... |
cardf 9251 | The cardinality function i... |
carden 9252 | Two sets are equinumerous ... |
cardeq0 9253 | Only the empty set has car... |
unsnen 9254 | Equinumerosity of a set wi... |
carddom 9255 | Two sets have the dominanc... |
cardsdom 9256 | Two sets have the strict d... |
domtri 9257 | Trichotomy law for dominan... |
entric 9258 | Trichotomy of equinumerosi... |
entri2 9259 | Trichotomy of dominance an... |
entri3 9260 | Trichotomy of dominance. ... |
sdomsdomcard 9261 | A set strictly dominates i... |
canth3 9262 | Cantor's theorem in terms ... |
infxpidm 9263 | The Cartesian product of a... |
ondomon 9264 | The collection of ordinal ... |
cardmin 9265 | The smallest ordinal that ... |
ficard 9266 | A set is finite iff its ca... |
infinf 9267 | Equivalence between two in... |
unirnfdomd 9268 | The union of the range of ... |
konigthlem 9269 | Lemma for ~ konigth . (Co... |
konigth 9270 | Konig's Theorem. If ` m (... |
alephsucpw 9271 | The power set of an aleph ... |
aleph1 9272 | The set exponentiation of ... |
alephval2 9273 | An alternate way to expres... |
dominfac 9274 | A nonempty set that is a s... |
iunctb 9275 | The countable union of cou... |
unictb 9276 | The countable union of cou... |
infmap 9277 | An exponentiation law for ... |
alephadd 9278 | The sum of two alephs is t... |
alephmul 9279 | The product of two alephs ... |
alephexp1 9280 | An exponentiation law for ... |
alephsuc3 9281 | An alternate representatio... |
alephexp2 9282 | An expression equinumerous... |
alephreg 9283 | A successor aleph is regul... |
pwcfsdom 9284 | A corollary of Konig's The... |
cfpwsdom 9285 | A corollary of Konig's The... |
alephom 9286 | From ~ canth2 , we know th... |
smobeth 9287 | The beth function is stric... |
nd1 9288 | A lemma for proving condit... |
nd2 9289 | A lemma for proving condit... |
nd3 9290 | A lemma for proving condit... |
nd4 9291 | A lemma for proving condit... |
axextnd 9292 | A version of the Axiom of ... |
axrepndlem1 9293 | Lemma for the Axiom of Rep... |
axrepndlem2 9294 | Lemma for the Axiom of Rep... |
axrepnd 9295 | A version of the Axiom of ... |
axunndlem1 9296 | Lemma for the Axiom of Uni... |
axunnd 9297 | A version of the Axiom of ... |
axpowndlem1 9298 | Lemma for the Axiom of Pow... |
axpowndlem2 9299 | Lemma for the Axiom of Pow... |
axpowndlem3 9300 | Lemma for the Axiom of Pow... |
axpowndlem4 9301 | Lemma for the Axiom of Pow... |
axpownd 9302 | A version of the Axiom of ... |
axregndlem1 9303 | Lemma for the Axiom of Reg... |
axregndlem2 9304 | Lemma for the Axiom of Reg... |
axregnd 9305 | A version of the Axiom of ... |
axinfndlem1 9306 | Lemma for the Axiom of Inf... |
axinfnd 9307 | A version of the Axiom of ... |
axacndlem1 9308 | Lemma for the Axiom of Cho... |
axacndlem2 9309 | Lemma for the Axiom of Cho... |
axacndlem3 9310 | Lemma for the Axiom of Cho... |
axacndlem4 9311 | Lemma for the Axiom of Cho... |
axacndlem5 9312 | Lemma for the Axiom of Cho... |
axacnd 9313 | A version of the Axiom of ... |
zfcndext 9314 | Axiom of Extensionality ~ ... |
zfcndrep 9315 | Axiom of Replacement ~ ax-... |
zfcndun 9316 | Axiom of Union ~ ax-un , r... |
zfcndpow 9317 | Axiom of Power Sets ~ ax-p... |
zfcndreg 9318 | Axiom of Regularity ~ ax-r... |
zfcndinf 9319 | Axiom of Infinity ~ ax-inf... |
zfcndac 9320 | Axiom of Choice ~ ax-ac , ... |
elgch 9323 | Elementhood in the collect... |
fingch 9324 | A finite set is a GCH-set.... |
gchi 9325 | The only GCH-sets which ha... |
gchen1 9326 | If ` A <_ B < ~P A ` , and... |
gchen2 9327 | If ` A < B <_ ~P A ` , and... |
gchor 9328 | If ` A <_ B <_ ~P A ` , an... |
engch 9329 | The property of being a GC... |
gchdomtri 9330 | Under certain conditions, ... |
fpwwe2cbv 9331 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem1 9332 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem2 9333 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem3 9334 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem5 9335 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem6 9336 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem7 9337 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem8 9338 | Lemma for ~ fpwwe2 . Show... |
fpwwe2lem9 9339 | Lemma for ~ fpwwe2 . Give... |
fpwwe2lem10 9340 | Lemma for ~ fpwwe2 . Give... |
fpwwe2lem11 9341 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem12 9342 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2lem13 9343 | Lemma for ~ fpwwe2 . (Con... |
fpwwe2 9344 | Given any function ` F ` f... |
fpwwecbv 9345 | Lemma for ~ fpwwe . (Cont... |
fpwwelem 9346 | Lemma for ~ fpwwe . (Cont... |
fpwwe 9347 | Given any function ` F ` f... |
canth4 9348 | An "effective" form of Can... |
canthnumlem 9349 | Lemma for ~ canthnum . (C... |
canthnum 9350 | The set of well-orderable ... |
canthwelem 9351 | Lemma for ~ canthnum . (C... |
canthwe 9352 | The set of well-orders of ... |
canthp1lem1 9353 | Lemma for ~ canthp1 . (Co... |
canthp1lem2 9354 | Lemma for ~ canthp1 . (Co... |
canthp1 9355 | A slightly stronger form o... |
finngch 9356 | The exclusion of finite se... |
gchcda1 9357 | An infinite GCH-set is ide... |
gchinf 9358 | An infinite GCH-set is Ded... |
pwfseqlem1 9359 | Lemma for ~ pwfseq . Deri... |
pwfseqlem2 9360 | Lemma for ~ pwfseq . (Con... |
pwfseqlem3 9361 | Lemma for ~ pwfseq . Usin... |
pwfseqlem4a 9362 | Lemma for ~ pwfseqlem4 . ... |
pwfseqlem4 9363 | Lemma for ~ pwfseq . Deri... |
pwfseqlem5 9364 | Lemma for ~ pwfseq . Alth... |
pwfseq 9365 | The powerset of a Dedekind... |
pwxpndom2 9366 | The powerset of a Dedekind... |
pwxpndom 9367 | The powerset of a Dedekind... |
pwcdandom 9368 | The powerset of a Dedekind... |
gchcdaidm 9369 | An infinite GCH-set is ide... |
gchxpidm 9370 | An infinite GCH-set is ide... |
gchpwdom 9371 | A relationship between dom... |
gchaleph 9372 | If ` ( aleph `` A ) ` is a... |
gchaleph2 9373 | If ` ( aleph `` A ) ` and ... |
hargch 9374 | If ` A + ~~ ~P A ` , then ... |
alephgch 9375 | If ` ( aleph `` suc A ) ` ... |
gch2 9376 | It is sufficient to requir... |
gch3 9377 | An equivalent formulation ... |
gch-kn 9378 | The equivalence of two ver... |
gchaclem 9379 | Lemma for ~ gchac (obsolet... |
gchhar 9380 | A "local" form of ~ gchac ... |
gchacg 9381 | A "local" form of ~ gchac ... |
gchac 9382 | The Generalized Continuum ... |
elwina 9387 | Conditions of weak inacces... |
elina 9388 | Conditions of strong inacc... |
winaon 9389 | A weakly inaccessible card... |
inawinalem 9390 | Lemma for ~ inawina . (Co... |
inawina 9391 | Every strongly inaccessibl... |
omina 9392 | ` _om ` is a strongly inac... |
winacard 9393 | A weakly inaccessible card... |
winainflem 9394 | A weakly inaccessible card... |
winainf 9395 | A weakly inaccessible card... |
winalim 9396 | A weakly inaccessible card... |
winalim2 9397 | A nontrivial weakly inacce... |
winafp 9398 | A nontrivial weakly inacce... |
winafpi 9399 | This theorem, which states... |
gchina 9400 | Assuming the GCH, weakly a... |
iswun 9405 | Properties of a weak unive... |
wuntr 9406 | A weak universe is transit... |
wununi 9407 | A weak universe is closed ... |
wunpw 9408 | A weak universe is closed ... |
wunelss 9409 | The elements of a weak uni... |
wunpr 9410 | A weak universe is closed ... |
wunun 9411 | A weak universe is closed ... |
wuntp 9412 | A weak universe is closed ... |
wunss 9413 | A weak universe is closed ... |
wunin 9414 | A weak universe is closed ... |
wundif 9415 | A weak universe is closed ... |
wunint 9416 | A weak universe is closed ... |
wunsn 9417 | A weak universe is closed ... |
wunsuc 9418 | A weak universe is closed ... |
wun0 9419 | A weak universe contains t... |
wunr1om 9420 | A weak universe is infinit... |
wunom 9421 | A weak universe contains a... |
wunfi 9422 | A weak universe contains a... |
wunop 9423 | A weak universe is closed ... |
wunot 9424 | A weak universe is closed ... |
wunxp 9425 | A weak universe is closed ... |
wunpm 9426 | A weak universe is closed ... |
wunmap 9427 | A weak universe is closed ... |
wunf 9428 | A weak universe is closed ... |
wundm 9429 | A weak universe is closed ... |
wunrn 9430 | A weak universe is closed ... |
wuncnv 9431 | A weak universe is closed ... |
wunres 9432 | A weak universe is closed ... |
wunfv 9433 | A weak universe is closed ... |
wunco 9434 | A weak universe is closed ... |
wuntpos 9435 | A weak universe is closed ... |
intwun 9436 | The intersection of a coll... |
r1limwun 9437 | Each limit stage in the cu... |
r1wunlim 9438 | The weak universes in the ... |
wunex2 9439 | Construct a weak universe ... |
wunex 9440 | Construct a weak universe ... |
uniwun 9441 | Every set is contained in ... |
wunex3 9442 | Construct a weak universe ... |
wuncval 9443 | Value of the weak universe... |
wuncid 9444 | The weak universe closure ... |
wunccl 9445 | The weak universe closure ... |
wuncss 9446 | The weak universe closure ... |
wuncidm 9447 | The weak universe closure ... |
wuncval2 9448 | Our earlier expression for... |
eltskg 9451 | Properties of a Tarski cla... |
eltsk2g 9452 | Properties of a Tarski cla... |
tskpwss 9453 | First axiom of a Tarski cl... |
tskpw 9454 | Second axiom of a Tarski c... |
tsken 9455 | Third axiom of a Tarski cl... |
0tsk 9456 | The empty set is a (transi... |
tsksdom 9457 | An element of a Tarski cla... |
tskssel 9458 | A part of a Tarski class s... |
tskss 9459 | The subsets of an element ... |
tskin 9460 | The intersection of two el... |
tsksn 9461 | A singleton of an element ... |
tsktrss 9462 | A transitive element of a ... |
tsksuc 9463 | If an element of a Tarski ... |
tsk0 9464 | A nonempty Tarski class co... |
tsk1 9465 | One is an element of a non... |
tsk2 9466 | Two is an element of a non... |
2domtsk 9467 | If a Tarski class is not e... |
tskr1om 9468 | A nonempty Tarski class is... |
tskr1om2 9469 | A nonempty Tarski class co... |
tskinf 9470 | A nonempty Tarski class is... |
tskpr 9471 | If ` A ` and ` B ` are mem... |
tskop 9472 | If ` A ` and ` B ` are mem... |
tskxpss 9473 | A Cartesian product of two... |
tskwe2 9474 | A Tarski class is well-ord... |
inttsk 9475 | The intersection of a coll... |
inar1 9476 | ` ( R1 `` A ) ` for ` A ` ... |
r1omALT 9477 | Alternate proof of ~ r1om ... |
rankcf 9478 | Any set must be at least a... |
inatsk 9479 | ` ( R1 `` A ) ` for ` A ` ... |
r1omtsk 9480 | The set of hereditarily fi... |
tskord 9481 | A Tarski class contains al... |
tskcard 9482 | An even more direct relati... |
r1tskina 9483 | There is a direct relation... |
tskuni 9484 | The union of an element of... |
tskwun 9485 | A nonempty transitive Tars... |
tskint 9486 | The intersection of an ele... |
tskun 9487 | The union of two elements ... |
tskxp 9488 | The Cartesian product of t... |
tskmap 9489 | Set exponentiation is an e... |
tskurn 9490 | A transitive Tarski class ... |
elgrug 9493 | Properties of a Grothendie... |
grutr 9494 | A Grothendieck universe is... |
gruelss 9495 | A Grothendieck universe is... |
grupw 9496 | A Grothendieck universe co... |
gruss 9497 | Any subset of an element o... |
grupr 9498 | A Grothendieck universe co... |
gruurn 9499 | A Grothendieck universe co... |
gruiun 9500 | If ` B ( x ) ` is a family... |
gruuni 9501 | A Grothendieck universe co... |
grurn 9502 | A Grothendieck universe co... |
gruima 9503 | A Grothendieck universe co... |
gruel 9504 | Any element of an element ... |
grusn 9505 | A Grothendieck universe co... |
gruop 9506 | A Grothendieck universe co... |
gruun 9507 | A Grothendieck universe co... |
gruxp 9508 | A Grothendieck universe co... |
grumap 9509 | A Grothendieck universe co... |
gruixp 9510 | A Grothendieck universe co... |
gruiin 9511 | A Grothendieck universe co... |
gruf 9512 | A Grothendieck universe co... |
gruen 9513 | A Grothendieck universe co... |
gruwun 9514 | A nonempty Grothendieck un... |
intgru 9515 | The intersection of a fami... |
ingru 9516 | The intersection of a univ... |
wfgru 9517 | The wellfounded part of a ... |
grudomon 9518 | Each ordinal that is compa... |
gruina 9519 | If a Grothendieck universe... |
grur1a 9520 | A characterization of Grot... |
grur1 9521 | A characterization of Grot... |
grutsk1 9522 | Grothendieck universes are... |
grutsk 9523 | Grothendieck universes are... |
axgroth5 9525 | The Tarski-Grothendieck ax... |
axgroth2 9526 | Alternate version of the T... |
grothpw 9527 | Derive the Axiom of Power ... |
grothpwex 9528 | Derive the Axiom of Power ... |
axgroth6 9529 | The Tarski-Grothendieck ax... |
grothomex 9530 | The Tarski-Grothendieck Ax... |
grothac 9531 | The Tarski-Grothendieck Ax... |
axgroth3 9532 | Alternate version of the T... |
axgroth4 9533 | Alternate version of the T... |
grothprimlem 9534 | Lemma for ~ grothprim . E... |
grothprim 9535 | The Tarski-Grothendieck Ax... |
grothtsk 9536 | The Tarski-Grothendieck Ax... |
inaprc 9537 | An equivalent to the Tarsk... |
tskmval 9540 | Value of our tarski map. ... |
tskmid 9541 | The set ` A ` is an elemen... |
tskmcl 9542 | A Tarski class that contai... |
sstskm 9543 | Being a part of ` ( tarski... |
eltskm 9544 | Belonging to ` ( tarskiMap... |
elni 9577 | Membership in the class of... |
elni2 9578 | Membership in the class of... |
pinn 9579 | A positive integer is a na... |
pion 9580 | A positive integer is an o... |
piord 9581 | A positive integer is ordi... |
niex 9582 | The class of positive inte... |
0npi 9583 | The empty set is not a pos... |
1pi 9584 | Ordinal 'one' is a positiv... |
addpiord 9585 | Positive integer addition ... |
mulpiord 9586 | Positive integer multiplic... |
mulidpi 9587 | 1 is an identity element f... |
ltpiord 9588 | Positive integer 'less tha... |
ltsopi 9589 | Positive integer 'less tha... |
ltrelpi 9590 | Positive integer 'less tha... |
dmaddpi 9591 | Domain of addition on posi... |
dmmulpi 9592 | Domain of multiplication o... |
addclpi 9593 | Closure of addition of pos... |
mulclpi 9594 | Closure of multiplication ... |
addcompi 9595 | Addition of positive integ... |
addasspi 9596 | Addition of positive integ... |
mulcompi 9597 | Multiplication of positive... |
mulasspi 9598 | Multiplication of positive... |
distrpi 9599 | Multiplication of positive... |
addcanpi 9600 | Addition cancellation law ... |
mulcanpi 9601 | Multiplication cancellatio... |
addnidpi 9602 | There is no identity eleme... |
ltexpi 9603 | Ordering on positive integ... |
ltapi 9604 | Ordering property of addit... |
ltmpi 9605 | Ordering property of multi... |
1lt2pi 9606 | One is less than two (one ... |
nlt1pi 9607 | No positive integer is les... |
indpi 9608 | Principle of Finite Induct... |
enqbreq 9620 | Equivalence relation for p... |
enqbreq2 9621 | Equivalence relation for p... |
enqer 9622 | The equivalence relation f... |
enqex 9623 | The equivalence relation f... |
nqex 9624 | The class of positive frac... |
0nnq 9625 | The empty set is not a pos... |
elpqn 9626 | Each positive fraction is ... |
ltrelnq 9627 | Positive fraction 'less th... |
pinq 9628 | The representatives of pos... |
1nq 9629 | The positive fraction 'one... |
nqereu 9630 | There is a unique element ... |
nqerf 9631 | Corollary of ~ nqereu : th... |
nqercl 9632 | Corollary of ~ nqereu : cl... |
nqerrel 9633 | Any member of ` ( N. X. N.... |
nqerid 9634 | Corollary of ~ nqereu : th... |
enqeq 9635 | Corollary of ~ nqereu : if... |
nqereq 9636 | The function ` /Q ` acts a... |
addpipq2 9637 | Addition of positive fract... |
addpipq 9638 | Addition of positive fract... |
addpqnq 9639 | Addition of positive fract... |
mulpipq2 9640 | Multiplication of positive... |
mulpipq 9641 | Multiplication of positive... |
mulpqnq 9642 | Multiplication of positive... |
ordpipq 9643 | Ordering of positive fract... |
ordpinq 9644 | Ordering of positive fract... |
addpqf 9645 | Closure of addition on pos... |
addclnq 9646 | Closure of addition on pos... |
mulpqf 9647 | Closure of multiplication ... |
mulclnq 9648 | Closure of multiplication ... |
addnqf 9649 | Domain of addition on posi... |
mulnqf 9650 | Domain of multiplication o... |
addcompq 9651 | Addition of positive fract... |
addcomnq 9652 | Addition of positive fract... |
mulcompq 9653 | Multiplication of positive... |
mulcomnq 9654 | Multiplication of positive... |
adderpqlem 9655 | Lemma for ~ adderpq . (Co... |
mulerpqlem 9656 | Lemma for ~ mulerpq . (Co... |
adderpq 9657 | Addition is compatible wit... |
mulerpq 9658 | Multiplication is compatib... |
addassnq 9659 | Addition of positive fract... |
mulassnq 9660 | Multiplication of positive... |
mulcanenq 9661 | Lemma for distributive law... |
distrnq 9662 | Multiplication of positive... |
1nqenq 9663 | The equivalence class of r... |
mulidnq 9664 | Multiplication identity el... |
recmulnq 9665 | Relationship between recip... |
recidnq 9666 | A positive fraction times ... |
recclnq 9667 | Closure law for positive f... |
recrecnq 9668 | Reciprocal of reciprocal o... |
dmrecnq 9669 | Domain of reciprocal on po... |
ltsonq 9670 | 'Less than' is a strict or... |
lterpq 9671 | Compatibility of ordering ... |
ltanq 9672 | Ordering property of addit... |
ltmnq 9673 | Ordering property of multi... |
1lt2nq 9674 | One is less than two (one ... |
ltaddnq 9675 | The sum of two fractions i... |
ltexnq 9676 | Ordering on positive fract... |
halfnq 9677 | One-half of any positive f... |
nsmallnq 9678 | The is no smallest positiv... |
ltbtwnnq 9679 | There exists a number betw... |
ltrnq 9680 | Ordering property of recip... |
archnq 9681 | For any fraction, there is... |
npex 9687 | The class of positive real... |
elnp 9688 | Membership in positive rea... |
elnpi 9689 | Membership in positive rea... |
prn0 9690 | A positive real is not emp... |
prpssnq 9691 | A positive real is a subse... |
elprnq 9692 | A positive real is a set o... |
0npr 9693 | The empty set is not a pos... |
prcdnq 9694 | A positive real is closed ... |
prub 9695 | A positive fraction not in... |
prnmax 9696 | A positive real has no lar... |
npomex 9697 | A simplifying observation,... |
prnmadd 9698 | A positive real has no lar... |
ltrelpr 9699 | Positive real 'less than' ... |
genpv 9700 | Value of general operation... |
genpelv 9701 | Membership in value of gen... |
genpprecl 9702 | Pre-closure law for genera... |
genpdm 9703 | Domain of general operatio... |
genpn0 9704 | The result of an operation... |
genpss 9705 | The result of an operation... |
genpnnp 9706 | The result of an operation... |
genpcd 9707 | Downward closure of an ope... |
genpnmax 9708 | An operation on positive r... |
genpcl 9709 | Closure of an operation on... |
genpass 9710 | Associativity of an operat... |
plpv 9711 | Value of addition on posit... |
mpv 9712 | Value of multiplication on... |
dmplp 9713 | Domain of addition on posi... |
dmmp 9714 | Domain of multiplication o... |
nqpr 9715 | The canonical embedding of... |
1pr 9716 | The positive real number '... |
addclprlem1 9717 | Lemma to prove downward cl... |
addclprlem2 9718 | Lemma to prove downward cl... |
addclpr 9719 | Closure of addition on pos... |
mulclprlem 9720 | Lemma to prove downward cl... |
mulclpr 9721 | Closure of multiplication ... |
addcompr 9722 | Addition of positive reals... |
addasspr 9723 | Addition of positive reals... |
mulcompr 9724 | Multiplication of positive... |
mulasspr 9725 | Multiplication of positive... |
distrlem1pr 9726 | Lemma for distributive law... |
distrlem4pr 9727 | Lemma for distributive law... |
distrlem5pr 9728 | Lemma for distributive law... |
distrpr 9729 | Multiplication of positive... |
1idpr 9730 | 1 is an identity element f... |
ltprord 9731 | Positive real 'less than' ... |
psslinpr 9732 | Proper subset is a linear ... |
ltsopr 9733 | Positive real 'less than' ... |
prlem934 9734 | Lemma 9-3.4 of [Gleason] p... |
ltaddpr 9735 | The sum of two positive re... |
ltaddpr2 9736 | The sum of two positive re... |
ltexprlem1 9737 | Lemma for Proposition 9-3.... |
ltexprlem2 9738 | Lemma for Proposition 9-3.... |
ltexprlem3 9739 | Lemma for Proposition 9-3.... |
ltexprlem4 9740 | Lemma for Proposition 9-3.... |
ltexprlem5 9741 | Lemma for Proposition 9-3.... |
ltexprlem6 9742 | Lemma for Proposition 9-3.... |
ltexprlem7 9743 | Lemma for Proposition 9-3.... |
ltexpri 9744 | Proposition 9-3.5(iv) of [... |
ltaprlem 9745 | Lemma for Proposition 9-3.... |
ltapr 9746 | Ordering property of addit... |
addcanpr 9747 | Addition cancellation law ... |
prlem936 9748 | Lemma 9-3.6 of [Gleason] p... |
reclem2pr 9749 | Lemma for Proposition 9-3.... |
reclem3pr 9750 | Lemma for Proposition 9-3.... |
reclem4pr 9751 | Lemma for Proposition 9-3.... |
recexpr 9752 | The reciprocal of a positi... |
suplem1pr 9753 | The union of a nonempty, b... |
suplem2pr 9754 | The union of a set of posi... |
supexpr 9755 | The union of a nonempty, b... |
enrbreq 9764 | Equivalence relation for s... |
enrer 9765 | The equivalence relation f... |
enreceq 9766 | Equivalence class equality... |
enrex 9767 | The equivalence relation f... |
ltrelsr 9768 | Signed real 'less than' is... |
addcmpblnr 9769 | Lemma showing compatibilit... |
mulcmpblnrlem 9770 | Lemma used in lemma showin... |
mulcmpblnr 9771 | Lemma showing compatibilit... |
prsrlem1 9772 | Decomposing signed reals i... |
addsrmo 9773 | There is at most one resul... |
mulsrmo 9774 | There is at most one resul... |
addsrpr 9775 | Addition of signed reals i... |
mulsrpr 9776 | Multiplication of signed r... |
ltsrpr 9777 | Ordering of signed reals i... |
gt0srpr 9778 | Greater than zero in terms... |
0nsr 9779 | The empty set is not a sig... |
0r 9780 | The constant ` 0R ` is a s... |
1sr 9781 | The constant ` 1R ` is a s... |
m1r 9782 | The constant ` -1R ` is a ... |
addclsr 9783 | Closure of addition on sig... |
mulclsr 9784 | Closure of multiplication ... |
dmaddsr 9785 | Domain of addition on sign... |
dmmulsr 9786 | Domain of multiplication o... |
addcomsr 9787 | Addition of signed reals i... |
addasssr 9788 | Addition of signed reals i... |
mulcomsr 9789 | Multiplication of signed r... |
mulasssr 9790 | Multiplication of signed r... |
distrsr 9791 | Multiplication of signed r... |
m1p1sr 9792 | Minus one plus one is zero... |
m1m1sr 9793 | Minus one times minus one ... |
ltsosr 9794 | Signed real 'less than' is... |
0lt1sr 9795 | 0 is less than 1 for signe... |
1ne0sr 9796 | 1 and 0 are distinct for s... |
0idsr 9797 | The signed real number 0 i... |
1idsr 9798 | 1 is an identity element f... |
00sr 9799 | A signed real times 0 is 0... |
ltasr 9800 | Ordering property of addit... |
pn0sr 9801 | A signed real plus its neg... |
negexsr 9802 | Existence of negative sign... |
recexsrlem 9803 | The reciprocal of a positi... |
addgt0sr 9804 | The sum of two positive si... |
mulgt0sr 9805 | The product of two positiv... |
sqgt0sr 9806 | The square of a nonzero si... |
recexsr 9807 | The reciprocal of a nonzer... |
mappsrpr 9808 | Mapping from positive sign... |
ltpsrpr 9809 | Mapping of order from posi... |
map2psrpr 9810 | Equivalence for positive s... |
supsrlem 9811 | Lemma for supremum theorem... |
supsr 9812 | A nonempty, bounded set of... |
opelcn 9829 | Ordered pair membership in... |
opelreal 9830 | Ordered pair membership in... |
elreal 9831 | Membership in class of rea... |
elreal2 9832 | Ordered pair membership in... |
0ncn 9833 | The empty set is not a com... |
ltrelre 9834 | 'Less than' is a relation ... |
addcnsr 9835 | Addition of complex number... |
mulcnsr 9836 | Multiplication of complex ... |
eqresr 9837 | Equality of real numbers i... |
addresr 9838 | Addition of real numbers i... |
mulresr 9839 | Multiplication of real num... |
ltresr 9840 | Ordering of real subset of... |
ltresr2 9841 | Ordering of real subset of... |
dfcnqs 9842 | Technical trick to permit ... |
addcnsrec 9843 | Technical trick to permit ... |
mulcnsrec 9844 | Technical trick to permit ... |
axaddf 9845 | Addition is an operation o... |
axmulf 9846 | Multiplication is an opera... |
axcnex 9847 | The complex numbers form a... |
axresscn 9848 | The real numbers are a sub... |
ax1cn 9849 | 1 is a complex number. Ax... |
axicn 9850 | ` _i ` is a complex number... |
axaddcl 9851 | Closure law for addition o... |
axaddrcl 9852 | Closure law for addition i... |
axmulcl 9853 | Closure law for multiplica... |
axmulrcl 9854 | Closure law for multiplica... |
axmulcom 9855 | Multiplication of complex ... |
axaddass 9856 | Addition of complex number... |
axmulass 9857 | Multiplication of complex ... |
axdistr 9858 | Distributive law for compl... |
axi2m1 9859 | i-squared equals -1 (expre... |
ax1ne0 9860 | 1 and 0 are distinct. Axi... |
ax1rid 9861 | ` 1 ` is an identity eleme... |
axrnegex 9862 | Existence of negative of r... |
axrrecex 9863 | Existence of reciprocal of... |
axcnre 9864 | A complex number can be ex... |
axpre-lttri 9865 | Ordering on reals satisfie... |
axpre-lttrn 9866 | Ordering on reals is trans... |
axpre-ltadd 9867 | Ordering property of addit... |
axpre-mulgt0 9868 | The product of two positiv... |
axpre-sup 9869 | A nonempty, bounded-above ... |
wuncn 9870 | A weak universe containing... |
cnex 9896 | Alias for ~ ax-cnex . See... |
addcl 9897 | Alias for ~ ax-addcl , for... |
readdcl 9898 | Alias for ~ ax-addrcl , fo... |
mulcl 9899 | Alias for ~ ax-mulcl , for... |
remulcl 9900 | Alias for ~ ax-mulrcl , fo... |
mulcom 9901 | Alias for ~ ax-mulcom , fo... |
addass 9902 | Alias for ~ ax-addass , fo... |
mulass 9903 | Alias for ~ ax-mulass , fo... |
adddi 9904 | Alias for ~ ax-distr , for... |
recn 9905 | A real number is a complex... |
reex 9906 | The real numbers form a se... |
reelprrecn 9907 | Reals are a subset of the ... |
cnelprrecn 9908 | Complex numbers are a subs... |
elimne0 9909 | Hypothesis for weak deduct... |
adddir 9910 | Distributive law for compl... |
0cn 9911 | 0 is a complex number. Se... |
0cnd 9912 | 0 is a complex number, ded... |
c0ex 9913 | 0 is a set (common case). ... |
1ex 9914 | 1 is a set. Common specia... |
cnre 9915 | Alias for ~ ax-cnre , for ... |
mulid1 9916 | ` 1 ` is an identity eleme... |
mulid2 9917 | Identity law for multiplic... |
1re 9918 | ` 1 ` is a real number. T... |
0re 9919 | ` 0 ` is a real number. S... |
0red 9920 | ` 0 ` is a real number, de... |
mulid1i 9921 | Identity law for multiplic... |
mulid2i 9922 | Identity law for multiplic... |
addcli 9923 | Closure law for addition. ... |
mulcli 9924 | Closure law for multiplica... |
mulcomi 9925 | Commutative law for multip... |
mulcomli 9926 | Commutative law for multip... |
addassi 9927 | Associative law for additi... |
mulassi 9928 | Associative law for multip... |
adddii 9929 | Distributive law (left-dis... |
adddiri 9930 | Distributive law (right-di... |
recni 9931 | A real number is a complex... |
readdcli 9932 | Closure law for addition o... |
remulcli 9933 | Closure law for multiplica... |
1red 9934 | 1 is an real number, deduc... |
1cnd 9935 | 1 is a complex number, ded... |
mulid1d 9936 | Identity law for multiplic... |
mulid2d 9937 | Identity law for multiplic... |
addcld 9938 | Closure law for addition. ... |
mulcld 9939 | Closure law for multiplica... |
mulcomd 9940 | Commutative law for multip... |
addassd 9941 | Associative law for additi... |
mulassd 9942 | Associative law for multip... |
adddid 9943 | Distributive law (left-dis... |
adddird 9944 | Distributive law (right-di... |
adddirp1d 9945 | Distributive law, plus 1 v... |
joinlmuladdmuld 9946 | Join AB+CB into (A+C) on L... |
recnd 9947 | Deduction from real number... |
readdcld 9948 | Closure law for addition o... |
remulcld 9949 | Closure law for multiplica... |
pnfnre 9960 | Plus infinity is not a rea... |
mnfnre 9961 | Minus infinity is not a re... |
ressxr 9962 | The standard reals are a s... |
rexpssxrxp 9963 | The Cartesian product of s... |
rexr 9964 | A standard real is an exte... |
0xr 9965 | Zero is an extended real. ... |
renepnf 9966 | No (finite) real equals pl... |
renemnf 9967 | No real equals minus infin... |
rexrd 9968 | A standard real is an exte... |
renepnfd 9969 | No (finite) real equals pl... |
renemnfd 9970 | No real equals minus infin... |
pnfxr 9971 | Plus infinity belongs to t... |
pnfex 9972 | Plus infinity exists (comm... |
pnfnemnf 9973 | Plus and minus infinity ar... |
mnfnepnf 9974 | Minus and plus infinity ar... |
mnfxr 9975 | Minus infinity belongs to ... |
rexri 9976 | A standard real is an exte... |
renfdisj 9977 | The reals and the infiniti... |
ltrelxr 9978 | 'Less than' is a relation ... |
ltrel 9979 | 'Less than' is a relation.... |
lerelxr 9980 | 'Less than or equal' is a ... |
lerel 9981 | 'Less or equal to' is a re... |
xrlenlt 9982 | 'Less than or equal to' ex... |
xrlenltd 9983 | 'Less than or equal to' ex... |
xrltnle 9984 | 'Less than' expressed in t... |
xrnltled 9985 | 'Not less than ' implies '... |
ssxr 9986 | The three (non-exclusive) ... |
ltxrlt 9987 | The standard less-than ` <... |
axlttri 9988 | Ordering on reals satisfie... |
axlttrn 9989 | Ordering on reals is trans... |
axltadd 9990 | Ordering property of addit... |
axmulgt0 9991 | The product of two positiv... |
axsup 9992 | A nonempty, bounded-above ... |
lttr 9993 | Alias for ~ axlttrn , for ... |
mulgt0 9994 | The product of two positiv... |
lenlt 9995 | 'Less than or equal to' ex... |
ltnle 9996 | 'Less than' expressed in t... |
ltso 9997 | 'Less than' is a strict or... |
gtso 9998 | 'Greater than' is a strict... |
lttri2 9999 | Consequence of trichotomy.... |
lttri3 10000 | Trichotomy law for 'less t... |
lttri4 10001 | Trichotomy law for 'less t... |
letri3 10002 | Trichotomy law. (Contribu... |
leloe 10003 | 'Less than or equal to' ex... |
eqlelt 10004 | Equality in terms of 'less... |
ltle 10005 | 'Less than' implies 'less ... |
leltne 10006 | 'Less than or equal to' im... |
lelttr 10007 | Transitive law. (Contribu... |
ltletr 10008 | Transitive law. (Contribu... |
ltleletr 10009 | Transitive law, weaker for... |
letr 10010 | Transitive law. (Contribu... |
ltnr 10011 | 'Less than' is irreflexive... |
leid 10012 | 'Less than or equal to' is... |
ltne 10013 | 'Less than' implies not eq... |
ltnsym 10014 | 'Less than' is not symmetr... |
ltnsym2 10015 | 'Less than' is antisymmetr... |
letric 10016 | Trichotomy law. (Contribu... |
ltlen 10017 | 'Less than' expressed in t... |
eqle 10018 | Equality implies 'less tha... |
eqled 10019 | Equality implies 'less tha... |
ltadd2 10020 | Addition to both sides of ... |
ne0gt0 10021 | A nonzero nonnegative numb... |
lecasei 10022 | Ordering elimination by ca... |
lelttric 10023 | Trichotomy law. (Contribu... |
ltlecasei 10024 | Ordering elimination by ca... |
ltnri 10025 | 'Less than' is irreflexive... |
eqlei 10026 | Equality implies 'less tha... |
eqlei2 10027 | Equality implies 'less tha... |
gtneii 10028 | 'Less than' implies not eq... |
ltneii 10029 | 'Greater than' implies not... |
lttri2i 10030 | Consequence of trichotomy.... |
lttri3i 10031 | Consequence of trichotomy.... |
letri3i 10032 | Consequence of trichotomy.... |
leloei 10033 | 'Less than or equal to' in... |
ltleni 10034 | 'Less than' expressed in t... |
ltnsymi 10035 | 'Less than' is not symmetr... |
lenlti 10036 | 'Less than or equal to' in... |
ltnlei 10037 | 'Less than' in terms of 'l... |
ltlei 10038 | 'Less than' implies 'less ... |
ltleii 10039 | 'Less than' implies 'less ... |
ltnei 10040 | 'Less than' implies not eq... |
letrii 10041 | Trichotomy law for 'less t... |
lttri 10042 | 'Less than' is transitive.... |
lelttri 10043 | 'Less than or equal to', '... |
ltletri 10044 | 'Less than', 'less than or... |
letri 10045 | 'Less than or equal to' is... |
le2tri3i 10046 | Extended trichotomy law fo... |
ltadd2i 10047 | Addition to both sides of ... |
mulgt0i 10048 | The product of two positiv... |
mulgt0ii 10049 | The product of two positiv... |
ltnrd 10050 | 'Less than' is irreflexive... |
gtned 10051 | 'Less than' implies not eq... |
ltned 10052 | 'Greater than' implies not... |
ne0gt0d 10053 | A nonzero nonnegative numb... |
lttrid 10054 | Ordering on reals satisfie... |
lttri2d 10055 | Consequence of trichotomy.... |
lttri3d 10056 | Consequence of trichotomy.... |
lttri4d 10057 | Trichotomy law for 'less t... |
letri3d 10058 | Consequence of trichotomy.... |
leloed 10059 | 'Less than or equal to' in... |
eqleltd 10060 | Equality in terms of 'less... |
ltlend 10061 | 'Less than' expressed in t... |
lenltd 10062 | 'Less than or equal to' in... |
ltnled 10063 | 'Less than' in terms of 'l... |
ltled 10064 | 'Less than' implies 'less ... |
ltnsymd 10065 | 'Less than' implies 'less ... |
nltled 10066 | 'Not less than ' implies '... |
lensymd 10067 | 'Less than or equal to' im... |
letrid 10068 | Trichotomy law for 'less t... |
leltned 10069 | 'Less than or equal to' im... |
leneltd 10070 | 'Less than or equal to' an... |
mulgt0d 10071 | The product of two positiv... |
ltadd2d 10072 | Addition to both sides of ... |
letrd 10073 | Transitive law deduction f... |
lelttrd 10074 | Transitive law deduction f... |
ltadd2dd 10075 | Addition to both sides of ... |
ltletrd 10076 | Transitive law deduction f... |
lttrd 10077 | Transitive law deduction f... |
lelttrdi 10078 | If a number is less than a... |
dedekind 10079 | The Dedekind cut theorem. ... |
dedekindle 10080 | The Dedekind cut theorem, ... |
mul12 10081 | Commutative/associative la... |
mul32 10082 | Commutative/associative la... |
mul31 10083 | Commutative/associative la... |
mul4 10084 | Rearrangement of 4 factors... |
muladd11 10085 | A simple product of sums e... |
1p1times 10086 | Two times a number. (Cont... |
peano2cn 10087 | A theorem for complex numb... |
peano2re 10088 | A theorem for reals analog... |
readdcan 10089 | Cancellation law for addit... |
00id 10090 | ` 0 ` is its own additive ... |
mul02lem1 10091 | Lemma for ~ mul02 . If an... |
mul02lem2 10092 | Lemma for ~ mul02 . Zero ... |
mul02 10093 | Multiplication by ` 0 ` . ... |
mul01 10094 | Multiplication by ` 0 ` . ... |
addid1 10095 | ` 0 ` is an additive ident... |
cnegex 10096 | Existence of the negative ... |
cnegex2 10097 | Existence of a left invers... |
addid2 10098 | ` 0 ` is a left identity f... |
addcan 10099 | Cancellation law for addit... |
addcan2 10100 | Cancellation law for addit... |
addcom 10101 | Addition commutes. This u... |
addid1i 10102 | ` 0 ` is an additive ident... |
addid2i 10103 | ` 0 ` is a left identity f... |
mul02i 10104 | Multiplication by 0. Theo... |
mul01i 10105 | Multiplication by ` 0 ` . ... |
addcomi 10106 | Addition commutes. Based ... |
addcomli 10107 | Addition commutes. (Contr... |
addcani 10108 | Cancellation law for addit... |
addcan2i 10109 | Cancellation law for addit... |
mul12i 10110 | Commutative/associative la... |
mul32i 10111 | Commutative/associative la... |
mul4i 10112 | Rearrangement of 4 factors... |
mul02d 10113 | Multiplication by 0. Theo... |
mul01d 10114 | Multiplication by ` 0 ` . ... |
addid1d 10115 | ` 0 ` is an additive ident... |
addid2d 10116 | ` 0 ` is a left identity f... |
addcomd 10117 | Addition commutes. Based ... |
addcand 10118 | Cancellation law for addit... |
addcan2d 10119 | Cancellation law for addit... |
addcanad 10120 | Cancelling a term on the l... |
addcan2ad 10121 | Cancelling a term on the r... |
addneintrd 10122 | Introducing a term on the ... |
addneintr2d 10123 | Introducing a term on the ... |
mul12d 10124 | Commutative/associative la... |
mul32d 10125 | Commutative/associative la... |
mul31d 10126 | Commutative/associative la... |
mul4d 10127 | Rearrangement of 4 factors... |
muladd11r 10128 | A simple product of sums e... |
comraddd 10129 | Commute RHS addition, in d... |
ltaddneg 10130 | Adding a negative number t... |
ltaddnegr 10131 | Adding a negative number t... |
add12 10132 | Commutative/associative la... |
add32 10133 | Commutative/associative la... |
add32r 10134 | Commutative/associative la... |
add4 10135 | Rearrangement of 4 terms i... |
add42 10136 | Rearrangement of 4 terms i... |
add12i 10137 | Commutative/associative la... |
add32i 10138 | Commutative/associative la... |
add4i 10139 | Rearrangement of 4 terms i... |
add42i 10140 | Rearrangement of 4 terms i... |
add12d 10141 | Commutative/associative la... |
add32d 10142 | Commutative/associative la... |
add4d 10143 | Rearrangement of 4 terms i... |
add42d 10144 | Rearrangement of 4 terms i... |
0cnALT 10149 | Alternate proof of ~ 0cn w... |
negeu 10150 | Existential uniqueness of ... |
subval 10151 | Value of subtraction, whic... |
negeq 10152 | Equality theorem for negat... |
negeqi 10153 | Equality inference for neg... |
negeqd 10154 | Equality deduction for neg... |
nfnegd 10155 | Deduction version of ~ nfn... |
nfneg 10156 | Bound-variable hypothesis ... |
csbnegg 10157 | Move class substitution in... |
negex 10158 | A negative is a set. (Con... |
subcl 10159 | Closure law for subtractio... |
negcl 10160 | Closure law for negative. ... |
negicn 10161 | ` -u _i ` is a complex num... |
subf 10162 | Subtraction is an operatio... |
subadd 10163 | Relationship between subtr... |
subadd2 10164 | Relationship between subtr... |
subsub23 10165 | Swap subtrahend and result... |
pncan 10166 | Cancellation law for subtr... |
pncan2 10167 | Cancellation law for subtr... |
pncan3 10168 | Subtraction and addition o... |
npcan 10169 | Cancellation law for subtr... |
addsubass 10170 | Associative-type law for a... |
addsub 10171 | Law for addition and subtr... |
subadd23 10172 | Commutative/associative la... |
addsub12 10173 | Commutative/associative la... |
2addsub 10174 | Law for subtraction and ad... |
addsubeq4 10175 | Relation between sums and ... |
pncan3oi 10176 | Subtraction and addition o... |
mvrraddi 10177 | Move RHS right addition to... |
mvlladdi 10178 | Move LHS left addition to ... |
subid 10179 | Subtraction of a number fr... |
subid1 10180 | Identity law for subtracti... |
npncan 10181 | Cancellation law for subtr... |
nppcan 10182 | Cancellation law for subtr... |
nnpcan 10183 | Cancellation law for subtr... |
nppcan3 10184 | Cancellation law for subtr... |
subcan2 10185 | Cancellation law for subtr... |
subeq0 10186 | If the difference between ... |
npncan2 10187 | Cancellation law for subtr... |
subsub2 10188 | Law for double subtraction... |
nncan 10189 | Cancellation law for subtr... |
subsub 10190 | Law for double subtraction... |
nppcan2 10191 | Cancellation law for subtr... |
subsub3 10192 | Law for double subtraction... |
subsub4 10193 | Law for double subtraction... |
sub32 10194 | Swap the second and third ... |
nnncan 10195 | Cancellation law for subtr... |
nnncan1 10196 | Cancellation law for subtr... |
nnncan2 10197 | Cancellation law for subtr... |
npncan3 10198 | Cancellation law for subtr... |
pnpcan 10199 | Cancellation law for mixed... |
pnpcan2 10200 | Cancellation law for mixed... |
pnncan 10201 | Cancellation law for mixed... |
ppncan 10202 | Cancellation law for mixed... |
addsub4 10203 | Rearrangement of 4 terms i... |
subadd4 10204 | Rearrangement of 4 terms i... |
sub4 10205 | Rearrangement of 4 terms i... |
neg0 10206 | Minus 0 equals 0. (Contri... |
negid 10207 | Addition of a number and i... |
negsub 10208 | Relationship between subtr... |
subneg 10209 | Relationship between subtr... |
negneg 10210 | A number is equal to the n... |
neg11 10211 | Negative is one-to-one. (... |
negcon1 10212 | Negative contraposition la... |
negcon2 10213 | Negative contraposition la... |
negeq0 10214 | A number is zero iff its n... |
subcan 10215 | Cancellation law for subtr... |
negsubdi 10216 | Distribution of negative o... |
negdi 10217 | Distribution of negative o... |
negdi2 10218 | Distribution of negative o... |
negsubdi2 10219 | Distribution of negative o... |
neg2sub 10220 | Relationship between subtr... |
renegcli 10221 | Closure law for negative o... |
resubcli 10222 | Closure law for subtractio... |
renegcl 10223 | Closure law for negative o... |
resubcl 10224 | Closure law for subtractio... |
negreb 10225 | The negative of a real is ... |
peano2cnm 10226 | "Reverse" second Peano pos... |
peano2rem 10227 | "Reverse" second Peano pos... |
negcli 10228 | Closure law for negative. ... |
negidi 10229 | Addition of a number and i... |
negnegi 10230 | A number is equal to the n... |
subidi 10231 | Subtraction of a number fr... |
subid1i 10232 | Identity law for subtracti... |
negne0bi 10233 | A number is nonzero iff it... |
negrebi 10234 | The negative of a real is ... |
negne0i 10235 | The negative of a nonzero ... |
subcli 10236 | Closure law for subtractio... |
pncan3i 10237 | Subtraction and addition o... |
negsubi 10238 | Relationship between subtr... |
subnegi 10239 | Relationship between subtr... |
subeq0i 10240 | If the difference between ... |
neg11i 10241 | Negative is one-to-one. (... |
negcon1i 10242 | Negative contraposition la... |
negcon2i 10243 | Negative contraposition la... |
negdii 10244 | Distribution of negative o... |
negsubdii 10245 | Distribution of negative o... |
negsubdi2i 10246 | Distribution of negative o... |
subaddi 10247 | Relationship between subtr... |
subadd2i 10248 | Relationship between subtr... |
subaddrii 10249 | Relationship between subtr... |
subsub23i 10250 | Swap subtrahend and result... |
addsubassi 10251 | Associative-type law for s... |
addsubi 10252 | Law for subtraction and ad... |
subcani 10253 | Cancellation law for subtr... |
subcan2i 10254 | Cancellation law for subtr... |
pnncani 10255 | Cancellation law for mixed... |
addsub4i 10256 | Rearrangement of 4 terms i... |
0reALT 10257 | Alternate proof of ~ 0re .... |
negcld 10258 | Closure law for negative. ... |
subidd 10259 | Subtraction of a number fr... |
subid1d 10260 | Identity law for subtracti... |
negidd 10261 | Addition of a number and i... |
negnegd 10262 | A number is equal to the n... |
negeq0d 10263 | A number is zero iff its n... |
negne0bd 10264 | A number is nonzero iff it... |
negcon1d 10265 | Contraposition law for una... |
negcon1ad 10266 | Contraposition law for una... |
neg11ad 10267 | The negatives of two compl... |
negned 10268 | If two complex numbers are... |
negne0d 10269 | The negative of a nonzero ... |
negrebd 10270 | The negative of a real is ... |
subcld 10271 | Closure law for subtractio... |
pncand 10272 | Cancellation law for subtr... |
pncan2d 10273 | Cancellation law for subtr... |
pncan3d 10274 | Subtraction and addition o... |
npcand 10275 | Cancellation law for subtr... |
nncand 10276 | Cancellation law for subtr... |
negsubd 10277 | Relationship between subtr... |
subnegd 10278 | Relationship between subtr... |
subeq0d 10279 | If the difference between ... |
subne0d 10280 | Two unequal numbers have n... |
subeq0ad 10281 | The difference of two comp... |
subne0ad 10282 | If the difference of two c... |
neg11d 10283 | If the difference between ... |
negdid 10284 | Distribution of negative o... |
negdi2d 10285 | Distribution of negative o... |
negsubdid 10286 | Distribution of negative o... |
negsubdi2d 10287 | Distribution of negative o... |
neg2subd 10288 | Relationship between subtr... |
subaddd 10289 | Relationship between subtr... |
subadd2d 10290 | Relationship between subtr... |
addsubassd 10291 | Associative-type law for s... |
addsubd 10292 | Law for subtraction and ad... |
subadd23d 10293 | Commutative/associative la... |
addsub12d 10294 | Commutative/associative la... |
npncand 10295 | Cancellation law for subtr... |
nppcand 10296 | Cancellation law for subtr... |
nppcan2d 10297 | Cancellation law for subtr... |
nppcan3d 10298 | Cancellation law for subtr... |
subsubd 10299 | Law for double subtraction... |
subsub2d 10300 | Law for double subtraction... |
subsub3d 10301 | Law for double subtraction... |
subsub4d 10302 | Law for double subtraction... |
sub32d 10303 | Swap the second and third ... |
nnncand 10304 | Cancellation law for subtr... |
nnncan1d 10305 | Cancellation law for subtr... |
nnncan2d 10306 | Cancellation law for subtr... |
npncan3d 10307 | Cancellation law for subtr... |
pnpcand 10308 | Cancellation law for mixed... |
pnpcan2d 10309 | Cancellation law for mixed... |
pnncand 10310 | Cancellation law for mixed... |
ppncand 10311 | Cancellation law for mixed... |
subcand 10312 | Cancellation law for subtr... |
subcan2d 10313 | Cancellation law for subtr... |
subcanad 10314 | Cancellation law for subtr... |
subneintrd 10315 | Introducing subtraction on... |
subcan2ad 10316 | Cancellation law for subtr... |
subneintr2d 10317 | Introducing subtraction on... |
addsub4d 10318 | Rearrangement of 4 terms i... |
subadd4d 10319 | Rearrangement of 4 terms i... |
sub4d 10320 | Rearrangement of 4 terms i... |
2addsubd 10321 | Law for subtraction and ad... |
addsubeq4d 10322 | Relation between sums and ... |
mvlraddd 10323 | Move LHS right addition to... |
mvrraddd 10324 | Move RHS right addition to... |
subaddeqd 10325 | Transfer two terms of a su... |
addlsub 10326 | Left-subtraction: Subtrac... |
addrsub 10327 | Right-subtraction: Subtra... |
subexsub 10328 | A subtraction law: Exchan... |
addid0 10329 | If adding a number to a an... |
addn0nid 10330 | Adding a nonzero number to... |
pnpncand 10331 | Addition/subtraction cance... |
subeqrev 10332 | Reverse the order of subtr... |
pncan1 10333 | Cancellation law for addit... |
npcan1 10334 | Cancellation law for subtr... |
subeq0bd 10335 | If two complex numbers are... |
renegcld 10336 | Closure law for negative o... |
resubcld 10337 | Closure law for subtractio... |
negn0 10338 | The image under negation o... |
negf1o 10339 | Negation is an isomorphism... |
kcnktkm1cn 10340 | k times k minus 1 is a com... |
muladd 10341 | Product of two sums. (Con... |
subdi 10342 | Distribution of multiplica... |
subdir 10343 | Distribution of multiplica... |
ine0 10344 | The imaginary unit ` _i ` ... |
mulneg1 10345 | Product with negative is n... |
mulneg2 10346 | The product with a negativ... |
mulneg12 10347 | Swap the negative sign in ... |
mul2neg 10348 | Product of two negatives. ... |
submul2 10349 | Convert a subtraction to a... |
mulm1 10350 | Product with minus one is ... |
addneg1mul 10351 | Addition with product with... |
mulsub 10352 | Product of two differences... |
mulsub2 10353 | Swap the order of subtract... |
mulm1i 10354 | Product with minus one is ... |
mulneg1i 10355 | Product with negative is n... |
mulneg2i 10356 | Product with negative is n... |
mul2negi 10357 | Product of two negatives. ... |
subdii 10358 | Distribution of multiplica... |
subdiri 10359 | Distribution of multiplica... |
muladdi 10360 | Product of two sums. (Con... |
mulm1d 10361 | Product with minus one is ... |
mulneg1d 10362 | Product with negative is n... |
mulneg2d 10363 | Product with negative is n... |
mul2negd 10364 | Product of two negatives. ... |
subdid 10365 | Distribution of multiplica... |
subdird 10366 | Distribution of multiplica... |
subdir2d 10367 | Distribution of multiplica... |
muladdd 10368 | Product of two sums. (Con... |
mulsubd 10369 | Product of two differences... |
muls1d 10370 | Multiplication by one minu... |
mulsubfacd 10371 | Multiplication followed by... |
gt0ne0 10372 | Positive implies nonzero. ... |
lt0ne0 10373 | A number which is less tha... |
ltadd1 10374 | Addition to both sides of ... |
leadd1 10375 | Addition to both sides of ... |
leadd2 10376 | Addition to both sides of ... |
ltsubadd 10377 | 'Less than' relationship b... |
ltsubadd2 10378 | 'Less than' relationship b... |
lesubadd 10379 | 'Less than or equal to' re... |
lesubadd2 10380 | 'Less than or equal to' re... |
ltaddsub 10381 | 'Less than' relationship b... |
ltaddsub2 10382 | 'Less than' relationship b... |
leaddsub 10383 | 'Less than or equal to' re... |
leaddsub2 10384 | 'Less than or equal to' re... |
suble 10385 | Swap subtrahends in an ine... |
lesub 10386 | Swap subtrahends in an ine... |
ltsub23 10387 | 'Less than' relationship b... |
ltsub13 10388 | 'Less than' relationship b... |
le2add 10389 | Adding both sides of two '... |
ltleadd 10390 | Adding both sides of two o... |
leltadd 10391 | Adding both sides of two o... |
lt2add 10392 | Adding both sides of two '... |
addgt0 10393 | The sum of 2 positive numb... |
addgegt0 10394 | The sum of nonnegative and... |
addgtge0 10395 | The sum of nonnegative and... |
addge0 10396 | The sum of 2 nonnegative n... |
ltaddpos 10397 | Adding a positive number t... |
ltaddpos2 10398 | Adding a positive number t... |
ltsubpos 10399 | Subtracting a positive num... |
posdif 10400 | Comparison of two numbers ... |
lesub1 10401 | Subtraction from both side... |
lesub2 10402 | Subtraction of both sides ... |
ltsub1 10403 | Subtraction from both side... |
ltsub2 10404 | Subtraction of both sides ... |
lt2sub 10405 | Subtracting both sides of ... |
le2sub 10406 | Subtracting both sides of ... |
ltneg 10407 | Negative of both sides of ... |
ltnegcon1 10408 | Contraposition of negative... |
ltnegcon2 10409 | Contraposition of negative... |
leneg 10410 | Negative of both sides of ... |
lenegcon1 10411 | Contraposition of negative... |
lenegcon2 10412 | Contraposition of negative... |
lt0neg1 10413 | Comparison of a number and... |
lt0neg2 10414 | Comparison of a number and... |
le0neg1 10415 | Comparison of a number and... |
le0neg2 10416 | Comparison of a number and... |
addge01 10417 | A number is less than or e... |
addge02 10418 | A number is less than or e... |
add20 10419 | Two nonnegative numbers ar... |
subge0 10420 | Nonnegative subtraction. ... |
suble0 10421 | Nonpositive subtraction. ... |
leaddle0 10422 | The sum of a real number a... |
subge02 10423 | Nonnegative subtraction. ... |
lesub0 10424 | Lemma to show a nonnegativ... |
mulge0 10425 | The product of two nonnega... |
mullt0 10426 | The product of two negativ... |
msqgt0 10427 | A nonzero square is positi... |
msqge0 10428 | A square is nonnegative. ... |
0lt1 10429 | 0 is less than 1. Theorem... |
0le1 10430 | 0 is less than or equal to... |
relin01 10431 | An interval law for less t... |
ltordlem 10432 | Lemma for ~ ltord1 . (Con... |
ltord1 10433 | Infer an ordering relation... |
leord1 10434 | Infer an ordering relation... |
eqord1 10435 | Infer an ordering relation... |
ltord2 10436 | Infer an ordering relation... |
leord2 10437 | Infer an ordering relation... |
eqord2 10438 | Infer an ordering relation... |
wloglei 10439 | Form of ~ wlogle where bot... |
wlogle 10440 | If the predicate ` ch ( x ... |
leidi 10441 | 'Less than or equal to' is... |
gt0ne0i 10442 | Positive means nonzero (us... |
gt0ne0ii 10443 | Positive implies nonzero. ... |
msqgt0i 10444 | A nonzero square is positi... |
msqge0i 10445 | A square is nonnegative. ... |
addgt0i 10446 | Addition of 2 positive num... |
addge0i 10447 | Addition of 2 nonnegative ... |
addgegt0i 10448 | Addition of nonnegative an... |
addgt0ii 10449 | Addition of 2 positive num... |
add20i 10450 | Two nonnegative numbers ar... |
ltnegi 10451 | Negative of both sides of ... |
lenegi 10452 | Negative of both sides of ... |
ltnegcon2i 10453 | Contraposition of negative... |
mulge0i 10454 | The product of two nonnega... |
lesub0i 10455 | Lemma to show a nonnegativ... |
ltaddposi 10456 | Adding a positive number t... |
posdifi 10457 | Comparison of two numbers ... |
ltnegcon1i 10458 | Contraposition of negative... |
lenegcon1i 10459 | Contraposition of negative... |
subge0i 10460 | Nonnegative subtraction. ... |
ltadd1i 10461 | Addition to both sides of ... |
leadd1i 10462 | Addition to both sides of ... |
leadd2i 10463 | Addition to both sides of ... |
ltsubaddi 10464 | 'Less than' relationship b... |
lesubaddi 10465 | 'Less than or equal to' re... |
ltsubadd2i 10466 | 'Less than' relationship b... |
lesubadd2i 10467 | 'Less than or equal to' re... |
ltaddsubi 10468 | 'Less than' relationship b... |
lt2addi 10469 | Adding both side of two in... |
le2addi 10470 | Adding both side of two in... |
gt0ne0d 10471 | Positive implies nonzero. ... |
lt0ne0d 10472 | Something less than zero i... |
leidd 10473 | 'Less than or equal to' is... |
msqgt0d 10474 | A nonzero square is positi... |
msqge0d 10475 | A square is nonnegative. ... |
lt0neg1d 10476 | Comparison of a number and... |
lt0neg2d 10477 | Comparison of a number and... |
le0neg1d 10478 | Comparison of a number and... |
le0neg2d 10479 | Comparison of a number and... |
addgegt0d 10480 | Addition of nonnegative an... |
addgt0d 10481 | Addition of 2 positive num... |
addge0d 10482 | Addition of 2 nonnegative ... |
mulge0d 10483 | The product of two nonnega... |
ltnegd 10484 | Negative of both sides of ... |
lenegd 10485 | Negative of both sides of ... |
ltnegcon1d 10486 | Contraposition of negative... |
ltnegcon2d 10487 | Contraposition of negative... |
lenegcon1d 10488 | Contraposition of negative... |
lenegcon2d 10489 | Contraposition of negative... |
ltaddposd 10490 | Adding a positive number t... |
ltaddpos2d 10491 | Adding a positive number t... |
ltsubposd 10492 | Subtracting a positive num... |
posdifd 10493 | Comparison of two numbers ... |
addge01d 10494 | A number is less than or e... |
addge02d 10495 | A number is less than or e... |
subge0d 10496 | Nonnegative subtraction. ... |
suble0d 10497 | Nonpositive subtraction. ... |
subge02d 10498 | Nonnegative subtraction. ... |
ltadd1d 10499 | Addition to both sides of ... |
leadd1d 10500 | Addition to both sides of ... |
leadd2d 10501 | Addition to both sides of ... |
ltsubaddd 10502 | 'Less than' relationship b... |
lesubaddd 10503 | 'Less than or equal to' re... |
ltsubadd2d 10504 | 'Less than' relationship b... |
lesubadd2d 10505 | 'Less than or equal to' re... |
ltaddsubd 10506 | 'Less than' relationship b... |
ltaddsub2d 10507 | 'Less than' relationship b... |
leaddsub2d 10508 | 'Less than or equal to' re... |
subled 10509 | Swap subtrahends in an ine... |
lesubd 10510 | Swap subtrahends in an ine... |
ltsub23d 10511 | 'Less than' relationship b... |
ltsub13d 10512 | 'Less than' relationship b... |
lesub1d 10513 | Subtraction from both side... |
lesub2d 10514 | Subtraction of both sides ... |
ltsub1d 10515 | Subtraction from both side... |
ltsub2d 10516 | Subtraction of both sides ... |
ltadd1dd 10517 | Addition to both sides of ... |
ltsub1dd 10518 | Subtraction from both side... |
ltsub2dd 10519 | Subtraction of both sides ... |
leadd1dd 10520 | Addition to both sides of ... |
leadd2dd 10521 | Addition to both sides of ... |
lesub1dd 10522 | Subtraction from both side... |
lesub2dd 10523 | Subtraction of both sides ... |
lesub3d 10524 | The result of subtracting ... |
le2addd 10525 | Adding both side of two in... |
le2subd 10526 | Subtracting both sides of ... |
ltleaddd 10527 | Adding both sides of two o... |
leltaddd 10528 | Adding both sides of two o... |
lt2addd 10529 | Adding both side of two in... |
lt2subd 10530 | Subtracting both sides of ... |
possumd 10531 | Condition for a positive s... |
sublt0d 10532 | When a subtraction gives a... |
ltaddsublt 10533 | Addition and subtraction o... |
1le1 10534 | ` 1 <_ 1 ` . Common speci... |
ixi 10535 | ` _i ` times itself is min... |
recextlem1 10536 | Lemma for ~ recex . (Cont... |
recextlem2 10537 | Lemma for ~ recex . (Cont... |
recex 10538 | Existence of reciprocal of... |
mulcand 10539 | Cancellation law for multi... |
mulcan2d 10540 | Cancellation law for multi... |
mulcanad 10541 | Cancellation of a nonzero ... |
mulcan2ad 10542 | Cancellation of a nonzero ... |
mulcan 10543 | Cancellation law for multi... |
mulcan2 10544 | Cancellation law for multi... |
mulcani 10545 | Cancellation law for multi... |
mul0or 10546 | If a product is zero, one ... |
mulne0b 10547 | The product of two nonzero... |
mulne0 10548 | The product of two nonzero... |
mulne0i 10549 | The product of two nonzero... |
muleqadd 10550 | Property of numbers whose ... |
receu 10551 | Existential uniqueness of ... |
mulnzcnopr 10552 | Multiplication maps nonzer... |
msq0i 10553 | A number is zero iff its s... |
mul0ori 10554 | If a product is zero, one ... |
msq0d 10555 | A number is zero iff its s... |
mul0ord 10556 | If a product is zero, one ... |
mulne0bd 10557 | The product of two nonzero... |
mulne0d 10558 | The product of two nonzero... |
mulcan1g 10559 | A generalized form of the ... |
mulcan2g 10560 | A generalized form of the ... |
mulne0bad 10561 | A factor of a nonzero comp... |
mulne0bbd 10562 | A factor of a nonzero comp... |
1div0 10565 | You can't divide by zero, ... |
divval 10566 | Value of division: if ` A ... |
divmul 10567 | Relationship between divis... |
divmul2 10568 | Relationship between divis... |
divmul3 10569 | Relationship between divis... |
divcl 10570 | Closure law for division. ... |
reccl 10571 | Closure law for reciprocal... |
divcan2 10572 | A cancellation law for div... |
divcan1 10573 | A cancellation law for div... |
diveq0 10574 | A ratio is zero iff the nu... |
divne0b 10575 | The ratio of nonzero numbe... |
divne0 10576 | The ratio of nonzero numbe... |
recne0 10577 | The reciprocal of a nonzer... |
recid 10578 | Multiplication of a number... |
recid2 10579 | Multiplication of a number... |
divrec 10580 | Relationship between divis... |
divrec2 10581 | Relationship between divis... |
divass 10582 | An associative law for div... |
div23 10583 | A commutative/associative ... |
div32 10584 | A commutative/associative ... |
div13 10585 | A commutative/associative ... |
div12 10586 | A commutative/associative ... |
divmulass 10587 | An associative law for div... |
divmulasscom 10588 | An associative/commutative... |
divdir 10589 | Distribution of division o... |
divcan3 10590 | A cancellation law for div... |
divcan4 10591 | A cancellation law for div... |
div11 10592 | One-to-one relationship fo... |
divid 10593 | A number divided by itself... |
div0 10594 | Division into zero is zero... |
div1 10595 | A number divided by 1 is i... |
1div1e1 10596 | 1 divided by 1 is 1 (commo... |
diveq1 10597 | Equality in terms of unit ... |
divneg 10598 | Move negative sign inside ... |
muldivdir 10599 | Distribution of division o... |
divsubdir 10600 | Distribution of division o... |
recrec 10601 | A number is equal to the r... |
rec11 10602 | Reciprocal is one-to-one. ... |
rec11r 10603 | Mutual reciprocals. (Cont... |
divmuldiv 10604 | Multiplication of two rati... |
divdivdiv 10605 | Division of two ratios. T... |
divcan5 10606 | Cancellation of common fac... |
divmul13 10607 | Swap the denominators in t... |
divmul24 10608 | Swap the numerators in the... |
divmuleq 10609 | Cross-multiply in an equal... |
recdiv 10610 | The reciprocal of a ratio.... |
divcan6 10611 | Cancellation of inverted f... |
divdiv32 10612 | Swap denominators in a div... |
divcan7 10613 | Cancel equal divisors in a... |
dmdcan 10614 | Cancellation law for divis... |
divdiv1 10615 | Division into a fraction. ... |
divdiv2 10616 | Division by a fraction. (... |
recdiv2 10617 | Division into a reciprocal... |
ddcan 10618 | Cancellation in a double d... |
divadddiv 10619 | Addition of two ratios. T... |
divsubdiv 10620 | Subtraction of two ratios.... |
conjmul 10621 | Two numbers whose reciproc... |
rereccl 10622 | Closure law for reciprocal... |
redivcl 10623 | Closure law for division o... |
eqneg 10624 | A number equal to its nega... |
eqnegd 10625 | A complex number equals it... |
eqnegad 10626 | If a complex number equals... |
div2neg 10627 | Quotient of two negatives.... |
divneg2 10628 | Move negative sign inside ... |
recclzi 10629 | Closure law for reciprocal... |
recne0zi 10630 | The reciprocal of a nonzer... |
recidzi 10631 | Multiplication of a number... |
div1i 10632 | A number divided by 1 is i... |
eqnegi 10633 | A number equal to its nega... |
reccli 10634 | Closure law for reciprocal... |
recidi 10635 | Multiplication of a number... |
recreci 10636 | A number is equal to the r... |
dividi 10637 | A number divided by itself... |
div0i 10638 | Division into zero is zero... |
divclzi 10639 | Closure law for division. ... |
divcan1zi 10640 | A cancellation law for div... |
divcan2zi 10641 | A cancellation law for div... |
divreczi 10642 | Relationship between divis... |
divcan3zi 10643 | A cancellation law for div... |
divcan4zi 10644 | A cancellation law for div... |
rec11i 10645 | Reciprocal is one-to-one. ... |
divcli 10646 | Closure law for division. ... |
divcan2i 10647 | A cancellation law for div... |
divcan1i 10648 | A cancellation law for div... |
divreci 10649 | Relationship between divis... |
divcan3i 10650 | A cancellation law for div... |
divcan4i 10651 | A cancellation law for div... |
divne0i 10652 | The ratio of nonzero numbe... |
rec11ii 10653 | Reciprocal is one-to-one. ... |
divasszi 10654 | An associative law for div... |
divmulzi 10655 | Relationship between divis... |
divdirzi 10656 | Distribution of division o... |
divdiv23zi 10657 | Swap denominators in a div... |
divmuli 10658 | Relationship between divis... |
divdiv32i 10659 | Swap denominators in a div... |
divassi 10660 | An associative law for div... |
divdiri 10661 | Distribution of division o... |
div23i 10662 | A commutative/associative ... |
div11i 10663 | One-to-one relationship fo... |
divmuldivi 10664 | Multiplication of two rati... |
divmul13i 10665 | Swap denominators of two r... |
divadddivi 10666 | Addition of two ratios. T... |
divdivdivi 10667 | Division of two ratios. T... |
rerecclzi 10668 | Closure law for reciprocal... |
rereccli 10669 | Closure law for reciprocal... |
redivclzi 10670 | Closure law for division o... |
redivcli 10671 | Closure law for division o... |
div1d 10672 | A number divided by 1 is i... |
reccld 10673 | Closure law for reciprocal... |
recne0d 10674 | The reciprocal of a nonzer... |
recidd 10675 | Multiplication of a number... |
recid2d 10676 | Multiplication of a number... |
recrecd 10677 | A number is equal to the r... |
dividd 10678 | A number divided by itself... |
div0d 10679 | Division into zero is zero... |
divcld 10680 | Closure law for division. ... |
divcan1d 10681 | A cancellation law for div... |
divcan2d 10682 | A cancellation law for div... |
divrecd 10683 | Relationship between divis... |
divrec2d 10684 | Relationship between divis... |
divcan3d 10685 | A cancellation law for div... |
divcan4d 10686 | A cancellation law for div... |
diveq0d 10687 | A ratio is zero iff the nu... |
diveq1d 10688 | Equality in terms of unit ... |
diveq1ad 10689 | The quotient of two comple... |
diveq0ad 10690 | A fraction of complex numb... |
divne1d 10691 | If two complex numbers are... |
divne0bd 10692 | A ratio is zero iff the nu... |
divnegd 10693 | Move negative sign inside ... |
divneg2d 10694 | Move negative sign inside ... |
div2negd 10695 | Quotient of two negatives.... |
divne0d 10696 | The ratio of nonzero numbe... |
recdivd 10697 | The reciprocal of a ratio.... |
recdiv2d 10698 | Division into a reciprocal... |
divcan6d 10699 | Cancellation of inverted f... |
ddcand 10700 | Cancellation in a double d... |
rec11d 10701 | Reciprocal is one-to-one. ... |
divmuld 10702 | Relationship between divis... |
div32d 10703 | A commutative/associative ... |
div13d 10704 | A commutative/associative ... |
divdiv32d 10705 | Swap denominators in a div... |
divcan5d 10706 | Cancellation of common fac... |
divcan5rd 10707 | Cancellation of common fac... |
divcan7d 10708 | Cancel equal divisors in a... |
dmdcand 10709 | Cancellation law for divis... |
dmdcan2d 10710 | Cancellation law for divis... |
divdiv1d 10711 | Division into a fraction. ... |
divdiv2d 10712 | Division by a fraction. (... |
divmul2d 10713 | Relationship between divis... |
divmul3d 10714 | Relationship between divis... |
divassd 10715 | An associative law for div... |
div12d 10716 | A commutative/associative ... |
div23d 10717 | A commutative/associative ... |
divdird 10718 | Distribution of division o... |
divsubdird 10719 | Distribution of division o... |
div11d 10720 | One-to-one relationship fo... |
divmuldivd 10721 | Multiplication of two rati... |
divmul13d 10722 | Swap denominators of two r... |
divmul24d 10723 | Swap the numerators in the... |
divadddivd 10724 | Addition of two ratios. T... |
divsubdivd 10725 | Subtraction of two ratios.... |
divmuleqd 10726 | Cross-multiply in an equal... |
divdivdivd 10727 | Division of two ratios. T... |
diveq1bd 10728 | If two complex numbers are... |
div2sub 10729 | Swap the order of subtract... |
div2subd 10730 | Swap subtrahend and minuen... |
rereccld 10731 | Closure law for reciprocal... |
redivcld 10732 | Closure law for division o... |
subrec 10733 | Subtraction of reciprocals... |
subreci 10734 | Subtraction of reciprocals... |
subrecd 10735 | Subtraction of reciprocals... |
mvllmuld 10736 | Move LHS left multiplicati... |
mvllmuli 10737 | Move LHS left multiplicati... |
elimgt0 10738 | Hypothesis for weak deduct... |
elimge0 10739 | Hypothesis for weak deduct... |
ltp1 10740 | A number is less than itse... |
lep1 10741 | A number is less than or e... |
ltm1 10742 | A number minus 1 is less t... |
lem1 10743 | A number minus 1 is less t... |
letrp1 10744 | A transitive property of '... |
p1le 10745 | A transitive property of p... |
recgt0 10746 | The reciprocal of a positi... |
prodgt0 10747 | Infer that a multiplicand ... |
prodgt02 10748 | Infer that a multiplier is... |
prodge0 10749 | Infer that a multiplicand ... |
prodge02 10750 | Infer that a multiplier is... |
ltmul1a 10751 | Lemma for ~ ltmul1 . Mult... |
ltmul1 10752 | Multiplication of both sid... |
ltmul2 10753 | Multiplication of both sid... |
lemul1 10754 | Multiplication of both sid... |
lemul2 10755 | Multiplication of both sid... |
lemul1a 10756 | Multiplication of both sid... |
lemul2a 10757 | Multiplication of both sid... |
ltmul12a 10758 | Comparison of product of t... |
lemul12b 10759 | Comparison of product of t... |
lemul12a 10760 | Comparison of product of t... |
mulgt1 10761 | The product of two numbers... |
ltmulgt11 10762 | Multiplication by a number... |
ltmulgt12 10763 | Multiplication by a number... |
lemulge11 10764 | Multiplication by a number... |
lemulge12 10765 | Multiplication by a number... |
ltdiv1 10766 | Division of both sides of ... |
lediv1 10767 | Division of both sides of ... |
gt0div 10768 | Division of a positive num... |
ge0div 10769 | Division of a nonnegative ... |
divgt0 10770 | The ratio of two positive ... |
divge0 10771 | The ratio of nonnegative a... |
mulge0b 10772 | A condition for multiplica... |
mulle0b 10773 | A condition for multiplica... |
mulsuble0b 10774 | A condition for multiplica... |
ltmuldiv 10775 | 'Less than' relationship b... |
ltmuldiv2 10776 | 'Less than' relationship b... |
ltdivmul 10777 | 'Less than' relationship b... |
ledivmul 10778 | 'Less than or equal to' re... |
ltdivmul2 10779 | 'Less than' relationship b... |
lt2mul2div 10780 | 'Less than' relationship b... |
ledivmul2 10781 | 'Less than or equal to' re... |
lemuldiv 10782 | 'Less than or equal' relat... |
lemuldiv2 10783 | 'Less than or equal' relat... |
ltrec 10784 | The reciprocal of both sid... |
lerec 10785 | The reciprocal of both sid... |
lt2msq1 10786 | Lemma for ~ lt2msq . (Con... |
lt2msq 10787 | Two nonnegative numbers co... |
ltdiv2 10788 | Division of a positive num... |
ltrec1 10789 | Reciprocal swap in a 'less... |
lerec2 10790 | Reciprocal swap in a 'less... |
ledivdiv 10791 | Invert ratios of positive ... |
lediv2 10792 | Division of a positive num... |
ltdiv23 10793 | Swap denominator with othe... |
lediv23 10794 | Swap denominator with othe... |
lediv12a 10795 | Comparison of ratio of two... |
lediv2a 10796 | Division of both sides of ... |
reclt1 10797 | The reciprocal of a positi... |
recgt1 10798 | The reciprocal of a positi... |
recgt1i 10799 | The reciprocal of a number... |
recp1lt1 10800 | Construct a number less th... |
recreclt 10801 | Given a positive number ` ... |
le2msq 10802 | The square function on non... |
msq11 10803 | The square of a nonnegativ... |
ledivp1 10804 | Less-than-or-equal-to and ... |
squeeze0 10805 | If a nonnegative number is... |
ltp1i 10806 | A number is less than itse... |
recgt0i 10807 | The reciprocal of a positi... |
recgt0ii 10808 | The reciprocal of a positi... |
prodgt0i 10809 | Infer that a multiplicand ... |
prodge0i 10810 | Infer that a multiplicand ... |
divgt0i 10811 | The ratio of two positive ... |
divge0i 10812 | The ratio of nonnegative a... |
ltreci 10813 | The reciprocal of both sid... |
lereci 10814 | The reciprocal of both sid... |
lt2msqi 10815 | The square function on non... |
le2msqi 10816 | The square function on non... |
msq11i 10817 | The square of a nonnegativ... |
divgt0i2i 10818 | The ratio of two positive ... |
ltrecii 10819 | The reciprocal of both sid... |
divgt0ii 10820 | The ratio of two positive ... |
ltmul1i 10821 | Multiplication of both sid... |
ltdiv1i 10822 | Division of both sides of ... |
ltmuldivi 10823 | 'Less than' relationship b... |
ltmul2i 10824 | Multiplication of both sid... |
lemul1i 10825 | Multiplication of both sid... |
lemul2i 10826 | Multiplication of both sid... |
ltdiv23i 10827 | Swap denominator with othe... |
ledivp1i 10828 | Less-than-or-equal-to and ... |
ltdivp1i 10829 | Less-than and division rel... |
ltdiv23ii 10830 | Swap denominator with othe... |
ltmul1ii 10831 | Multiplication of both sid... |
ltdiv1ii 10832 | Division of both sides of ... |
ltp1d 10833 | A number is less than itse... |
lep1d 10834 | A number is less than or e... |
ltm1d 10835 | A number minus 1 is less t... |
lem1d 10836 | A number minus 1 is less t... |
recgt0d 10837 | The reciprocal of a positi... |
divgt0d 10838 | The ratio of two positive ... |
mulgt1d 10839 | The product of two numbers... |
lemulge11d 10840 | Multiplication by a number... |
lemulge12d 10841 | Multiplication by a number... |
lemul1ad 10842 | Multiplication of both sid... |
lemul2ad 10843 | Multiplication of both sid... |
ltmul12ad 10844 | Comparison of product of t... |
lemul12ad 10845 | Comparison of product of t... |
lemul12bd 10846 | Comparison of product of t... |
fimaxre 10847 | A finite set of real numbe... |
fimaxre2 10848 | A nonempty finite set of r... |
fimaxre3 10849 | A nonempty finite set of r... |
negfi 10850 | The negation of a finite s... |
fiminre 10851 | A nonempty finite set of r... |
lbreu 10852 | If a set of reals contains... |
lbcl 10853 | If a set of reals contains... |
lble 10854 | If a set of reals contains... |
lbinf 10855 | If a set of reals contains... |
lbinfcl 10856 | If a set of reals contains... |
lbinfle 10857 | If a set of reals contains... |
sup2 10858 | A nonempty, bounded-above ... |
sup3 10859 | A version of the completen... |
infm3lem 10860 | Lemma for ~ infm3 . (Cont... |
infm3 10861 | The completeness axiom for... |
suprcl 10862 | Closure of supremum of a n... |
suprub 10863 | A member of a nonempty bou... |
suprlub 10864 | The supremum of a nonempty... |
suprnub 10865 | An upper bound is not less... |
suprleub 10866 | The supremum of a nonempty... |
supaddc 10867 | The supremum function dist... |
supadd 10868 | The supremum function dist... |
supmul1 10869 | The supremum function dist... |
supmullem1 10870 | Lemma for ~ supmul . (Con... |
supmullem2 10871 | Lemma for ~ supmul . (Con... |
supmul 10872 | The supremum function dist... |
sup3ii 10873 | A version of the completen... |
suprclii 10874 | Closure of supremum of a n... |
suprubii 10875 | A member of a nonempty bou... |
suprlubii 10876 | The supremum of a nonempty... |
suprnubii 10877 | An upper bound is not less... |
suprleubii 10878 | The supremum of a nonempty... |
riotaneg 10879 | The negative of the unique... |
negiso 10880 | Negation is an order anti-... |
dfinfre 10881 | The infimum of a set of re... |
infrecl 10882 | Closure of infimum of a no... |
infrenegsup 10883 | The infimum of a set of re... |
infregelb 10884 | Any lower bound of a nonem... |
infrelb 10885 | If a nonempty set of real ... |
supfirege 10886 | The supremum of a finite s... |
inelr 10887 | The imaginary unit ` _i ` ... |
rimul 10888 | A real number times the im... |
cru 10889 | The representation of comp... |
crne0 10890 | The real representation of... |
creur 10891 | The real part of a complex... |
creui 10892 | The imaginary part of a co... |
cju 10893 | The complex conjugate of a... |
ofsubeq0 10894 | Function analogue of ~ sub... |
ofnegsub 10895 | Function analogue of ~ neg... |
ofsubge0 10896 | Function analogue of ~ sub... |
nnexALT 10899 | Alternate proof of ~ nnex ... |
peano5nni 10900 | Peano's inductive postulat... |
nnssre 10901 | The positive integers are ... |
nnsscn 10902 | The positive integers are ... |
nnex 10903 | The set of positive intege... |
nnre 10904 | A positive integer is a re... |
nncn 10905 | A positive integer is a co... |
nnrei 10906 | A positive integer is a re... |
nncni 10907 | A positive integer is a co... |
1nn 10908 | Peano postulate: 1 is a po... |
peano2nn 10909 | Peano postulate: a success... |
dfnn2 10910 | Alternate definition of th... |
dfnn3 10911 | Alternate definition of th... |
nnred 10912 | A positive integer is a re... |
nncnd 10913 | A positive integer is a co... |
peano2nnd 10914 | Peano postulate: a success... |
nnind 10915 | Principle of Mathematical ... |
nnindALT 10916 | Principle of Mathematical ... |
nn1m1nn 10917 | Every positive integer is ... |
nn1suc 10918 | If a statement holds for 1... |
nnaddcl 10919 | Closure of addition of pos... |
nnmulcl 10920 | Closure of multiplication ... |
nnmulcli 10921 | Closure of multiplication ... |
nn2ge 10922 | There exists a positive in... |
nnge1 10923 | A positive integer is one ... |
nngt1ne1 10924 | A positive integer is grea... |
nnle1eq1 10925 | A positive integer is less... |
nngt0 10926 | A positive integer is posi... |
nnnlt1 10927 | A positive integer is not ... |
nnnle0 10928 | A positive integer is not ... |
0nnn 10929 | Zero is not a positive int... |
nnne0 10930 | A positive integer is nonz... |
nngt0i 10931 | A positive integer is posi... |
nnne0i 10932 | A positive integer is nonz... |
nndivre 10933 | The quotient of a real and... |
nnrecre 10934 | The reciprocal of a positi... |
nnrecgt0 10935 | The reciprocal of a positi... |
nnsub 10936 | Subtraction of positive in... |
nnsubi 10937 | Subtraction of positive in... |
nndiv 10938 | Two ways to express " ` A ... |
nndivtr 10939 | Transitive property of div... |
nnge1d 10940 | A positive integer is one ... |
nngt0d 10941 | A positive integer is posi... |
nnne0d 10942 | A positive integer is nonz... |
nnrecred 10943 | The reciprocal of a positi... |
nnaddcld 10944 | Closure of addition of pos... |
nnmulcld 10945 | Closure of multiplication ... |
nndivred 10946 | A positive integer is one ... |
0ne1 10965 | ` 0 =/= 1 ` (common case);... |
1m1e0 10966 | ` ( 1 - 1 ) = 0 ` (common ... |
2re 10967 | The number 2 is real. (Co... |
2cn 10968 | The number 2 is a complex ... |
2ex 10969 | 2 is a set (common case). ... |
2cnd 10970 | 2 is a complex number, ded... |
3re 10971 | The number 3 is real. (Co... |
3cn 10972 | The number 3 is a complex ... |
3ex 10973 | 3 is a set (common case). ... |
4re 10974 | The number 4 is real. (Co... |
4cn 10975 | The number 4 is a complex ... |
5re 10976 | The number 5 is real. (Co... |
5cn 10977 | The number 5 is complex. ... |
6re 10978 | The number 6 is real. (Co... |
6cn 10979 | The number 6 is complex. ... |
7re 10980 | The number 7 is real. (Co... |
7cn 10981 | The number 7 is complex. ... |
8re 10982 | The number 8 is real. (Co... |
8cn 10983 | The number 8 is complex. ... |
9re 10984 | The number 9 is real. (Co... |
9cn 10985 | The number 9 is complex. ... |
10reOLD 10986 | Obsolete version of ~ 10re... |
0le0 10987 | Zero is nonnegative. (Con... |
0le2 10988 | 0 is less than or equal to... |
2pos 10989 | The number 2 is positive. ... |
2ne0 10990 | The number 2 is nonzero. ... |
3pos 10991 | The number 3 is positive. ... |
3ne0 10992 | The number 3 is nonzero. ... |
4pos 10993 | The number 4 is positive. ... |
4ne0 10994 | The number 4 is nonzero. ... |
5pos 10995 | The number 5 is positive. ... |
6pos 10996 | The number 6 is positive. ... |
7pos 10997 | The number 7 is positive. ... |
8pos 10998 | The number 8 is positive. ... |
9pos 10999 | The number 9 is positive. ... |
10posOLD 11000 | The number 10 is positive.... |
neg1cn 11001 | -1 is a complex number (co... |
neg1rr 11002 | -1 is a real number (commo... |
neg1ne0 11003 | -1 is nonzero (common case... |
neg1lt0 11004 | -1 is less than 0 (common ... |
negneg1e1 11005 | ` -u -u 1 ` is 1 (common c... |
1pneg1e0 11006 | ` 1 + -u 1 ` is 0 (common ... |
0m0e0 11007 | 0 minus 0 equals 0 (common... |
1m0e1 11008 | 1 - 0 = 1 (common case). ... |
0p1e1 11009 | 0 + 1 = 1. (Contributed b... |
1p0e1 11010 | 1 + 0 = 1. (Contributed b... |
1p1e2 11011 | 1 + 1 = 2. (Contributed b... |
2m1e1 11012 | 2 - 1 = 1. The result is ... |
1e2m1 11013 | 1 = 2 - 1 (common case). ... |
3m1e2 11014 | 3 - 1 = 2. (Contributed b... |
4m1e3 11015 | 4 - 1 = 3. (Contributed b... |
5m1e4 11016 | 5 - 1 = 4. (Contributed b... |
6m1e5 11017 | 6 - 1 = 5. (Contributed b... |
7m1e6 11018 | 7 - 1 = 6. (Contributed b... |
8m1e7 11019 | 8 - 1 = 7. (Contributed b... |
9m1e8 11020 | 9 - 1 = 8. (Contributed b... |
2p2e4 11021 | Two plus two equals four. ... |
2times 11022 | Two times a number. (Cont... |
times2 11023 | A number times 2. (Contri... |
2timesi 11024 | Two times a number. (Cont... |
times2i 11025 | A number times 2. (Contri... |
2txmxeqx 11026 | Two times a complex number... |
2div2e1 11027 | 2 divided by 2 is 1 (commo... |
2p1e3 11028 | 2 + 1 = 3. (Contributed b... |
1p2e3 11029 | 1 + 2 = 3 (common case). ... |
3p1e4 11030 | 3 + 1 = 4. (Contributed b... |
4p1e5 11031 | 4 + 1 = 5. (Contributed b... |
5p1e6 11032 | 5 + 1 = 6. (Contributed b... |
6p1e7 11033 | 6 + 1 = 7. (Contributed b... |
7p1e8 11034 | 7 + 1 = 8. (Contributed b... |
8p1e9 11035 | 8 + 1 = 9. (Contributed b... |
9p1e10OLD 11036 | 9 + 1 = 10. (Contributed ... |
3p2e5 11037 | 3 + 2 = 5. (Contributed b... |
3p3e6 11038 | 3 + 3 = 6. (Contributed b... |
4p2e6 11039 | 4 + 2 = 6. (Contributed b... |
4p3e7 11040 | 4 + 3 = 7. (Contributed b... |
4p4e8 11041 | 4 + 4 = 8. (Contributed b... |
5p2e7 11042 | 5 + 2 = 7. (Contributed b... |
5p3e8 11043 | 5 + 3 = 8. (Contributed b... |
5p4e9 11044 | 5 + 4 = 9. (Contributed b... |
5p5e10OLD 11045 | 5 + 5 = 10. (Contributed ... |
6p2e8 11046 | 6 + 2 = 8. (Contributed b... |
6p3e9 11047 | 6 + 3 = 9. (Contributed b... |
6p4e10OLD 11048 | 6 + 4 = 10. (Contributed ... |
7p2e9 11049 | 7 + 2 = 9. (Contributed b... |
7p3e10OLD 11050 | 7 + 3 = 10. (Contributed ... |
8p2e10OLD 11051 | 8 + 2 = 10. (Contributed ... |
1t1e1 11052 | 1 times 1 equals 1. (Cont... |
2t1e2 11053 | 2 times 1 equals 2. (Cont... |
2t2e4 11054 | 2 times 2 equals 4. (Cont... |
3t1e3 11055 | 3 times 1 equals 3. (Cont... |
3t2e6 11056 | 3 times 2 equals 6. (Cont... |
3t3e9 11057 | 3 times 3 equals 9. (Cont... |
4t2e8 11058 | 4 times 2 equals 8. (Cont... |
5t2e10OLD 11059 | 5 times 2 equals 10. (Con... |
2t0e0 11060 | 2 times 0 equals 0. (Cont... |
4d2e2 11061 | One half of four is two. ... |
2nn 11062 | 2 is a positive integer. ... |
3nn 11063 | 3 is a positive integer. ... |
4nn 11064 | 4 is a positive integer. ... |
5nn 11065 | 5 is a positive integer. ... |
6nn 11066 | 6 is a positive integer. ... |
7nn 11067 | 7 is a positive integer. ... |
8nn 11068 | 8 is a positive integer. ... |
9nn 11069 | 9 is a positive integer. ... |
10nnOLD 11070 | Obsolete version of ~ 10nn... |
1lt2 11071 | 1 is less than 2. (Contri... |
2lt3 11072 | 2 is less than 3. (Contri... |
1lt3 11073 | 1 is less than 3. (Contri... |
3lt4 11074 | 3 is less than 4. (Contri... |
2lt4 11075 | 2 is less than 4. (Contri... |
1lt4 11076 | 1 is less than 4. (Contri... |
4lt5 11077 | 4 is less than 5. (Contri... |
3lt5 11078 | 3 is less than 5. (Contri... |
2lt5 11079 | 2 is less than 5. (Contri... |
1lt5 11080 | 1 is less than 5. (Contri... |
5lt6 11081 | 5 is less than 6. (Contri... |
4lt6 11082 | 4 is less than 6. (Contri... |
3lt6 11083 | 3 is less than 6. (Contri... |
2lt6 11084 | 2 is less than 6. (Contri... |
1lt6 11085 | 1 is less than 6. (Contri... |
6lt7 11086 | 6 is less than 7. (Contri... |
5lt7 11087 | 5 is less than 7. (Contri... |
4lt7 11088 | 4 is less than 7. (Contri... |
3lt7 11089 | 3 is less than 7. (Contri... |
2lt7 11090 | 2 is less than 7. (Contri... |
1lt7 11091 | 1 is less than 7. (Contri... |
7lt8 11092 | 7 is less than 8. (Contri... |
6lt8 11093 | 6 is less than 8. (Contri... |
5lt8 11094 | 5 is less than 8. (Contri... |
4lt8 11095 | 4 is less than 8. (Contri... |
3lt8 11096 | 3 is less than 8. (Contri... |
2lt8 11097 | 2 is less than 8. (Contri... |
1lt8 11098 | 1 is less than 8. (Contri... |
8lt9 11099 | 8 is less than 9. (Contri... |
7lt9 11100 | 7 is less than 9. (Contri... |
6lt9 11101 | 6 is less than 9. (Contri... |
5lt9 11102 | 5 is less than 9. (Contri... |
4lt9 11103 | 4 is less than 9. (Contri... |
3lt9 11104 | 3 is less than 9. (Contri... |
2lt9 11105 | 2 is less than 9. (Contri... |
1lt9 11106 | 1 is less than 9. (Contri... |
9lt10OLD 11107 | 9 is less than 10. (Contr... |
8lt10OLD 11108 | 8 is less than 10. (Contr... |
7lt10OLD 11109 | 7 is less than 10. (Contr... |
6lt10OLD 11110 | 6 is less than 10. (Contr... |
5lt10OLD 11111 | 5 is less than 10. (Contr... |
4lt10OLD 11112 | 4 is less than 10. (Contr... |
3lt10OLD 11113 | 3 is less than 10. (Contr... |
2lt10OLD 11114 | 2 is less than 10. (Contr... |
1lt10OLD 11115 | 1 is less than 10. (Contr... |
0ne2 11116 | 0 is not equal to 2. (Con... |
1ne2 11117 | 1 is not equal to 2. (Con... |
1le2 11118 | 1 is less than or equal to... |
2cnne0 11119 | 2 is a nonzero complex num... |
2rene0 11120 | 2 is a nonzero real number... |
1le3 11121 | 1 is less than or equal to... |
neg1mulneg1e1 11122 | ` -u 1 x. -u 1 ` is 1 (com... |
halfre 11123 | One-half is real. (Contri... |
halfcn 11124 | One-half is complex. (Con... |
halfgt0 11125 | One-half is greater than z... |
halfge0 11126 | One-half is not negative. ... |
halflt1 11127 | One-half is less than one.... |
1mhlfehlf 11128 | Prove that 1 - 1/2 = 1/2. ... |
8th4div3 11129 | An eighth of four thirds i... |
halfpm6th 11130 | One half plus or minus one... |
it0e0 11131 | i times 0 equals 0 (common... |
2mulicn 11132 | ` ( 2 x. _i ) e. CC ` (com... |
2muline0 11133 | ` ( 2 x. _i ) =/= 0 ` (com... |
halfcl 11134 | Closure of half of a numbe... |
rehalfcl 11135 | Real closure of half. (Co... |
half0 11136 | Half of a number is zero i... |
2halves 11137 | Two halves make a whole. ... |
halfpos2 11138 | A number is positive iff i... |
halfpos 11139 | A positive number is great... |
halfnneg2 11140 | A number is nonnegative if... |
halfaddsubcl 11141 | Closure of half-sum and ha... |
halfaddsub 11142 | Sum and difference of half... |
subhalfhalf 11143 | Subtracting the half of a ... |
lt2halves 11144 | A sum is less than the who... |
addltmul 11145 | Sum is less than product f... |
nominpos 11146 | There is no smallest posit... |
avglt1 11147 | Ordering property for aver... |
avglt2 11148 | Ordering property for aver... |
avgle1 11149 | Ordering property for aver... |
avgle2 11150 | Ordering property for aver... |
avgle 11151 | The average of two numbers... |
2timesd 11152 | Two times a number. (Cont... |
times2d 11153 | A number times 2. (Contri... |
halfcld 11154 | Closure of half of a numbe... |
2halvesd 11155 | Two halves make a whole. ... |
rehalfcld 11156 | Real closure of half. (Co... |
lt2halvesd 11157 | A sum is less than the who... |
rehalfcli 11158 | Half a real number is real... |
lt2addmuld 11159 | If two real numbers are le... |
add1p1 11160 | Adding two times 1 to a nu... |
sub1m1 11161 | Subtracting two times 1 fr... |
cnm2m1cnm3 11162 | Subtracting 2 and afterwar... |
xp1d2m1eqxm1d2 11163 | A complex number increased... |
div4p1lem1div2 11164 | An integer greater than 5,... |
nnunb 11165 | The set of positive intege... |
arch 11166 | Archimedean property of re... |
nnrecl 11167 | There exists a positive in... |
bndndx 11168 | A bounded real sequence ` ... |
elnn0 11171 | Nonnegative integers expre... |
nnssnn0 11172 | Positive naturals are a su... |
nn0ssre 11173 | Nonnegative integers are a... |
nn0sscn 11174 | Nonnegative integers are a... |
nn0ex 11175 | The set of nonnegative int... |
nnnn0 11176 | A positive integer is a no... |
nnnn0i 11177 | A positive integer is a no... |
nn0re 11178 | A nonnegative integer is a... |
nn0cn 11179 | A nonnegative integer is a... |
nn0rei 11180 | A nonnegative integer is a... |
nn0cni 11181 | A nonnegative integer is a... |
dfn2 11182 | The set of positive intege... |
elnnne0 11183 | The positive integer prope... |
0nn0 11184 | 0 is a nonnegative integer... |
1nn0 11185 | 1 is a nonnegative integer... |
2nn0 11186 | 2 is a nonnegative integer... |
3nn0 11187 | 3 is a nonnegative integer... |
4nn0 11188 | 4 is a nonnegative integer... |
5nn0 11189 | 5 is a nonnegative integer... |
6nn0 11190 | 6 is a nonnegative integer... |
7nn0 11191 | 7 is a nonnegative integer... |
8nn0 11192 | 8 is a nonnegative integer... |
9nn0 11193 | 9 is a nonnegative integer... |
10nn0OLD 11194 | Obsolete version of ~ 10nn... |
nn0ge0 11195 | A nonnegative integer is g... |
nn0nlt0 11196 | A nonnegative integer is n... |
nn0ge0i 11197 | Nonnegative integers are n... |
nn0le0eq0 11198 | A nonnegative integer is l... |
nn0p1gt0 11199 | A nonnegative integer incr... |
nnnn0addcl 11200 | A positive integer plus a ... |
nn0nnaddcl 11201 | A nonnegative integer plus... |
0mnnnnn0 11202 | The result of subtracting ... |
un0addcl 11203 | If ` S ` is closed under a... |
un0mulcl 11204 | If ` S ` is closed under m... |
nn0addcl 11205 | Closure of addition of non... |
nn0mulcl 11206 | Closure of multiplication ... |
nn0addcli 11207 | Closure of addition of non... |
nn0mulcli 11208 | Closure of multiplication ... |
nn0p1nn 11209 | A nonnegative integer plus... |
peano2nn0 11210 | Second Peano postulate for... |
nnm1nn0 11211 | A positive integer minus 1... |
elnn0nn 11212 | The nonnegative integer pr... |
elnnnn0 11213 | The positive integer prope... |
elnnnn0b 11214 | The positive integer prope... |
elnnnn0c 11215 | The positive integer prope... |
nn0addge1 11216 | A number is less than or e... |
nn0addge2 11217 | A number is less than or e... |
nn0addge1i 11218 | A number is less than or e... |
nn0addge2i 11219 | A number is less than or e... |
nn0sub 11220 | Subtraction of nonnegative... |
ltsubnn0 11221 | Subtracting a nonnegative ... |
nn0negleid 11222 | A nonnegative integer is g... |
difgtsumgt 11223 | If the difference of a rea... |
nn0le2xi 11224 | A nonnegative integer is l... |
nn0lele2xi 11225 | 'Less than or equal to' im... |
frnnn0supp 11226 | Two ways to write the supp... |
frnnn0fsupp 11227 | A function on ` NN0 ` is f... |
nnnn0d 11228 | A positive integer is a no... |
nn0red 11229 | A nonnegative integer is a... |
nn0cnd 11230 | A nonnegative integer is a... |
nn0ge0d 11231 | A nonnegative integer is g... |
nn0addcld 11232 | Closure of addition of non... |
nn0mulcld 11233 | Closure of multiplication ... |
nn0readdcl 11234 | Closure law for addition o... |
nn0n0n1ge2 11235 | A nonnegative integer whic... |
nn0n0n1ge2b 11236 | A nonnegative integer is n... |
nn0ge2m1nn 11237 | If a nonnegative integer i... |
nn0ge2m1nn0 11238 | If a nonnegative integer i... |
nn0nndivcl 11239 | Closure law for dividing o... |
elxnn0 11242 | An extended nonnegative in... |
nn0ssxnn0 11243 | The standard nonnegative i... |
nn0xnn0 11244 | A standard nonnegative int... |
xnn0xr 11245 | An extended nonnegative in... |
0xnn0 11246 | Zero is an extended nonneg... |
pnf0xnn0 11247 | Positive infinity is an ex... |
nn0nepnf 11248 | No standard nonnegative in... |
nn0xnn0d 11249 | A standard nonnegative int... |
nn0nepnfd 11250 | No standard nonnegative in... |
xnn0nemnf 11251 | No extended nonnegative in... |
xnn0xrnemnf 11252 | The extended nonnegative i... |
xnn0nnn0pnf 11253 | An extended nonnegative in... |
elz 11256 | Membership in the set of i... |
nnnegz 11257 | The negative of a positive... |
zre 11258 | An integer is a real. (Co... |
zcn 11259 | An integer is a complex nu... |
zrei 11260 | An integer is a real numbe... |
zssre 11261 | The integers are a subset ... |
zsscn 11262 | The integers are a subset ... |
zex 11263 | The set of integers exists... |
elnnz 11264 | Positive integer property ... |
0z 11265 | Zero is an integer. (Cont... |
0zd 11266 | Zero is an integer, deduct... |
elnn0z 11267 | Nonnegative integer proper... |
elznn0nn 11268 | Integer property expressed... |
elznn0 11269 | Integer property expressed... |
elznn 11270 | Integer property expressed... |
elz2 11271 | Membership in the set of i... |
dfz2 11272 | Alternative definition of ... |
zexALT 11273 | Alternate proof of ~ zex .... |
nnssz 11274 | Positive integers are a su... |
nn0ssz 11275 | Nonnegative integers are a... |
nnz 11276 | A positive integer is an i... |
nn0z 11277 | A nonnegative integer is a... |
nnzi 11278 | A positive integer is an i... |
nn0zi 11279 | A nonnegative integer is a... |
elnnz1 11280 | Positive integer property ... |
znnnlt1 11281 | An integer is not a positi... |
nnzrab 11282 | Positive integers expresse... |
nn0zrab 11283 | Nonnegative integers expre... |
1z 11284 | One is an integer. (Contr... |
1zzd 11285 | 1 is an integer, deductive... |
2z 11286 | 2 is an integer. (Contrib... |
3z 11287 | 3 is an integer. (Contrib... |
4z 11288 | 4 is an integer. (Contrib... |
znegcl 11289 | Closure law for negative i... |
neg1z 11290 | -1 is an integer (common c... |
znegclb 11291 | A complex number is an int... |
nn0negz 11292 | The negative of a nonnegat... |
nn0negzi 11293 | The negative of a nonnegat... |
zaddcl 11294 | Closure of addition of int... |
peano2z 11295 | Second Peano postulate gen... |
zsubcl 11296 | Closure of subtraction of ... |
peano2zm 11297 | "Reverse" second Peano pos... |
zletr 11298 | Transitive law of ordering... |
zrevaddcl 11299 | Reverse closure law for ad... |
znnsub 11300 | The positive difference of... |
znn0sub 11301 | The nonnegative difference... |
nzadd 11302 | The sum of a real number n... |
zmulcl 11303 | Closure of multiplication ... |
zltp1le 11304 | Integer ordering relation.... |
zleltp1 11305 | Integer ordering relation.... |
zlem1lt 11306 | Integer ordering relation.... |
zltlem1 11307 | Integer ordering relation.... |
zgt0ge1 11308 | An integer greater than ` ... |
nnleltp1 11309 | Positive integer ordering ... |
nnltp1le 11310 | Positive integer ordering ... |
nnaddm1cl 11311 | Closure of addition of pos... |
nn0ltp1le 11312 | Nonnegative integer orderi... |
nn0leltp1 11313 | Nonnegative integer orderi... |
nn0ltlem1 11314 | Nonnegative integer orderi... |
nn0sub2 11315 | Subtraction of nonnegative... |
nn0lt10b 11316 | A nonnegative integer less... |
nn0lt2 11317 | A nonnegative integer less... |
nn0lem1lt 11318 | Nonnegative integer orderi... |
nnlem1lt 11319 | Positive integer ordering ... |
nnltlem1 11320 | Positive integer ordering ... |
nnm1ge0 11321 | A positive integer decreas... |
nn0ge0div 11322 | Division of a nonnegative ... |
zdiv 11323 | Two ways to express " ` M ... |
zdivadd 11324 | Property of divisibility: ... |
zdivmul 11325 | Property of divisibility: ... |
zextle 11326 | An extensionality-like pro... |
zextlt 11327 | An extensionality-like pro... |
recnz 11328 | The reciprocal of a number... |
btwnnz 11329 | A number between an intege... |
gtndiv 11330 | A larger number does not d... |
halfnz 11331 | One-half is not an integer... |
3halfnz 11332 | Three halves is not an int... |
suprzcl 11333 | The supremum of a bounded-... |
prime 11334 | Two ways to express " ` A ... |
msqznn 11335 | The square of a nonzero in... |
zneo 11336 | No even integer equals an ... |
nneo 11337 | A positive integer is even... |
nneoi 11338 | A positive integer is even... |
zeo 11339 | An integer is even or odd.... |
zeo2 11340 | An integer is even or odd ... |
peano2uz2 11341 | Second Peano postulate for... |
peano5uzi 11342 | Peano's inductive postulat... |
peano5uzti 11343 | Peano's inductive postulat... |
dfuzi 11344 | An expression for the uppe... |
uzind 11345 | Induction on the upper int... |
uzind2 11346 | Induction on the upper int... |
uzind3 11347 | Induction on the upper int... |
nn0ind 11348 | Principle of Mathematical ... |
nn0indALT 11349 | Principle of Mathematical ... |
nn0indd 11350 | Principle of Mathematical ... |
fzind 11351 | Induction on the integers ... |
fnn0ind 11352 | Induction on the integers ... |
nn0ind-raph 11353 | Principle of Mathematical ... |
zindd 11354 | Principle of Mathematical ... |
btwnz 11355 | Any real number can be san... |
nn0zd 11356 | A positive integer is an i... |
nnzd 11357 | A nonnegative integer is a... |
zred 11358 | An integer is a real numbe... |
zcnd 11359 | An integer is a complex nu... |
znegcld 11360 | Closure law for negative i... |
peano2zd 11361 | Deduction from second Pean... |
zaddcld 11362 | Closure of addition of int... |
zsubcld 11363 | Closure of subtraction of ... |
zmulcld 11364 | Closure of multiplication ... |
znnn0nn 11365 | The negative of a negative... |
zadd2cl 11366 | Increasing an integer by 2... |
zriotaneg 11367 | The negative of the unique... |
suprfinzcl 11368 | The supremum of a nonempty... |
dfdecOLD 11371 | Define the "decimal constr... |
9p1e10 11372 | 9 + 1 = 10. (Contributed ... |
dfdec10 11373 | Version of the definition ... |
decex 11374 | A decimal number is a set.... |
decexOLD 11375 | Obsolete proof of ~ decex ... |
deceq1 11376 | Equality theorem for the d... |
deceq1OLD 11377 | Obsolete proof of ~ deceq1... |
deceq2 11378 | Equality theorem for the d... |
deceq2OLD 11379 | Obsolete proof of ~ deceq1... |
deceq1i 11380 | Equality theorem for the d... |
deceq2i 11381 | Equality theorem for the d... |
deceq12i 11382 | Equality theorem for the d... |
numnncl 11383 | Closure for a numeral (wit... |
num0u 11384 | Add a zero in the units pl... |
num0h 11385 | Add a zero in the higher p... |
numcl 11386 | Closure for a decimal inte... |
numsuc 11387 | The successor of a decimal... |
deccl 11388 | Closure for a numeral. (C... |
decclOLD 11389 | Obsolete proof of ~ deccl ... |
10nn 11390 | 10 is a positive integer. ... |
10pos 11391 | The number 10 is positive.... |
10nn0 11392 | 10 is a nonnegative intege... |
10re 11393 | The number 10 is real. (C... |
decnncl 11394 | Closure for a numeral. (C... |
decnnclOLD 11395 | Obsolete proof of ~ decnnc... |
dec0u 11396 | Add a zero in the units pl... |
dec0uOLD 11397 | Obsolete version of ~ dec0... |
dec0h 11398 | Add a zero in the higher p... |
dec0hOLD 11399 | Obsolete proof of ~ dec0h ... |
numnncl2 11400 | Closure for a decimal inte... |
decnncl2 11401 | Closure for a decimal inte... |
decnncl2OLD 11402 | Obsolete proof of ~ decnnc... |
numlt 11403 | Comparing two decimal inte... |
numltc 11404 | Comparing two decimal inte... |
le9lt10 11405 | A "decimal digit" (i.e. a ... |
declt 11406 | Comparing two decimal inte... |
decltOLD 11407 | Obsolete proof of ~ declt ... |
decltc 11408 | Comparing two decimal inte... |
decltcOLD 11409 | Obsolete version of ~ decl... |
declth 11410 | Comparing two decimal inte... |
decsuc 11411 | The successor of a decimal... |
decsucOLD 11412 | Obsolete proof of ~ decsuc... |
3declth 11413 | Comparing two decimal inte... |
3decltc 11414 | Comparing two decimal inte... |
3decltcOLD 11415 | Obsolete version of ~ 3dec... |
decle 11416 | Comparing two decimal inte... |
decleh 11417 | Comparing two decimal inte... |
declei 11418 | Comparing a digit to a dec... |
decleOLD 11419 | Obsolete version of ~ decl... |
declecOLD 11420 | Obsolete version of ~ decl... |
numlti 11421 | Comparing a digit to a dec... |
declti 11422 | Comparing a digit to a dec... |
decltdi 11423 | Comparing a digit to a dec... |
decltiOLD 11424 | Obsolete version of ~ decl... |
numsucc 11425 | The successor of a decimal... |
decsucc 11426 | The successor of a decimal... |
decsuccOLD 11427 | Obsolete version of ~ decs... |
1e0p1 11428 | The successor of zero. (C... |
dec10p 11429 | Ten plus an integer. (Con... |
dec10pOLD 11430 | Obsolete version of ~ dec1... |
dec10OLD 11431 | The decimal form of 10. N... |
9p1e10bOLD 11432 | Obsolete proof of ~ 9p1e10... |
numma 11433 | Perform a multiply-add of ... |
nummac 11434 | Perform a multiply-add of ... |
numma2c 11435 | Perform a multiply-add of ... |
numadd 11436 | Add two decimal integers `... |
numaddc 11437 | Add two decimal integers `... |
nummul1c 11438 | The product of a decimal i... |
nummul2c 11439 | The product of a decimal i... |
decma 11440 | Perform a multiply-add of ... |
decmaOLD 11441 | Obsolete proof of ~ decma ... |
decmac 11442 | Perform a multiply-add of ... |
decmacOLD 11443 | Obsolete proof of ~ decmac... |
decma2c 11444 | Perform a multiply-add of ... |
decma2cOLD 11445 | Obsolete proof of ~ decma2... |
decadd 11446 | Add two numerals ` M ` and... |
decaddOLD 11447 | Obsolete proof of ~ decadd... |
decaddc 11448 | Add two numerals ` M ` and... |
decaddcOLD 11449 | Obsolete proof of ~ decadd... |
decaddc2OLD 11450 | Obsolete version of ~ deca... |
decaddc2 11451 | Add two numerals ` M ` and... |
decrmanc 11452 | Perform a multiply-add of ... |
decrmac 11453 | Perform a multiply-add of ... |
decaddm10 11454 | The sum of two multiples o... |
decaddi 11455 | Add two numerals ` M ` and... |
decaddci 11456 | Add two numerals ` M ` and... |
decaddci2 11457 | Add two numerals ` M ` and... |
decaddci2OLD 11458 | Obsolete version of ~ deca... |
decsubi 11459 | Difference between a numer... |
decsubiOLD 11460 | Obsolete proof of ~ decsub... |
decmul1 11461 | The product of a numeral w... |
decmul1OLD 11462 | Obsolete proof of ~ decmul... |
decmul1c 11463 | The product of a numeral w... |
decmul1cOLD 11464 | Obsolete proof of ~ decmul... |
decmul2c 11465 | The product of a numeral w... |
decmul2cOLD 11466 | Obsolete proof of ~ decmul... |
decmulnc 11467 | The product of a numeral w... |
11multnc 11468 | The product of 11 (as nume... |
decmul10add 11469 | A multiplication of a numb... |
decmul10addOLD 11470 | Obsolete proof of ~ decmul... |
6p5lem 11471 | Lemma for ~ 6p5e11 and rel... |
5p5e10 11472 | 5 + 5 = 10. (Contributed ... |
5p5e10bOLD 11473 | Obsolete proof of ~ 5p5e10... |
6p4e10 11474 | 6 + 4 = 10. (Contributed ... |
6p4e10bOLD 11475 | Obsolete proof of ~ 6p4e10... |
6p5e11 11476 | 6 + 5 = 11. (Contributed ... |
6p5e11OLD 11477 | Obsolete proof of ~ 6p5e11... |
6p6e12 11478 | 6 + 6 = 12. (Contributed ... |
7p3e10 11479 | 7 + 3 = 10. (Contributed ... |
7p3e10bOLD 11480 | Obsolete proof of ~ 7p3e10... |
7p4e11 11481 | 7 + 4 = 11. (Contributed ... |
7p4e11OLD 11482 | Obsolete proof of ~ 7p4e11... |
7p5e12 11483 | 7 + 5 = 12. (Contributed ... |
7p6e13 11484 | 7 + 6 = 13. (Contributed ... |
7p7e14 11485 | 7 + 7 = 14. (Contributed ... |
8p2e10 11486 | 8 + 2 = 10. (Contributed ... |
8p2e10bOLD 11487 | Obsolete proof of ~ 8p2e10... |
8p3e11 11488 | 8 + 3 = 11. (Contributed ... |
8p3e11OLD 11489 | Obsolete proof of ~ 8p3e11... |
8p4e12 11490 | 8 + 4 = 12. (Contributed ... |
8p5e13 11491 | 8 + 5 = 13. (Contributed ... |
8p6e14 11492 | 8 + 6 = 14. (Contributed ... |
8p7e15 11493 | 8 + 7 = 15. (Contributed ... |
8p8e16 11494 | 8 + 8 = 16. (Contributed ... |
9p2e11 11495 | 9 + 2 = 11. (Contributed ... |
9p2e11OLD 11496 | Obsolete proof of ~ 9p2e11... |
9p3e12 11497 | 9 + 3 = 12. (Contributed ... |
9p4e13 11498 | 9 + 4 = 13. (Contributed ... |
9p5e14 11499 | 9 + 5 = 14. (Contributed ... |
9p6e15 11500 | 9 + 6 = 15. (Contributed ... |
9p7e16 11501 | 9 + 7 = 16. (Contributed ... |
9p8e17 11502 | 9 + 8 = 17. (Contributed ... |
9p9e18 11503 | 9 + 9 = 18. (Contributed ... |
10p10e20 11504 | 10 + 10 = 20. (Contribute... |
10p10e20OLD 11505 | Obsolete version of ~ 10p1... |
10m1e9 11506 | 10 - 1 = 9. (Contributed ... |
4t3lem 11507 | Lemma for ~ 4t3e12 and rel... |
4t3e12 11508 | 4 times 3 equals 12. (Con... |
4t4e16 11509 | 4 times 4 equals 16. (Con... |
5t2e10 11510 | 5 times 2 equals 10. (Con... |
5t3e15 11511 | 5 times 3 equals 15. (Con... |
5t3e15OLD 11512 | Obsolete proof of ~ 5t3e15... |
5t4e20 11513 | 5 times 4 equals 20. (Con... |
5t4e20OLD 11514 | Obsolete proof of ~ 5t4e20... |
5t5e25 11515 | 5 times 5 equals 25. (Con... |
5t5e25OLD 11516 | Obsolete proof of ~ 5t5e25... |
6t2e12 11517 | 6 times 2 equals 12. (Con... |
6t3e18 11518 | 6 times 3 equals 18. (Con... |
6t4e24 11519 | 6 times 4 equals 24. (Con... |
6t5e30 11520 | 6 times 5 equals 30. (Con... |
6t5e30OLD 11521 | Obsolete proof of ~ 6t5e30... |
6t6e36 11522 | 6 times 6 equals 36. (Con... |
6t6e36OLD 11523 | Obsolete proof of ~ 6t6e36... |
7t2e14 11524 | 7 times 2 equals 14. (Con... |
7t3e21 11525 | 7 times 3 equals 21. (Con... |
7t4e28 11526 | 7 times 4 equals 28. (Con... |
7t5e35 11527 | 7 times 5 equals 35. (Con... |
7t6e42 11528 | 7 times 6 equals 42. (Con... |
7t7e49 11529 | 7 times 7 equals 49. (Con... |
8t2e16 11530 | 8 times 2 equals 16. (Con... |
8t3e24 11531 | 8 times 3 equals 24. (Con... |
8t4e32 11532 | 8 times 4 equals 32. (Con... |
8t5e40 11533 | 8 times 5 equals 40. (Con... |
8t5e40OLD 11534 | Obsolete proof of ~ 8t5e40... |
8t6e48 11535 | 8 times 6 equals 48. (Con... |
8t6e48OLD 11536 | Obsolete proof of ~ 8t6e48... |
8t7e56 11537 | 8 times 7 equals 56. (Con... |
8t8e64 11538 | 8 times 8 equals 64. (Con... |
9t2e18 11539 | 9 times 2 equals 18. (Con... |
9t3e27 11540 | 9 times 3 equals 27. (Con... |
9t4e36 11541 | 9 times 4 equals 36. (Con... |
9t5e45 11542 | 9 times 5 equals 45. (Con... |
9t6e54 11543 | 9 times 6 equals 54. (Con... |
9t7e63 11544 | 9 times 7 equals 63. (Con... |
9t8e72 11545 | 9 times 8 equals 72. (Con... |
9t9e81 11546 | 9 times 9 equals 81. (Con... |
9t11e99 11547 | 9 times 11 equals 99. (Co... |
9t11e99OLD 11548 | Obsolete proof of ~ 9t11e9... |
9lt10 11549 | 9 is less than 10. (Contr... |
8lt10 11550 | 8 is less than 10. (Contr... |
7lt10 11551 | 7 is less than 10. (Contr... |
6lt10 11552 | 6 is less than 10. (Contr... |
5lt10 11553 | 5 is less than 10. (Contr... |
4lt10 11554 | 4 is less than 10. (Contr... |
3lt10 11555 | 3 is less than 10. (Contr... |
2lt10 11556 | 2 is less than 10. (Contr... |
1lt10 11557 | 1 is less than 10. (Contr... |
decbin0 11558 | Decompose base 4 into base... |
decbin2 11559 | Decompose base 4 into base... |
decbin3 11560 | Decompose base 4 into base... |
halfthird 11561 | Half minus a third. (Cont... |
5recm6rec 11562 | One fifth minus one sixth.... |
uzval 11565 | The value of the upper int... |
uzf 11566 | The domain and range of th... |
eluz1 11567 | Membership in the upper se... |
eluzel2 11568 | Implication of membership ... |
eluz2 11569 | Membership in an upper set... |
eluzmn 11570 | Membership in an earlier u... |
eluz1i 11571 | Membership in an upper set... |
eluzuzle 11572 | An integer in an upper set... |
eluzelz 11573 | A member of an upper set o... |
eluzelre 11574 | A member of an upper set o... |
eluzelcn 11575 | A member of an upper set o... |
eluzle 11576 | Implication of membership ... |
eluz 11577 | Membership in an upper set... |
uzid 11578 | Membership of the least me... |
uzn0 11579 | The upper integers are all... |
uztrn 11580 | Transitive law for sets of... |
uztrn2 11581 | Transitive law for sets of... |
uzneg 11582 | Contraposition law for upp... |
uzssz 11583 | An upper set of integers i... |
uzss 11584 | Subset relationship for tw... |
uztric 11585 | Totality of the ordering r... |
uz11 11586 | The upper integers functio... |
eluzp1m1 11587 | Membership in the next upp... |
eluzp1l 11588 | Strict ordering implied by... |
eluzp1p1 11589 | Membership in the next upp... |
eluzaddi 11590 | Membership in a later uppe... |
eluzsubi 11591 | Membership in an earlier u... |
eluzadd 11592 | Membership in a later uppe... |
eluzsub 11593 | Membership in an earlier u... |
uzm1 11594 | Choices for an element of ... |
uznn0sub 11595 | The nonnegative difference... |
uzin 11596 | Intersection of two upper ... |
uzp1 11597 | Choices for an element of ... |
nn0uz 11598 | Nonnegative integers expre... |
nnuz 11599 | Positive integers expresse... |
elnnuz 11600 | A positive integer express... |
elnn0uz 11601 | A nonnegative integer expr... |
eluz2nn 11602 | An integer is greater than... |
eluzge2nn0 11603 | If an integer is greater t... |
eluz2n0 11604 | An integer greater than or... |
uzuzle23 11605 | An integer in the upper se... |
eluzge3nn 11606 | If an integer is greater t... |
uz3m2nn 11607 | An integer greater than or... |
1eluzge0 11608 | 1 is an integer greater th... |
2eluzge0 11609 | 2 is an integer greater th... |
2eluzge1 11610 | 2 is an integer greater th... |
uznnssnn 11611 | The upper integers startin... |
raluz 11612 | Restricted universal quant... |
raluz2 11613 | Restricted universal quant... |
rexuz 11614 | Restricted existential qua... |
rexuz2 11615 | Restricted existential qua... |
2rexuz 11616 | Double existential quantif... |
peano2uz 11617 | Second Peano postulate for... |
peano2uzs 11618 | Second Peano postulate for... |
peano2uzr 11619 | Reversed second Peano axio... |
uzaddcl 11620 | Addition closure law for a... |
nn0pzuz 11621 | The sum of a nonnegative i... |
uzind4 11622 | Induction on the upper set... |
uzind4ALT 11623 | Induction on the upper set... |
uzind4s 11624 | Induction on the upper set... |
uzind4s2 11625 | Induction on the upper set... |
uzind4i 11626 | Induction on the upper int... |
uzwo 11627 | Well-ordering principle: a... |
uzwo2 11628 | Well-ordering principle: a... |
nnwo 11629 | Well-ordering principle: a... |
nnwof 11630 | Well-ordering principle: a... |
nnwos 11631 | Well-ordering principle: a... |
indstr 11632 | Strong Mathematical Induct... |
eluznn0 11633 | Membership in a nonnegativ... |
eluznn 11634 | Membership in a positive u... |
eluz2b1 11635 | Two ways to say "an intege... |
eluz2gt1 11636 | An integer greater than or... |
eluz2b2 11637 | Two ways to say "an intege... |
eluz2b3 11638 | Two ways to say "an intege... |
uz2m1nn 11639 | One less than an integer g... |
1nuz2 11640 | 1 is not in ` ( ZZ>= `` 2 ... |
elnn1uz2 11641 | A positive integer is eith... |
uz2mulcl 11642 | Closure of multiplication ... |
indstr2 11643 | Strong Mathematical Induct... |
uzinfi 11644 | Extract the lower bound of... |
nninf 11645 | The infimum of the set of ... |
nn0inf 11646 | The infimum of the set of ... |
infssuzle 11647 | The infimum of a subset of... |
infssuzcl 11648 | The infimum of a subset of... |
ublbneg 11649 | The image under negation o... |
eqreznegel 11650 | Two ways to express the im... |
supminf 11651 | The supremum of a bounded-... |
lbzbi 11652 | If a set of reals is bound... |
zsupss 11653 | Any nonempty bounded subse... |
suprzcl2 11654 | The supremum of a bounded-... |
suprzub 11655 | The supremum of a bounded-... |
uzsupss 11656 | Any bounded subset of an u... |
nn01to3 11657 | A (nonnegative) integer be... |
nn0ge2m1nnALT 11658 | Alternate proof of ~ nn0ge... |
uzwo3 11659 | Well-ordering principle: a... |
zmin 11660 | There is a unique smallest... |
zmax 11661 | There is a unique largest ... |
zbtwnre 11662 | There is a unique integer ... |
rebtwnz 11663 | There is a unique greatest... |
elq 11666 | Membership in the set of r... |
qmulz 11667 | If ` A ` is rational, then... |
znq 11668 | The ratio of an integer an... |
qre 11669 | A rational number is a rea... |
zq 11670 | An integer is a rational n... |
zssq 11671 | The integers are a subset ... |
nn0ssq 11672 | The nonnegative integers a... |
nnssq 11673 | The positive integers are ... |
qssre 11674 | The rationals are a subset... |
qsscn 11675 | The rationals are a subset... |
qex 11676 | The set of rational number... |
nnq 11677 | A positive integer is rati... |
qcn 11678 | A rational number is a com... |
qexALT 11679 | Alternate proof of ~ qex .... |
qaddcl 11680 | Closure of addition of rat... |
qnegcl 11681 | Closure law for the negati... |
qmulcl 11682 | Closure of multiplication ... |
qsubcl 11683 | Closure of subtraction of ... |
qreccl 11684 | Closure of reciprocal of r... |
qdivcl 11685 | Closure of division of rat... |
qrevaddcl 11686 | Reverse closure law for ad... |
nnrecq 11687 | The reciprocal of a positi... |
irradd 11688 | The sum of an irrational n... |
irrmul 11689 | The product of an irration... |
rpnnen1lem2 11690 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem1 11691 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem3 11692 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem4 11693 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem5 11694 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1lem6 11695 | Lemma for ~ rpnnen1 . (Co... |
rpnnen1 11696 | One half of ~ rpnnen , whe... |
rpnnen1lem1OLD 11697 | Lemma for ~ rpnnen1OLD . ... |
rpnnen1lem3OLD 11698 | Lemma for ~ rpnnen1OLD . ... |
rpnnen1lem4OLD 11699 | Lemma for ~ rpnnen1OLD . ... |
rpnnen1lem5OLD 11700 | Lemma for ~ rpnnen1OLD . ... |
rpnnen1OLD 11701 | One half of ~ rpnnen , whe... |
reexALT 11702 | Alternate proof of ~ reex ... |
cnref1o 11703 | There is a natural one-to-... |
cnexALT 11704 | The set of complex numbers... |
xrex 11705 | The set of extended reals ... |
addex 11706 | The addition operation is ... |
mulex 11707 | The multiplication operati... |
elrp 11710 | Membership in the set of p... |
elrpii 11711 | Membership in the set of p... |
1rp 11712 | 1 is a positive real. (Co... |
2rp 11713 | 2 is a positive real. (Co... |
3rp 11714 | 3 is a positive real. (Co... |
rpre 11715 | A positive real is a real.... |
rpxr 11716 | A positive real is an exte... |
rpcn 11717 | A positive real is a compl... |
nnrp 11718 | A positive integer is a po... |
rpssre 11719 | The positive reals are a s... |
rpgt0 11720 | A positive real is greater... |
rpge0 11721 | A positive real is greater... |
rpregt0 11722 | A positive real is a posit... |
rprege0 11723 | A positive real is a nonne... |
rpne0 11724 | A positive real is nonzero... |
rprene0 11725 | A positive real is a nonze... |
rpcnne0 11726 | A positive real is a nonze... |
rpcndif0 11727 | A positive real number is ... |
ralrp 11728 | Quantification over positi... |
rexrp 11729 | Quantification over positi... |
rpaddcl 11730 | Closure law for addition o... |
rpmulcl 11731 | Closure law for multiplica... |
rpdivcl 11732 | Closure law for division o... |
rpreccl 11733 | Closure law for reciprocat... |
rphalfcl 11734 | Closure law for half of a ... |
rpgecl 11735 | A number greater or equal ... |
rphalflt 11736 | Half of a positive real is... |
rerpdivcl 11737 | Closure law for division o... |
ge0p1rp 11738 | A nonnegative number plus ... |
rpneg 11739 | Either a nonzero real or i... |
negelrp 11740 | Elementhood of a negation ... |
0nrp 11741 | Zero is not a positive rea... |
ltsubrp 11742 | Subtracting a positive rea... |
ltaddrp 11743 | Adding a positive number t... |
difrp 11744 | Two ways to say one number... |
elrpd 11745 | Membership in the set of p... |
nnrpd 11746 | A positive integer is a po... |
zgt1rpn0n1 11747 | An integer greater than 1 ... |
rpred 11748 | A positive real is a real.... |
rpxrd 11749 | A positive real is an exte... |
rpcnd 11750 | A positive real is a compl... |
rpgt0d 11751 | A positive real is greater... |
rpge0d 11752 | A positive real is greater... |
rpne0d 11753 | A positive real is nonzero... |
rpregt0d 11754 | A positive real is real an... |
rprege0d 11755 | A positive real is real an... |
rprene0d 11756 | A positive real is a nonze... |
rpcnne0d 11757 | A positive real is a nonze... |
rpreccld 11758 | Closure law for reciprocat... |
rprecred 11759 | Closure law for reciprocat... |
rphalfcld 11760 | Closure law for half of a ... |
reclt1d 11761 | The reciprocal of a positi... |
recgt1d 11762 | The reciprocal of a positi... |
rpaddcld 11763 | Closure law for addition o... |
rpmulcld 11764 | Closure law for multiplica... |
rpdivcld 11765 | Closure law for division o... |
ltrecd 11766 | The reciprocal of both sid... |
lerecd 11767 | The reciprocal of both sid... |
ltrec1d 11768 | Reciprocal swap in a 'less... |
lerec2d 11769 | Reciprocal swap in a 'less... |
lediv2ad 11770 | Division of both sides of ... |
ltdiv2d 11771 | Division of a positive num... |
lediv2d 11772 | Division of a positive num... |
ledivdivd 11773 | Invert ratios of positive ... |
divge1 11774 | The ratio of a number over... |
divlt1lt 11775 | A real number divided by a... |
divle1le 11776 | A real number divided by a... |
ledivge1le 11777 | If a number is less than o... |
ge0p1rpd 11778 | A nonnegative number plus ... |
rerpdivcld 11779 | Closure law for division o... |
ltsubrpd 11780 | Subtracting a positive rea... |
ltaddrpd 11781 | Adding a positive number t... |
ltaddrp2d 11782 | Adding a positive number t... |
ltmulgt11d 11783 | Multiplication by a number... |
ltmulgt12d 11784 | Multiplication by a number... |
gt0divd 11785 | Division of a positive num... |
ge0divd 11786 | Division of a nonnegative ... |
rpgecld 11787 | A number greater or equal ... |
divge0d 11788 | The ratio of nonnegative a... |
ltmul1d 11789 | The ratio of nonnegative a... |
ltmul2d 11790 | Multiplication of both sid... |
lemul1d 11791 | Multiplication of both sid... |
lemul2d 11792 | Multiplication of both sid... |
ltdiv1d 11793 | Division of both sides of ... |
lediv1d 11794 | Division of both sides of ... |
ltmuldivd 11795 | 'Less than' relationship b... |
ltmuldiv2d 11796 | 'Less than' relationship b... |
lemuldivd 11797 | 'Less than or equal to' re... |
lemuldiv2d 11798 | 'Less than or equal to' re... |
ltdivmuld 11799 | 'Less than' relationship b... |
ltdivmul2d 11800 | 'Less than' relationship b... |
ledivmuld 11801 | 'Less than or equal to' re... |
ledivmul2d 11802 | 'Less than or equal to' re... |
ltmul1dd 11803 | The ratio of nonnegative a... |
ltmul2dd 11804 | Multiplication of both sid... |
ltdiv1dd 11805 | Division of both sides of ... |
lediv1dd 11806 | Division of both sides of ... |
lediv12ad 11807 | Comparison of ratio of two... |
mul2lt0rlt0 11808 | If the result of a multipl... |
mul2lt0rgt0 11809 | If the result of a multipl... |
mul2lt0llt0 11810 | If the result of a multipl... |
mul2lt0lgt0 11811 | If the result of a multipl... |
mul2lt0bi 11812 | If the result of a multipl... |
ltdiv23d 11813 | Swap denominator with othe... |
lediv23d 11814 | Swap denominator with othe... |
lt2mul2divd 11815 | The ratio of nonnegative a... |
nnledivrp 11816 | Division of a positive int... |
nn0ledivnn 11817 | Division of a nonnegative ... |
addlelt 11818 | If the sum of a real numbe... |
ltxr 11825 | The 'less than' binary rel... |
elxr 11826 | Membership in the set of e... |
xrnemnf 11827 | An extended real other tha... |
xrnepnf 11828 | An extended real other tha... |
xrltnr 11829 | The extended real 'less th... |
ltpnf 11830 | Any (finite) real is less ... |
ltpnfd 11831 | Any (finite) real is less ... |
0ltpnf 11832 | Zero is less than plus inf... |
mnflt 11833 | Minus infinity is less tha... |
mnfltd 11834 | Minus infinity is less tha... |
mnflt0 11835 | Minus infinity is less tha... |
mnfltpnf 11836 | Minus infinity is less tha... |
mnfltxr 11837 | Minus infinity is less tha... |
pnfnlt 11838 | No extended real is greate... |
nltmnf 11839 | No extended real is less t... |
pnfge 11840 | Plus infinity is an upper ... |
xnn0n0n1ge2b 11841 | An extended nonnegative in... |
0lepnf 11842 | 0 less than or equal to po... |
xnn0ge0 11843 | An extended nonnegative in... |
nn0pnfge0OLD 11844 | Obsolete version of ~ xnn0... |
mnfle 11845 | Minus infinity is less tha... |
xrltnsym 11846 | Ordering on the extended r... |
xrltnsym2 11847 | 'Less than' is antisymmetr... |
xrlttri 11848 | Ordering on the extended r... |
xrlttr 11849 | Ordering on the extended r... |
xrltso 11850 | 'Less than' is a strict or... |
xrlttri2 11851 | Trichotomy law for 'less t... |
xrlttri3 11852 | Trichotomy law for 'less t... |
xrleloe 11853 | 'Less than or equal' expre... |
xrleltne 11854 | 'Less than or equal to' im... |
xrltlen 11855 | 'Less than' expressed in t... |
dfle2 11856 | Alternative definition of ... |
dflt2 11857 | Alternative definition of ... |
xrltle 11858 | 'Less than' implies 'less ... |
xrleid 11859 | 'Less than or equal to' is... |
xrletri 11860 | Trichotomy law for extende... |
xrletri3 11861 | Trichotomy law for extende... |
xrletrid 11862 | Trichotomy law for extende... |
xrlelttr 11863 | Transitive law for orderin... |
xrltletr 11864 | Transitive law for orderin... |
xrletr 11865 | Transitive law for orderin... |
xrlttrd 11866 | Transitive law for orderin... |
xrlelttrd 11867 | Transitive law for orderin... |
xrltletrd 11868 | Transitive law for orderin... |
xrletrd 11869 | Transitive law for orderin... |
xrltne 11870 | 'Less than' implies not eq... |
nltpnft 11871 | An extended real is not le... |
ngtmnft 11872 | An extended real is not gr... |
xrrebnd 11873 | An extended real is real i... |
xrre 11874 | A way of proving that an e... |
xrre2 11875 | An extended real between t... |
xrre3 11876 | A way of proving that an e... |
ge0gtmnf 11877 | A nonnegative extended rea... |
ge0nemnf 11878 | A nonnegative extended rea... |
xrrege0 11879 | A nonnegative extended rea... |
xrmax1 11880 | An extended real is less t... |
xrmax2 11881 | An extended real is less t... |
xrmin1 11882 | The minimum of two extende... |
xrmin2 11883 | The minimum of two extende... |
xrmaxeq 11884 | The maximum of two extende... |
xrmineq 11885 | The minimum of two extende... |
xrmaxlt 11886 | Two ways of saying the max... |
xrltmin 11887 | Two ways of saying an exte... |
xrmaxle 11888 | Two ways of saying the max... |
xrlemin 11889 | Two ways of saying a numbe... |
max1 11890 | A number is less than or e... |
max1ALT 11891 | A number is less than or e... |
max2 11892 | A number is less than or e... |
2resupmax 11893 | The supremum of two real n... |
min1 11894 | The minimum of two numbers... |
min2 11895 | The minimum of two numbers... |
maxle 11896 | Two ways of saying the max... |
lemin 11897 | Two ways of saying a numbe... |
maxlt 11898 | Two ways of saying the max... |
ltmin 11899 | Two ways of saying a numbe... |
lemaxle 11900 | A real number which is les... |
max0sub 11901 | Decompose a real number in... |
ifle 11902 | An if statement transforms... |
z2ge 11903 | There exists an integer gr... |
qbtwnre 11904 | The rational numbers are d... |
qbtwnxr 11905 | The rational numbers are d... |
qsqueeze 11906 | If a nonnegative real is l... |
qextltlem 11907 | Lemma for ~ qextlt and qex... |
qextlt 11908 | An extensionality-like pro... |
qextle 11909 | An extensionality-like pro... |
xralrple 11910 | Show that ` A ` is less th... |
alrple 11911 | Show that ` A ` is less th... |
xnegeq 11912 | Equality of two extended n... |
xnegex 11913 | A negative extended real e... |
xnegpnf 11914 | Minus ` +oo ` . Remark of... |
xnegmnf 11915 | Minus ` -oo ` . Remark of... |
rexneg 11916 | Minus a real number. Rema... |
xneg0 11917 | The negative of zero. (Co... |
xnegcl 11918 | Closure of extended real n... |
xnegneg 11919 | Extended real version of ~... |
xneg11 11920 | Extended real version of ~... |
xltnegi 11921 | Forward direction of ~ xlt... |
xltneg 11922 | Extended real version of ~... |
xleneg 11923 | Extended real version of ~... |
xlt0neg1 11924 | Extended real version of ~... |
xlt0neg2 11925 | Extended real version of ~... |
xle0neg1 11926 | Extended real version of ~... |
xle0neg2 11927 | Extended real version of ~... |
xaddval 11928 | Value of the extended real... |
xaddf 11929 | The extended real addition... |
xmulval 11930 | Value of the extended real... |
xaddpnf1 11931 | Addition of positive infin... |
xaddpnf2 11932 | Addition of positive infin... |
xaddmnf1 11933 | Addition of negative infin... |
xaddmnf2 11934 | Addition of negative infin... |
pnfaddmnf 11935 | Addition of positive and n... |
mnfaddpnf 11936 | Addition of negative and p... |
rexadd 11937 | The extended real addition... |
rexsub 11938 | Extended real subtraction ... |
rexaddd 11939 | The extended real addition... |
xnn0xaddcl 11940 | The extended nonnegative i... |
xaddnemnf 11941 | Closure of extended real a... |
xaddnepnf 11942 | Closure of extended real a... |
xnegid 11943 | Extended real version of ~... |
xaddcl 11944 | The extended real addition... |
xaddcom 11945 | The extended real addition... |
xaddid1 11946 | Extended real version of ~... |
xaddid2 11947 | Extended real version of ~... |
xaddid1d 11948 | ` 0 ` is a right identity ... |
xnn0xadd0 11949 | The sum of two extended no... |
xnegdi 11950 | Extended real version of ~... |
xaddass 11951 | Associativity of extended ... |
xaddass2 11952 | Associativity of extended ... |
xpncan 11953 | Extended real version of ~... |
xnpcan 11954 | Extended real version of ~... |
xleadd1a 11955 | Extended real version of ~... |
xleadd2a 11956 | Commuted form of ~ xleadd1... |
xleadd1 11957 | Weakened version of ~ xlea... |
xltadd1 11958 | Extended real version of ~... |
xltadd2 11959 | Extended real version of ~... |
xaddge0 11960 | The sum of nonnegative ext... |
xle2add 11961 | Extended real version of ~... |
xlt2add 11962 | Extended real version of ~... |
xsubge0 11963 | Extended real version of ~... |
xposdif 11964 | Extended real version of ~... |
xlesubadd 11965 | Under certain conditions, ... |
xmullem 11966 | Lemma for ~ rexmul . (Con... |
xmullem2 11967 | Lemma for ~ xmulneg1 . (C... |
xmulcom 11968 | Extended real multiplicati... |
xmul01 11969 | Extended real version of ~... |
xmul02 11970 | Extended real version of ~... |
xmulneg1 11971 | Extended real version of ~... |
xmulneg2 11972 | Extended real version of ~... |
rexmul 11973 | The extended real multipli... |
xmulf 11974 | The extended real multipli... |
xmulcl 11975 | Closure of extended real m... |
xmulpnf1 11976 | Multiplication by plus inf... |
xmulpnf2 11977 | Multiplication by plus inf... |
xmulmnf1 11978 | Multiplication by minus in... |
xmulmnf2 11979 | Multiplication by minus in... |
xmulpnf1n 11980 | Multiplication by plus inf... |
xmulid1 11981 | Extended real version of ~... |
xmulid2 11982 | Extended real version of ~... |
xmulm1 11983 | Extended real version of ~... |
xmulasslem2 11984 | Lemma for ~ xmulass . (Co... |
xmulgt0 11985 | Extended real version of ~... |
xmulge0 11986 | Extended real version of ~... |
xmulasslem 11987 | Lemma for ~ xmulass . (Co... |
xmulasslem3 11988 | Lemma for ~ xmulass . (Co... |
xmulass 11989 | Associativity of the exten... |
xlemul1a 11990 | Extended real version of ~... |
xlemul2a 11991 | Extended real version of ~... |
xlemul1 11992 | Extended real version of ~... |
xlemul2 11993 | Extended real version of ~... |
xltmul1 11994 | Extended real version of ~... |
xltmul2 11995 | Extended real version of ~... |
xadddilem 11996 | Lemma for ~ xadddi . (Con... |
xadddi 11997 | Distributive property for ... |
xadddir 11998 | Commuted version of ~ xadd... |
xadddi2 11999 | The assumption that the mu... |
xadddi2r 12000 | Commuted version of ~ xadd... |
x2times 12001 | Extended real version of ~... |
xnegcld 12002 | Closure of extended real n... |
xaddcld 12003 | The extended real addition... |
xmulcld 12004 | Closure of extended real m... |
xadd4d 12005 | Rearrangement of 4 terms i... |
xnn0add4d 12006 | Rearrangement of 4 terms i... |
xrsupexmnf 12007 | Adding minus infinity to a... |
xrinfmexpnf 12008 | Adding plus infinity to a ... |
xrsupsslem 12009 | Lemma for ~ xrsupss . (Co... |
xrinfmsslem 12010 | Lemma for ~ xrinfmss . (C... |
xrsupss 12011 | Any subset of extended rea... |
xrinfmss 12012 | Any subset of extended rea... |
xrinfmss2 12013 | Any subset of extended rea... |
xrub 12014 | By quantifying only over r... |
supxr 12015 | The supremum of a set of e... |
supxr2 12016 | The supremum of a set of e... |
supxrcl 12017 | The supremum of an arbitra... |
supxrun 12018 | The supremum of the union ... |
supxrmnf 12019 | Adding minus infinity to a... |
supxrpnf 12020 | The supremum of a set of e... |
supxrunb1 12021 | The supremum of an unbound... |
supxrunb2 12022 | The supremum of an unbound... |
supxrbnd1 12023 | The supremum of a bounded-... |
supxrbnd2 12024 | The supremum of a bounded-... |
xrsup0 12025 | The supremum of an empty s... |
supxrub 12026 | A member of a set of exten... |
supxrlub 12027 | The supremum of a set of e... |
supxrleub 12028 | The supremum of a set of e... |
supxrre 12029 | The real and extended real... |
supxrbnd 12030 | The supremum of a bounded-... |
supxrgtmnf 12031 | The supremum of a nonempty... |
supxrre1 12032 | The supremum of a nonempty... |
supxrre2 12033 | The supremum of a nonempty... |
supxrss 12034 | Smaller sets of extended r... |
infxrcl 12035 | The infimum of an arbitrar... |
infxrlb 12036 | A member of a set of exten... |
infxrgelb 12037 | The infimum of a set of ex... |
infxrre 12038 | The real and extended real... |
xrinf0 12039 | The infimum of the empty s... |
infxrss 12040 | Larger sets of extended re... |
reltre 12041 | For all real numbers there... |
rpltrp 12042 | For all positive real numb... |
reltxrnmnf 12043 | For all extended real numb... |
infmremnf 12044 | The infimum of the reals i... |
infmrp1 12045 | The infimum of the positiv... |
ixxval 12054 | Value of the interval func... |
elixx1 12055 | Membership in an interval ... |
ixxf 12056 | The set of intervals of ex... |
ixxex 12057 | The set of intervals of ex... |
ixxssxr 12058 | The set of intervals of ex... |
elixx3g 12059 | Membership in a set of ope... |
ixxssixx 12060 | An interval is a subset of... |
ixxdisj 12061 | Split an interval into dis... |
ixxun 12062 | Split an interval into two... |
ixxin 12063 | Intersection of two interv... |
ixxss1 12064 | Subset relationship for in... |
ixxss2 12065 | Subset relationship for in... |
ixxss12 12066 | Subset relationship for in... |
ixxub 12067 | Extract the upper bound of... |
ixxlb 12068 | Extract the lower bound of... |
iooex 12069 | The set of open intervals ... |
iooval 12070 | Value of the open interval... |
ioo0 12071 | An empty open interval of ... |
ioon0 12072 | An open interval of extend... |
ndmioo 12073 | The open interval function... |
iooid 12074 | An open interval with iden... |
elioo3g 12075 | Membership in a set of ope... |
elioore 12076 | A member of an open interv... |
lbioo 12077 | An open interval does not ... |
ubioo 12078 | An open interval does not ... |
iooval2 12079 | Value of the open interval... |
iooin 12080 | Intersection of two open i... |
iooss1 12081 | Subset relationship for op... |
iooss2 12082 | Subset relationship for op... |
iocval 12083 | Value of the open-below, c... |
icoval 12084 | Value of the closed-below,... |
iccval 12085 | Value of the closed interv... |
elioo1 12086 | Membership in an open inte... |
elioo2 12087 | Membership in an open inte... |
elioc1 12088 | Membership in an open-belo... |
elico1 12089 | Membership in a closed-bel... |
elicc1 12090 | Membership in a closed int... |
iccid 12091 | A closed interval with ide... |
ico0 12092 | An empty open interval of ... |
ioc0 12093 | An empty open interval of ... |
icc0 12094 | An empty closed interval o... |
elicod 12095 | Membership in a left close... |
icogelb 12096 | An element of a left close... |
elicore 12097 | A member of a left closed,... |
ubioc1 12098 | The upper bound belongs to... |
lbico1 12099 | The lower bound belongs to... |
iccleub 12100 | An element of a closed int... |
iccgelb 12101 | An element of a closed int... |
elioo5 12102 | Membership in an open inte... |
eliooxr 12103 | A nonempty open interval s... |
eliooord 12104 | Ordering implied by a memb... |
elioo4g 12105 | Membership in an open inte... |
ioossre 12106 | An open interval is a set ... |
elioc2 12107 | Membership in an open-belo... |
elico2 12108 | Membership in a closed-bel... |
elicc2 12109 | Membership in a closed rea... |
elicc2i 12110 | Inference for membership i... |
elicc4 12111 | Membership in a closed rea... |
iccss 12112 | Condition for a closed int... |
iccssioo 12113 | Condition for a closed int... |
icossico 12114 | Condition for a closed-bel... |
iccss2 12115 | Condition for a closed int... |
iccssico 12116 | Condition for a closed int... |
iccssioo2 12117 | Condition for a closed int... |
iccssico2 12118 | Condition for a closed int... |
ioomax 12119 | The open interval from min... |
iccmax 12120 | The closed interval from m... |
ioopos 12121 | The set of positive reals ... |
ioorp 12122 | The set of positive reals ... |
iooshf 12123 | Shift the arguments of the... |
iocssre 12124 | A closed-above interval wi... |
icossre 12125 | A closed-below interval wi... |
iccssre 12126 | A closed real interval is ... |
iccssxr 12127 | A closed interval is a set... |
iocssxr 12128 | An open-below, closed-abov... |
icossxr 12129 | A closed-below, open-above... |
ioossicc 12130 | An open interval is a subs... |
icossicc 12131 | A closed-below, open-above... |
iocssicc 12132 | A closed-above, open-below... |
ioossico 12133 | An open interval is a subs... |
iocssioo 12134 | Condition for a closed int... |
icossioo 12135 | Condition for a closed int... |
ioossioo 12136 | Condition for an open inte... |
iccsupr 12137 | A nonempty subset of a clo... |
elioopnf 12138 | Membership in an unbounded... |
elioomnf 12139 | Membership in an unbounded... |
elicopnf 12140 | Membership in a closed unb... |
repos 12141 | Two ways of saying that a ... |
ioof 12142 | The set of open intervals ... |
iccf 12143 | The set of closed interval... |
unirnioo 12144 | The union of the range of ... |
dfioo2 12145 | Alternate definition of th... |
ioorebas 12146 | Open intervals are element... |
xrge0neqmnf 12147 | An extended nonnegative re... |
xrge0nre 12148 | An extended real which is ... |
elrege0 12149 | The predicate "is a nonneg... |
nn0rp0 12150 | A nonnegative integer is a... |
rge0ssre 12151 | Nonnegative real numbers a... |
elxrge0 12152 | Elementhood in the set of ... |
0e0icopnf 12153 | 0 is a member of ` ( 0 [,)... |
0e0iccpnf 12154 | 0 is a member of ` ( 0 [,]... |
ge0addcl 12155 | The nonnegative reals are ... |
ge0mulcl 12156 | The nonnegative reals are ... |
ge0xaddcl 12157 | The nonnegative reals are ... |
ge0xmulcl 12158 | The nonnegative extended r... |
lbicc2 12159 | The lower bound of a close... |
ubicc2 12160 | The upper bound of a close... |
0elunit 12161 | Zero is an element of the ... |
1elunit 12162 | One is an element of the c... |
iooneg 12163 | Membership in a negated op... |
iccneg 12164 | Membership in a negated cl... |
icoshft 12165 | A shifted real is a member... |
icoshftf1o 12166 | Shifting a closed-below, o... |
icoun 12167 | The union of end-to-end cl... |
icodisj 12168 | End-to-end closed-below, o... |
snunioo 12169 | The closure of one end of ... |
snunico 12170 | The closure of the open en... |
snunioc 12171 | The closure of the open en... |
prunioo 12172 | The closure of an open rea... |
ioodisj 12173 | If the upper bound of one ... |
ioojoin 12174 | Join two open intervals to... |
difreicc 12175 | The class difference of ` ... |
iccsplit 12176 | Split a closed interval in... |
iccshftr 12177 | Membership in a shifted in... |
iccshftri 12178 | Membership in a shifted in... |
iccshftl 12179 | Membership in a shifted in... |
iccshftli 12180 | Membership in a shifted in... |
iccdil 12181 | Membership in a dilated in... |
iccdili 12182 | Membership in a dilated in... |
icccntr 12183 | Membership in a contracted... |
icccntri 12184 | Membership in a contracted... |
divelunit 12185 | A condition for a ratio to... |
lincmb01cmp 12186 | A linear combination of tw... |
iccf1o 12187 | Describe a bijection from ... |
iccen 12188 | Any nontrivial closed inte... |
xov1plusxeqvd 12189 | A complex number ` X ` is ... |
unitssre 12190 | ` ( 0 [,] 1 ) ` is a subse... |
supicc 12191 | Supremum of a bounded set ... |
supiccub 12192 | The supremum of a bounded ... |
supicclub 12193 | The supremum of a bounded ... |
supicclub2 12194 | The supremum of a bounded ... |
zltaddlt1le 12195 | The sum of an integer and ... |
xnn0xrge0 12196 | An extended nonnegative in... |
fzval 12199 | The value of a finite set ... |
fzval2 12200 | An alternative way of expr... |
fzf 12201 | Establish the domain and c... |
elfz1 12202 | Membership in a finite set... |
elfz 12203 | Membership in a finite set... |
elfz2 12204 | Membership in a finite set... |
elfz5 12205 | Membership in a finite set... |
elfz4 12206 | Membership in a finite set... |
elfzuzb 12207 | Membership in a finite set... |
eluzfz 12208 | Membership in a finite set... |
elfzuz 12209 | A member of a finite set o... |
elfzuz3 12210 | Membership in a finite set... |
elfzel2 12211 | Membership in a finite set... |
elfzel1 12212 | Membership in a finite set... |
elfzelz 12213 | A member of a finite set o... |
fzssz 12214 | A finite sequence of integ... |
elfzle1 12215 | A member of a finite set o... |
elfzle2 12216 | A member of a finite set o... |
elfzuz2 12217 | Implication of membership ... |
elfzle3 12218 | Membership in a finite set... |
eluzfz1 12219 | Membership in a finite set... |
eluzfz2 12220 | Membership in a finite set... |
eluzfz2b 12221 | Membership in a finite set... |
elfz3 12222 | Membership in a finite set... |
elfz1eq 12223 | Membership in a finite set... |
elfzubelfz 12224 | If there is a member in a ... |
peano2fzr 12225 | A Peano-postulate-like the... |
fzn0 12226 | Properties of a finite int... |
fz0 12227 | A finite set of sequential... |
fzn 12228 | A finite set of sequential... |
fzen 12229 | A shifted finite set of se... |
fz1n 12230 | A 1-based finite set of se... |
0nelfz1 12231 | 0 is not an element of a f... |
0fz1 12232 | Two ways to say a finite 1... |
fz10 12233 | There are no integers betw... |
uzsubsubfz 12234 | Membership of an integer g... |
uzsubsubfz1 12235 | Membership of an integer g... |
ige3m2fz 12236 | Membership of an integer g... |
fzsplit2 12237 | Split a finite interval of... |
fzsplit 12238 | Split a finite interval of... |
fzdisj 12239 | Condition for two finite i... |
fz01en 12240 | 0-based and 1-based finite... |
elfznn 12241 | A member of a finite set o... |
elfz1end 12242 | A nonempty finite range of... |
fz1ssnn 12243 | A finite set of positive i... |
fznn0sub 12244 | Subtraction closure for a ... |
fzmmmeqm 12245 | Subtracting the difference... |
fzaddel 12246 | Membership of a sum in a f... |
fzadd2 12247 | Membership of a sum in a f... |
fzsubel 12248 | Membership of a difference... |
fzopth 12249 | A finite set of sequential... |
fzass4 12250 | Two ways to express a nond... |
fzss1 12251 | Subset relationship for fi... |
fzss2 12252 | Subset relationship for fi... |
fzssuz 12253 | A finite set of sequential... |
fzsn 12254 | A finite interval of integ... |
fzssp1 12255 | Subset relationship for fi... |
fzssnn 12256 | Finite sets of sequential ... |
ssfzunsn 12257 | A subset of a finite seque... |
fzsuc 12258 | Join a successor to the en... |
fzpred 12259 | Join a predecessor to the ... |
fzpreddisj 12260 | A finite set of sequential... |
elfzp1 12261 | Append an element to a fin... |
fzp1ss 12262 | Subset relationship for fi... |
fzelp1 12263 | Membership in a set of seq... |
fzp1elp1 12264 | Add one to an element of a... |
fznatpl1 12265 | Shift membership in a fini... |
fzpr 12266 | A finite interval of integ... |
fztp 12267 | A finite interval of integ... |
fzsuc2 12268 | Join a successor to the en... |
fzp1disj 12269 | ` ( M ... ( N + 1 ) ) ` is... |
fzdifsuc 12270 | Remove a successor from th... |
fzprval 12271 | Two ways of defining the f... |
fztpval 12272 | Two ways of defining the f... |
fzrev 12273 | Reversal of start and end ... |
fzrev2 12274 | Reversal of start and end ... |
fzrev2i 12275 | Reversal of start and end ... |
fzrev3 12276 | The "complement" of a memb... |
fzrev3i 12277 | The "complement" of a memb... |
fznn 12278 | Finite set of sequential i... |
elfz1b 12279 | Membership in a 1 based fi... |
elfzm11 12280 | Membership in a finite set... |
uzsplit 12281 | Express an upper integer s... |
uzdisj 12282 | The first ` N ` elements o... |
fseq1p1m1 12283 | Add/remove an item to/from... |
fseq1m1p1 12284 | Add/remove an item to/from... |
fz1sbc 12285 | Quantification over a one-... |
elfzp1b 12286 | An integer is a member of ... |
elfzm1b 12287 | An integer is a member of ... |
elfzp12 12288 | Options for membership in ... |
fzm1 12289 | Choices for an element of ... |
fzneuz 12290 | No finite set of sequentia... |
fznuz 12291 | Disjointness of the upper ... |
uznfz 12292 | Disjointness of the upper ... |
fzp1nel 12293 | One plus the upper bound o... |
fzrevral 12294 | Reversal of scanning order... |
fzrevral2 12295 | Reversal of scanning order... |
fzrevral3 12296 | Reversal of scanning order... |
fzshftral 12297 | Shift the scanning order i... |
ige2m1fz1 12298 | Membership of an integer g... |
ige2m1fz 12299 | Membership in a 0 based fi... |
elfz2nn0 12300 | Membership in a finite set... |
fznn0 12301 | Characterization of a fini... |
elfznn0 12302 | A member of a finite set o... |
elfz3nn0 12303 | The upper bound of a nonem... |
fz0ssnn0 12304 | Finite sets of sequential ... |
0elfz 12305 | 0 is an element of a finit... |
nn0fz0 12306 | A nonnegative integer is a... |
elfz0add 12307 | An element of a finite set... |
fz0sn 12308 | An integer range from 0 to... |
fz0tp 12309 | An integer range from 0 to... |
fz0to3un2pr 12310 | An integer range from 0 to... |
fz0to4untppr 12311 | An integer range from 0 to... |
elfz0ubfz0 12312 | An element of a finite set... |
elfz0fzfz0 12313 | A member of a finite set o... |
fz0fzelfz0 12314 | If a member of a finite se... |
fznn0sub2 12315 | Subtraction closure for a ... |
uzsubfz0 12316 | Membership of an integer g... |
fz0fzdiffz0 12317 | The difference of an integ... |
elfzmlbm 12318 | Subtracting the lower boun... |
elfzmlbp 12319 | Subtracting the lower boun... |
fzctr 12320 | Lemma for theorems about t... |
difelfzle 12321 | The difference of two inte... |
difelfznle 12322 | The difference of two inte... |
nn0split 12323 | Express the set of nonnega... |
nn0disj 12324 | The first ` N + 1 ` elemen... |
fz0sn0fz1 12325 | A finite set of sequential... |
fvffz0 12326 | The function value of a fu... |
1fv 12327 | A one value function. (Co... |
4fvwrd4 12328 | The first four function va... |
2ffzeq 12329 | Two functions over 0 based... |
preduz 12330 | The value of the predecess... |
prednn 12331 | The value of the predecess... |
prednn0 12332 | The value of the predecess... |
predfz 12333 | Calculate the predecessor ... |
fzof 12336 | Functionality of the half-... |
elfzoel1 12337 | Reverse closure for half-o... |
elfzoel2 12338 | Reverse closure for half-o... |
elfzoelz 12339 | Reverse closure for half-o... |
fzoval 12340 | Value of the half-open int... |
elfzo 12341 | Membership in a half-open ... |
elfzo2 12342 | Membership in a half-open ... |
elfzouz 12343 | Membership in a half-open ... |
nelfzo 12344 | An integer not being a mem... |
fzolb 12345 | The left endpoint of a hal... |
fzolb2 12346 | The left endpoint of a hal... |
elfzole1 12347 | A member in a half-open in... |
elfzolt2 12348 | A member in a half-open in... |
elfzolt3 12349 | Membership in a half-open ... |
elfzolt2b 12350 | A member in a half-open in... |
elfzolt3b 12351 | Membership in a half-open ... |
fzonel 12352 | A half-open range does not... |
elfzouz2 12353 | The upper bound of a half-... |
elfzofz 12354 | A half-open range is conta... |
elfzo3 12355 | Express membership in a ha... |
fzon0 12356 | A half-open integer interv... |
fzossfz 12357 | A half-open range is conta... |
fzon 12358 | A half-open set of sequent... |
fzo0n 12359 | A half-open range of nonne... |
fzonlt0 12360 | A half-open integer range ... |
fzo0 12361 | Half-open sets with equal ... |
fzonnsub 12362 | If ` K < N ` then ` N - K ... |
fzonnsub2 12363 | If ` M < N ` then ` N - M ... |
fzoss1 12364 | Subset relationship for ha... |
fzoss2 12365 | Subset relationship for ha... |
fzossrbm1 12366 | Subset of a half open rang... |
fzo0ss1 12367 | Subset relationship for ha... |
fzossnn0 12368 | A half-open integer range ... |
fzospliti 12369 | One direction of splitting... |
fzosplit 12370 | Split a half-open integer ... |
fzodisj 12371 | Abutting half-open integer... |
fzouzsplit 12372 | Split an upper integer set... |
fzouzdisj 12373 | A half-open integer range ... |
fzodisjsn 12374 | A half-open integer range ... |
lbfzo0 12375 | An integer is strictly gre... |
elfzo0 12376 | Membership in a half-open ... |
elfzo0z 12377 | Membership in a half-open ... |
nn0p1elfzo 12378 | A nonnegative integer incr... |
elfzo0le 12379 | A member in a half-open ra... |
elfzonn0 12380 | A member of a half-open ra... |
fzonmapblen 12381 | The result of subtracting ... |
fzofzim 12382 | If a nonnegative integer i... |
fz1fzo0m1 12383 | Translation of one between... |
fzossnn 12384 | Half-open integer ranges s... |
elfzo1 12385 | Membership in a half-open ... |
fzo1fzo0n0 12386 | An integer between 1 and a... |
fzo0n0 12387 | A half-open integer range ... |
fzoaddel 12388 | Translate membership in a ... |
fzo0addel 12389 | Translate membership in a ... |
fzo0addelr 12390 | Translate membership in a ... |
fzoaddel2 12391 | Translate membership in a ... |
elfzoext 12392 | Membership of an integer i... |
elincfzoext 12393 | Membership of an increased... |
fzosubel 12394 | Translate membership in a ... |
fzosubel2 12395 | Membership in a translated... |
fzosubel3 12396 | Membership in a translated... |
eluzgtdifelfzo 12397 | Membership of the differen... |
ige2m2fzo 12398 | Membership of an integer g... |
fzocatel 12399 | Translate membership in a ... |
ubmelfzo 12400 | If an integer in a 1 based... |
elfzodifsumelfzo 12401 | If an integer is in a half... |
elfzom1elp1fzo 12402 | Membership of an integer i... |
elfzom1elfzo 12403 | Membership in a half-open ... |
fzval3 12404 | Expressing a closed intege... |
fzosn 12405 | Expressing a singleton as ... |
elfzomin 12406 | Membership of an integer i... |
zpnn0elfzo 12407 | Membership of an integer i... |
zpnn0elfzo1 12408 | Membership of an integer i... |
fzosplitsnm1 12409 | Removing a singleton from ... |
elfzonlteqm1 12410 | If an element of a half-op... |
fzonn0p1 12411 | A nonnegative integer is e... |
fzossfzop1 12412 | A half-open range of nonne... |
fzonn0p1p1 12413 | If a nonnegative integer i... |
elfzom1p1elfzo 12414 | Increasing an element of a... |
fzo0ssnn0 12415 | Half-open integer ranges s... |
fzo0ssnn0OLD 12416 | Obsolete proof of ~ fzo0ss... |
fzo01 12417 | Expressing the singleton o... |
fzo12sn 12418 | A 1-based half-open intege... |
fzo13pr 12419 | A 1-based half-open intege... |
fzo0to2pr 12420 | A half-open integer range ... |
fzo0to3tp 12421 | A half-open integer range ... |
fzo0to42pr 12422 | A half-open integer range ... |
fzo1to4tp 12423 | A half-open integer range ... |
fzo0sn0fzo1 12424 | A half-open range of nonne... |
fzoend 12425 | The endpoint of a half-ope... |
fzo0end 12426 | The endpoint of a zero-bas... |
ssfzo12 12427 | Subset relationship for ha... |
ssfzoulel 12428 | If a half-open integer ran... |
ssfzo12bi 12429 | Subset relationship for ha... |
ubmelm1fzo 12430 | The result of subtracting ... |
fzofzp1 12431 | If a point is in a half-op... |
fzofzp1b 12432 | If a point is in a half-op... |
elfzom1b 12433 | An integer is a member of ... |
elfzom1elp1fzo1 12434 | Membership of a nonnegativ... |
elfzo1elm1fzo0 12435 | Membership of a positive i... |
elfzonelfzo 12436 | If an element of a half-op... |
fzonfzoufzol 12437 | If an element of a half-op... |
elfzomelpfzo 12438 | An integer increased by an... |
elfznelfzo 12439 | A value in a finite set of... |
elfznelfzob 12440 | A value in a finite set of... |
peano2fzor 12441 | A Peano-postulate-like the... |
fzosplitsn 12442 | Extending a half-open rang... |
fzosplitprm1 12443 | Extending a half-open inte... |
fzosplitsni 12444 | Membership in a half-open ... |
fzisfzounsn 12445 | A finite interval of integ... |
fzostep1 12446 | Two possibilities for a nu... |
fzoshftral 12447 | Shift the scanning order i... |
fzind2 12448 | Induction on the integers ... |
fvinim0ffz 12449 | The function values for th... |
injresinjlem 12450 | Lemma for ~ injresinj . (... |
injresinj 12451 | A function whose restricti... |
subfzo0 12452 | The difference between two... |
flval 12457 | Value of the floor (greate... |
flcl 12458 | The floor (greatest intege... |
reflcl 12459 | The floor (greatest intege... |
fllelt 12460 | A basic property of the fl... |
flcld 12461 | The floor (greatest intege... |
flle 12462 | A basic property of the fl... |
flltp1 12463 | A basic property of the fl... |
fllep1 12464 | A basic property of the fl... |
fraclt1 12465 | The fractional part of a r... |
fracle1 12466 | The fractional part of a r... |
fracge0 12467 | The fractional part of a r... |
flge 12468 | The floor function value i... |
fllt 12469 | The floor function value i... |
flflp1 12470 | Move floor function betwee... |
flid 12471 | An integer is its own floo... |
flidm 12472 | The floor function is idem... |
flidz 12473 | A real number equals its f... |
flltnz 12474 | If A is not an integer, th... |
flwordi 12475 | Ordering relationship for ... |
flword2 12476 | Ordering relationship for ... |
flval2 12477 | An alternate way to define... |
flval3 12478 | An alternate way to define... |
flbi 12479 | A condition equivalent to ... |
flbi2 12480 | A condition equivalent to ... |
adddivflid 12481 | The floor of a sum of an i... |
ico01fl0 12482 | The floor of a real number... |
flge0nn0 12483 | The floor of a number grea... |
flge1nn 12484 | The floor of a number grea... |
fldivnn0 12485 | The floor function of a di... |
refldivcl 12486 | The floor function of a di... |
divfl0 12487 | The floor of a fraction is... |
fladdz 12488 | An integer can be moved in... |
flzadd 12489 | An integer can be moved in... |
flmulnn0 12490 | Move a nonnegative integer... |
btwnzge0 12491 | A real bounded between an ... |
2tnp1ge0ge0 12492 | Two times an integer plus ... |
flhalf 12493 | Ordering relation for the ... |
fldivle 12494 | The floor function of a di... |
fldivnn0le 12495 | The floor function of a di... |
flltdivnn0lt 12496 | The floor function of a di... |
ltdifltdiv 12497 | If the dividend of a divis... |
fldiv4p1lem1div2 12498 | The floor of an integer eq... |
fldiv4lem1div2uz2 12499 | The floor of an integer gr... |
fldiv4lem1div2 12500 | The floor of a positive in... |
ceilval 12501 | The value of the ceiling f... |
dfceil2 12502 | Alternative definition of ... |
ceilval2 12503 | The value of the ceiling f... |
ceicl 12504 | The ceiling function retur... |
ceilcl 12505 | Closure of the ceiling fun... |
ceige 12506 | The ceiling of a real numb... |
ceilge 12507 | The ceiling of a real numb... |
ceim1l 12508 | One less than the ceiling ... |
ceilm1lt 12509 | One less than the ceiling ... |
ceile 12510 | The ceiling of a real numb... |
ceille 12511 | The ceiling of a real numb... |
ceilid 12512 | An integer is its own ceil... |
ceilidz 12513 | A real number equals its c... |
flleceil 12514 | The floor of a real number... |
fleqceilz 12515 | A real number is an intege... |
quoremz 12516 | Quotient and remainder of ... |
quoremnn0 12517 | Quotient and remainder of ... |
quoremnn0ALT 12518 | Alternate proof of ~ quore... |
intfrac2 12519 | Decompose a real into inte... |
intfracq 12520 | Decompose a rational numbe... |
fldiv 12521 | Cancellation of the embedd... |
fldiv2 12522 | Cancellation of an embedde... |
fznnfl 12523 | Finite set of sequential i... |
uzsup 12524 | An upper set of integers i... |
ioopnfsup 12525 | An upper set of reals is u... |
icopnfsup 12526 | An upper set of reals is u... |
rpsup 12527 | The positive reals are unb... |
resup 12528 | The real numbers are unbou... |
xrsup 12529 | The extended real numbers ... |
modval 12532 | The value of the modulo op... |
modvalr 12533 | The value of the modulo op... |
modcl 12534 | Closure law for the modulo... |
flpmodeq 12535 | Partition of a division in... |
modcld 12536 | Closure law for the modulo... |
mod0 12537 | ` A mod B ` is zero iff ` ... |
mulmod0 12538 | The product of an integer ... |
negmod0 12539 | ` A ` is divisible by ` B ... |
modge0 12540 | The modulo operation is no... |
modlt 12541 | The modulo operation is le... |
modelico 12542 | Modular reduction produces... |
moddiffl 12543 | The modulo operation diffe... |
moddifz 12544 | The modulo operation diffe... |
modfrac 12545 | The fractional part of a n... |
flmod 12546 | The floor function express... |
intfrac 12547 | Break a number into its in... |
zmod10 12548 | An integer modulo 1 is 0. ... |
zmod1congr 12549 | Two arbitrary integers are... |
modmulnn 12550 | Move a positive integer in... |
modvalp1 12551 | The value of the modulo op... |
zmodcl 12552 | Closure law for the modulo... |
zmodcld 12553 | Closure law for the modulo... |
zmodfz 12554 | An integer mod ` B ` lies ... |
zmodfzo 12555 | An integer mod ` B ` lies ... |
zmodfzp1 12556 | An integer mod ` B ` lies ... |
modid 12557 | Identity law for modulo. ... |
modid0 12558 | A positive real number mod... |
modid2 12559 | Identity law for modulo. ... |
zmodid2 12560 | Identity law for modulo re... |
zmodidfzo 12561 | Identity law for modulo re... |
zmodidfzoimp 12562 | Identity law for modulo re... |
0mod 12563 | Special case: 0 modulo a p... |
1mod 12564 | Special case: 1 modulo a r... |
modabs 12565 | Absorption law for modulo.... |
modabs2 12566 | Absorption law for modulo.... |
modcyc 12567 | The modulo operation is pe... |
modcyc2 12568 | The modulo operation is pe... |
modadd1 12569 | Addition property of the m... |
modaddabs 12570 | Absorption law for modulo.... |
modaddmod 12571 | The sum of a real number m... |
muladdmodid 12572 | The sum of a positive real... |
mulp1mod1 12573 | The product of an integer ... |
modmuladd 12574 | Decomposition of an intege... |
modmuladdim 12575 | Implication of a decomposi... |
modmuladdnn0 12576 | Implication of a decomposi... |
negmod 12577 | The negation of a number m... |
m1modnnsub1 12578 | Minus one modulo a positiv... |
m1modge3gt1 12579 | Minus one modulo an intege... |
addmodid 12580 | The sum of a positive inte... |
addmodidr 12581 | The sum of a positive inte... |
modadd2mod 12582 | The sum of a real number m... |
modm1p1mod0 12583 | If an real number modulo a... |
modltm1p1mod 12584 | If a real number modulo a ... |
modmul1 12585 | Multiplication property of... |
modmul12d 12586 | Multiplication property of... |
modnegd 12587 | Negation property of the m... |
modadd12d 12588 | Additive property of the m... |
modsub12d 12589 | Subtraction property of th... |
modsubmod 12590 | The difference of a real n... |
modsubmodmod 12591 | The difference of a real n... |
2txmodxeq0 12592 | Two times a positive real ... |
2submod 12593 | If a real number is betwee... |
modifeq2int 12594 | If a nonnegative integer i... |
modaddmodup 12595 | The sum of an integer modu... |
modaddmodlo 12596 | The sum of an integer modu... |
modmulmod 12597 | The product of a real numb... |
modmulmodr 12598 | The product of an integer ... |
modaddmulmod 12599 | The sum of a real number a... |
moddi 12600 | Distribute multiplication ... |
modsubdir 12601 | Distribute the modulo oper... |
modeqmodmin 12602 | A real number equals the d... |
modirr 12603 | A number modulo an irratio... |
modfzo0difsn 12604 | For a number within a half... |
modsumfzodifsn 12605 | The sum of a number within... |
modlteq 12606 | Two nonnegative integers l... |
addmodlteq 12607 | Two nonnegative integers l... |
om2uz0i 12608 | The mapping ` G ` is a one... |
om2uzsuci 12609 | The value of ` G ` (see ~ ... |
om2uzuzi 12610 | The value ` G ` (see ~ om2... |
om2uzlti 12611 | Less-than relation for ` G... |
om2uzlt2i 12612 | The mapping ` G ` (see ~ o... |
om2uzrani 12613 | Range of ` G ` (see ~ om2u... |
om2uzf1oi 12614 | ` G ` (see ~ om2uz0i ) is ... |
om2uzisoi 12615 | ` G ` (see ~ om2uz0i ) is ... |
om2uzoi 12616 | An alternative definition ... |
om2uzrdg 12617 | A helper lemma for the val... |
uzrdglem 12618 | A helper lemma for the val... |
uzrdgfni 12619 | The recursive definition g... |
uzrdg0i 12620 | Initial value of a recursi... |
uzrdgsuci 12621 | Successor value of a recur... |
ltweuz 12622 | ` < ` is a well-founded re... |
ltwenn 12623 | Less than well-orders the ... |
ltwefz 12624 | Less than well-orders a se... |
uzenom 12625 | An upper integer set is de... |
uzinf 12626 | An upper integer set is in... |
nnnfi 12627 | The set of positive intege... |
uzrdgxfr 12628 | Transfer the value of the ... |
fzennn 12629 | The cardinality of a finit... |
fzen2 12630 | The cardinality of a finit... |
cardfz 12631 | The cardinality of a finit... |
hashgf1o 12632 | ` G ` maps ` _om ` one-to-... |
fzfi 12633 | A finite interval of integ... |
fzfid 12634 | Commonly used special case... |
fzofi 12635 | Half-open integer sets are... |
fsequb 12636 | The values of a finite rea... |
fsequb2 12637 | The values of a finite rea... |
fseqsupcl 12638 | The values of a finite rea... |
fseqsupubi 12639 | The values of a finite rea... |
nn0ennn 12640 | The nonnegative integers a... |
nnenom 12641 | The set of positive intege... |
nnct 12642 | ` NN ` is countable. (Con... |
uzindi 12643 | Indirect strong induction ... |
axdc4uzlem 12644 | Lemma for ~ axdc4uz . (Co... |
axdc4uz 12645 | A version of ~ axdc4 that ... |
ssnn0fi 12646 | A subset of the nonnegativ... |
rabssnn0fi 12647 | A subset of the nonnegativ... |
uzsinds 12648 | Strong (or "total") induct... |
nnsinds 12649 | Strong (or "total") induct... |
nn0sinds 12650 | Strong (or "total") induct... |
fsuppmapnn0fiublem 12651 | Lemma for ~ fsuppmapnn0fiu... |
fsuppmapnn0fiub 12652 | If all functions of a fini... |
fsuppmapnn0fiubOLD 12653 | Obsolete proof of ~ fsuppm... |
fsuppmapnn0fiubex 12654 | If all functions of a fini... |
fsuppmapnn0fiub0 12655 | If all functions of a fini... |
suppssfz 12656 | Condition for a function o... |
fsuppmapnn0ub 12657 | If a function over the non... |
fsuppmapnn0fz 12658 | If a function over the non... |
mptnn0fsupp 12659 | A mapping from the nonnega... |
mptnn0fsuppd 12660 | A mapping from the nonnega... |
mptnn0fsuppr 12661 | A finitely supported mappi... |
f13idfv 12662 | A one-to-one function with... |
seqex 12665 | Existence of the sequence ... |
seqeq1 12666 | Equality theorem for the s... |
seqeq2 12667 | Equality theorem for the s... |
seqeq3 12668 | Equality theorem for the s... |
seqeq1d 12669 | Equality deduction for the... |
seqeq2d 12670 | Equality deduction for the... |
seqeq3d 12671 | Equality deduction for the... |
seqeq123d 12672 | Equality deduction for the... |
nfseq 12673 | Hypothesis builder for the... |
seqval 12674 | Value of the sequence buil... |
seqfn 12675 | The sequence builder funct... |
seq1 12676 | Value of the sequence buil... |
seq1i 12677 | Value of the sequence buil... |
seqp1 12678 | Value of the sequence buil... |
seqp1i 12679 | Value of the sequence buil... |
seqm1 12680 | Value of the sequence buil... |
seqcl2 12681 | Closure properties of the ... |
seqf2 12682 | Range of the recursive seq... |
seqcl 12683 | Closure properties of the ... |
seqf 12684 | Range of the recursive seq... |
seqfveq2 12685 | Equality of sequences. (C... |
seqfeq2 12686 | Equality of sequences. (C... |
seqfveq 12687 | Equality of sequences. (C... |
seqfeq 12688 | Equality of sequences. (C... |
seqshft2 12689 | Shifting the index set of ... |
seqres 12690 | Restricting its characteri... |
serf 12691 | An infinite series of comp... |
serfre 12692 | An infinite series of real... |
monoord 12693 | Ordering relation for a mo... |
monoord2 12694 | Ordering relation for a mo... |
sermono 12695 | The partial sums in an inf... |
seqsplit 12696 | Split a sequence into two ... |
seq1p 12697 | Removing the first term fr... |
seqcaopr3 12698 | Lemma for ~ seqcaopr2 . (... |
seqcaopr2 12699 | The sum of two infinite se... |
seqcaopr 12700 | The sum of two infinite se... |
seqf1olem2a 12701 | Lemma for ~ seqf1o . (Con... |
seqf1olem1 12702 | Lemma for ~ seqf1o . (Con... |
seqf1olem2 12703 | Lemma for ~ seqf1o . (Con... |
seqf1o 12704 | Rearrange a sum via an arb... |
seradd 12705 | The sum of two infinite se... |
sersub 12706 | The difference of two infi... |
seqid3 12707 | A sequence that consists e... |
seqid 12708 | Discard the first few term... |
seqid2 12709 | The last few terms of a se... |
seqhomo 12710 | Apply a homomorphism to a ... |
seqz 12711 | If the operation ` .+ ` ha... |
seqfeq4 12712 | Equality of series under d... |
seqfeq3 12713 | Equality of series under d... |
seqdistr 12714 | The distributive property ... |
ser0 12715 | The value of the partial s... |
ser0f 12716 | A zero-valued infinite ser... |
serge0 12717 | A finite sum of nonnegativ... |
serle 12718 | Comparison of partial sums... |
ser1const 12719 | Value of the partial serie... |
seqof 12720 | Distribute function operat... |
seqof2 12721 | Distribute function operat... |
expval 12724 | Value of exponentiation to... |
expnnval 12725 | Value of exponentiation to... |
exp0 12726 | Value of a complex number ... |
0exp0e1 12727 | ` 0 ^ 0 = 1 ` (common case... |
exp1 12728 | Value of a complex number ... |
expp1 12729 | Value of a complex number ... |
expneg 12730 | Value of a complex number ... |
expneg2 12731 | Value of a complex number ... |
expn1 12732 | A number to the negative o... |
expcllem 12733 | Lemma for proving nonnegat... |
expcl2lem 12734 | Lemma for proving integer ... |
nnexpcl 12735 | Closure of exponentiation ... |
nn0expcl 12736 | Closure of exponentiation ... |
zexpcl 12737 | Closure of exponentiation ... |
qexpcl 12738 | Closure of exponentiation ... |
reexpcl 12739 | Closure of exponentiation ... |
expcl 12740 | Closure law for nonnegativ... |
rpexpcl 12741 | Closure law for exponentia... |
reexpclz 12742 | Closure of exponentiation ... |
qexpclz 12743 | Closure of exponentiation ... |
m1expcl2 12744 | Closure of exponentiation ... |
m1expcl 12745 | Closure of exponentiation ... |
expclzlem 12746 | Closure law for integer ex... |
expclz 12747 | Closure law for integer ex... |
nn0expcli 12748 | Closure of exponentiation ... |
nn0sqcl 12749 | The square of a nonnegativ... |
expm1t 12750 | Exponentiation in terms of... |
1exp 12751 | Value of one raised to a n... |
expeq0 12752 | Positive integer exponenti... |
expne0 12753 | Positive integer exponenti... |
expne0i 12754 | Nonnegative integer expone... |
expgt0 12755 | Nonnegative integer expone... |
expnegz 12756 | Value of a complex number ... |
0exp 12757 | Value of zero raised to a ... |
expge0 12758 | Nonnegative integer expone... |
expge1 12759 | Nonnegative integer expone... |
expgt1 12760 | Positive integer exponenti... |
mulexp 12761 | Positive integer exponenti... |
mulexpz 12762 | Integer exponentiation of ... |
exprec 12763 | Nonnegative integer expone... |
expadd 12764 | Sum of exponents law for n... |
expaddzlem 12765 | Lemma for ~ expaddz . (Co... |
expaddz 12766 | Sum of exponents law for i... |
expmul 12767 | Product of exponents law f... |
expmulz 12768 | Product of exponents law f... |
m1expeven 12769 | Exponentiation of negative... |
expsub 12770 | Exponent subtraction law f... |
expp1z 12771 | Value of a nonzero complex... |
expm1 12772 | Value of a complex number ... |
expdiv 12773 | Nonnegative integer expone... |
ltexp2a 12774 | Ordering relationship for ... |
expcan 12775 | Cancellation law for expon... |
ltexp2 12776 | Ordering law for exponenti... |
leexp2 12777 | Ordering law for exponenti... |
leexp2a 12778 | Weak ordering relationship... |
ltexp2r 12779 | The power of a positive nu... |
leexp2r 12780 | Weak ordering relationship... |
leexp1a 12781 | Weak mantissa ordering rel... |
exple1 12782 | Nonnegative integer expone... |
expubnd 12783 | An upper bound on ` A ^ N ... |
sqval 12784 | Value of the square of a c... |
sqneg 12785 | The square of the negative... |
sqsubswap 12786 | Swap the order of subtract... |
sqcl 12787 | Closure of square. (Contr... |
sqmul 12788 | Distribution of square ove... |
sqeq0 12789 | A number is zero iff its s... |
sqdiv 12790 | Distribution of square ove... |
sqdivid 12791 | The square of a nonzero nu... |
sqne0 12792 | A number is nonzero iff it... |
resqcl 12793 | Closure of the square of a... |
sqgt0 12794 | The square of a nonzero re... |
nnsqcl 12795 | The naturals are closed un... |
zsqcl 12796 | Integers are closed under ... |
qsqcl 12797 | The square of a rational i... |
sq11 12798 | The square function is one... |
lt2sq 12799 | The square function on non... |
le2sq 12800 | The square function on non... |
le2sq2 12801 | The square of a 'less than... |
sqge0 12802 | A square of a real is nonn... |
zsqcl2 12803 | The square of an integer i... |
sumsqeq0 12804 | Two real numbers are equal... |
sqvali 12805 | Value of square. Inferenc... |
sqcli 12806 | Closure of square. (Contr... |
sqeq0i 12807 | A number is zero iff its s... |
sqrecii 12808 | Square of reciprocal. (Co... |
sqmuli 12809 | Distribution of square ove... |
sqdivi 12810 | Distribution of square ove... |
resqcli 12811 | Closure of square in reals... |
sqgt0i 12812 | The square of a nonzero re... |
sqge0i 12813 | A square of a real is nonn... |
lt2sqi 12814 | The square function on non... |
le2sqi 12815 | The square function on non... |
sq11i 12816 | The square function is one... |
sq0 12817 | The square of 0 is 0. (Co... |
sq0i 12818 | If a number is zero, its s... |
sq0id 12819 | If a number is zero, its s... |
sq1 12820 | The square of 1 is 1. (Co... |
neg1sqe1 12821 | ` -u 1 ` squared is 1 (com... |
sq2 12822 | The square of 2 is 4. (Co... |
sq3 12823 | The square of 3 is 9. (Co... |
sq4e2t8 12824 | The square of 4 is 2 times... |
cu2 12825 | The cube of 2 is 8. (Cont... |
irec 12826 | The reciprocal of ` _i ` .... |
i2 12827 | ` _i ` squared. (Contribu... |
i3 12828 | ` _i ` cubed. (Contribute... |
i4 12829 | ` _i ` to the fourth power... |
nnlesq 12830 | A positive integer is less... |
iexpcyc 12831 | Taking ` _i ` to the ` K `... |
expnass 12832 | A counterexample showing t... |
sqlecan 12833 | Cancel one factor of a squ... |
subsq 12834 | Factor the difference of t... |
subsq2 12835 | Express the difference of ... |
binom2i 12836 | The square of a binomial. ... |
subsqi 12837 | Factor the difference of t... |
sqeqori 12838 | The squares of two complex... |
subsq0i 12839 | The two solutions to the d... |
sqeqor 12840 | The squares of two complex... |
binom2 12841 | The square of a binomial. ... |
binom21 12842 | Special case of ~ binom2 w... |
binom2sub 12843 | Expand the square of a sub... |
binom2sub1 12844 | Special case of ~ binom2su... |
binom2subi 12845 | Expand the square of a sub... |
mulbinom2 12846 | The square of a binomial w... |
binom3 12847 | The cube of a binomial. (... |
sq01 12848 | If a complex number equals... |
zesq 12849 | An integer is even iff its... |
nnesq 12850 | A positive integer is even... |
crreczi 12851 | Reciprocal of a complex nu... |
bernneq 12852 | Bernoulli's inequality, du... |
bernneq2 12853 | Variation of Bernoulli's i... |
bernneq3 12854 | A corollary of ~ bernneq .... |
expnbnd 12855 | Exponentiation with a mant... |
expnlbnd 12856 | The reciprocal of exponent... |
expnlbnd2 12857 | The reciprocal of exponent... |
expmulnbnd 12858 | Exponentiation with a mant... |
digit2 12859 | Two ways to express the ` ... |
digit1 12860 | Two ways to express the ` ... |
modexp 12861 | Exponentiation property of... |
discr1 12862 | A nonnegative quadratic fo... |
discr 12863 | If a quadratic polynomial ... |
exp0d 12864 | Value of a complex number ... |
exp1d 12865 | Value of a complex number ... |
expeq0d 12866 | Positive integer exponenti... |
sqvald 12867 | Value of square. Inferenc... |
sqcld 12868 | Closure of square. (Contr... |
sqeq0d 12869 | A number is zero iff its s... |
expcld 12870 | Closure law for nonnegativ... |
expp1d 12871 | Value of a complex number ... |
expaddd 12872 | Sum of exponents law for n... |
expmuld 12873 | Product of exponents law f... |
sqrecd 12874 | Square of reciprocal. (Co... |
expclzd 12875 | Closure law for integer ex... |
expne0d 12876 | Nonnegative integer expone... |
expnegd 12877 | Value of a complex number ... |
exprecd 12878 | Nonnegative integer expone... |
expp1zd 12879 | Value of a nonzero complex... |
expm1d 12880 | Value of a complex number ... |
expsubd 12881 | Exponent subtraction law f... |
sqmuld 12882 | Distribution of square ove... |
sqdivd 12883 | Distribution of square ove... |
expdivd 12884 | Nonnegative integer expone... |
mulexpd 12885 | Positive integer exponenti... |
0expd 12886 | Value of zero raised to a ... |
reexpcld 12887 | Closure of exponentiation ... |
expge0d 12888 | Nonnegative integer expone... |
expge1d 12889 | Nonnegative integer expone... |
sqoddm1div8 12890 | A squared odd number minus... |
nnsqcld 12891 | The naturals are closed un... |
nnexpcld 12892 | Closure of exponentiation ... |
nn0expcld 12893 | Closure of exponentiation ... |
rpexpcld 12894 | Closure law for exponentia... |
ltexp2rd 12895 | The power of a positive nu... |
reexpclzd 12896 | Closure of exponentiation ... |
resqcld 12897 | Closure of square in reals... |
sqge0d 12898 | A square of a real is nonn... |
sqgt0d 12899 | The square of a nonzero re... |
ltexp2d 12900 | Ordering relationship for ... |
leexp2d 12901 | Ordering law for exponenti... |
expcand 12902 | Ordering relationship for ... |
leexp2ad 12903 | Ordering relationship for ... |
leexp2rd 12904 | Ordering relationship for ... |
lt2sqd 12905 | The square function on non... |
le2sqd 12906 | The square function on non... |
sq11d 12907 | The square function is one... |
mulsubdivbinom2 12908 | The square of a binomial w... |
muldivbinom2 12909 | The square of a binomial w... |
sq10 12910 | The square of 10 is 100. ... |
sq10e99m1 12911 | The square of 10 is 99 plu... |
3dec 12912 | A "decimal constructor" wh... |
sq10OLD 12913 | Old version of ~ sq10 . O... |
sq10e99m1OLD 12914 | Old version of ~ sq10e99m1... |
3decOLD 12915 | Old version of ~ 3dec . O... |
nn0le2msqi 12916 | The square function on non... |
nn0opthlem1 12917 | A rather pretty lemma for ... |
nn0opthlem2 12918 | Lemma for ~ nn0opthi . (C... |
nn0opthi 12919 | An ordered pair theorem fo... |
nn0opth2i 12920 | An ordered pair theorem fo... |
nn0opth2 12921 | An ordered pair theorem fo... |
facnn 12924 | Value of the factorial fun... |
fac0 12925 | The factorial of 0. (Cont... |
fac1 12926 | The factorial of 1. (Cont... |
facp1 12927 | The factorial of a success... |
fac2 12928 | The factorial of 2. (Cont... |
fac3 12929 | The factorial of 3. (Cont... |
fac4 12930 | The factorial of 4. (Cont... |
facnn2 12931 | Value of the factorial fun... |
faccl 12932 | Closure of the factorial f... |
faccld 12933 | Closure of the factorial f... |
facmapnn 12934 | The factorial function res... |
facne0 12935 | The factorial function is ... |
facdiv 12936 | A positive integer divides... |
facndiv 12937 | No positive integer (great... |
facwordi 12938 | Ordering property of facto... |
faclbnd 12939 | A lower bound for the fact... |
faclbnd2 12940 | A lower bound for the fact... |
faclbnd3 12941 | A lower bound for the fact... |
faclbnd4lem1 12942 | Lemma for ~ faclbnd4 . Pr... |
faclbnd4lem2 12943 | Lemma for ~ faclbnd4 . Us... |
faclbnd4lem3 12944 | Lemma for ~ faclbnd4 . Th... |
faclbnd4lem4 12945 | Lemma for ~ faclbnd4 . Pr... |
faclbnd4 12946 | Variant of ~ faclbnd5 prov... |
faclbnd5 12947 | The factorial function gro... |
faclbnd6 12948 | Geometric lower bound for ... |
facubnd 12949 | An upper bound for the fac... |
facavg 12950 | The product of two factori... |
bcval 12953 | Value of the binomial coef... |
bcval2 12954 | Value of the binomial coef... |
bcval3 12955 | Value of the binomial coef... |
bcval4 12956 | Value of the binomial coef... |
bcrpcl 12957 | Closure of the binomial co... |
bccmpl 12958 | "Complementing" its second... |
bcn0 12959 | ` N ` choose 0 is 1. Rema... |
bc0k 12960 | The binomial coefficient "... |
bcnn 12961 | ` N ` choose ` N ` is 1. ... |
bcn1 12962 | Binomial coefficient: ` N ... |
bcnp1n 12963 | Binomial coefficient: ` N ... |
bcm1k 12964 | The proportion of one bino... |
bcp1n 12965 | The proportion of one bino... |
bcp1nk 12966 | The proportion of one bino... |
bcval5 12967 | Write out the top and bott... |
bcn2 12968 | Binomial coefficient: ` N ... |
bcp1m1 12969 | Compute the binomial coeff... |
bcpasc 12970 | Pascal's rule for the bino... |
bccl 12971 | A binomial coefficient, in... |
bccl2 12972 | A binomial coefficient, in... |
bcn2m1 12973 | Compute the binomial coeff... |
bcn2p1 12974 | Compute the binomial coeff... |
permnn 12975 | The number of permutations... |
bcnm1 12976 | The binomial coefficent of... |
4bc3eq4 12977 | The value of four choose t... |
4bc2eq6 12978 | The value of four choose t... |
hashkf 12981 | The finite part of the siz... |
hashgval 12982 | The value of the ` # ` fun... |
hashginv 12983 | ` ``' G ` maps the size fu... |
hashinf 12984 | The value of the ` # ` fun... |
hashbnd 12985 | If ` A ` has size bounded ... |
hashfxnn0 12986 | The size function is a fun... |
hashf 12987 | The size function maps all... |
hashfOLD 12988 | Obsolete version of ~ hash... |
hashxnn0 12989 | The value of the hash func... |
hashresfn 12990 | Restriction of the domain ... |
dmhashres 12991 | Restriction of the domain ... |
hashnn0pnf 12992 | The value of the hash func... |
hashnnn0genn0 12993 | If the size of a set is no... |
hashnemnf 12994 | The size of a set is never... |
hashv01gt1 12995 | The size of a set is eithe... |
hashfz1 12996 | The set ` ( 1 ... N ) ` ha... |
hashen 12997 | Two finite sets have the s... |
hasheni 12998 | Equinumerous sets have the... |
hasheqf1o 12999 | The size of two finite set... |
fiinfnf1o 13000 | There is no bijection betw... |
focdmex 13001 | The codomain of an onto fu... |
hasheqf1oi 13002 | The size of two sets is eq... |
hasheqf1oiOLD 13003 | Obsolete version of ~ hash... |
hashf1rn 13004 | The size of a finite set w... |
hashf1rnOLD 13005 | Obsolete version of ~ hash... |
hasheqf1od 13006 | The size of two sets is eq... |
fz1eqb 13007 | Two possibly-empty 1-based... |
hashcard 13008 | The size function of the c... |
hashcl 13009 | Closure of the ` # ` funct... |
hashxrcl 13010 | Extended real closure of t... |
hashclb 13011 | Reverse closure of the ` #... |
hashvnfin 13012 | A set of finite size is a ... |
hashnfinnn0 13013 | The size of an infinite se... |
isfinite4 13014 | A finite set is equinumero... |
hasheq0 13015 | Two ways of saying a finit... |
hashneq0 13016 | Two ways of saying a set i... |
hashgt0n0 13017 | If the size of a set is gr... |
hashnncl 13018 | Positive natural closure o... |
hash0 13019 | The empty set has size zer... |
hashsng 13020 | The size of a singleton. ... |
hashen1 13021 | A set with only one elemen... |
hashrabrsn 13022 | The size of a restricted c... |
hashrabsn01 13023 | The size of a restricted c... |
hashrabsn1 13024 | If the size of a restricte... |
hashfn 13025 | A function is equinumerous... |
fseq1hash 13026 | The value of the size func... |
hashgadd 13027 | ` G ` maps ordinal additio... |
hashgval2 13028 | A short expression for the... |
hashdom 13029 | Dominance relation for the... |
hashdomi 13030 | Non-strict order relation ... |
hashsdom 13031 | Strict dominance relation ... |
hashun 13032 | The size of the union of d... |
hashun2 13033 | The size of the union of f... |
hashun3 13034 | The size of the union of f... |
hashinfxadd 13035 | The extended real addition... |
hashunx 13036 | The size of the union of d... |
hashge0 13037 | The cardinality of a set i... |
hashgt0 13038 | The cardinality of a nonem... |
hashge1 13039 | The cardinality of a nonem... |
1elfz0hash 13040 | 1 is an element of the fin... |
hashnn0n0nn 13041 | If a nonnegative integer i... |
hashunsng 13042 | The size of the union of a... |
hashprg 13043 | The size of an unordered p... |
hashprgOLD 13044 | Obsolete version of ~ hash... |
elprchashprn2 13045 | If one element of an unord... |
hashprb 13046 | The size of an unordered p... |
hashprdifel 13047 | The elements of an unorder... |
prhash2ex 13048 | There is (at least) one se... |
hashle00 13049 | If the size of a set is le... |
hashgt0elex 13050 | If the size of a set is gr... |
hashgt0elexb 13051 | The size of a set is great... |
hashp1i 13052 | Size of a finite ordinal. ... |
hash1 13053 | Size of a finite ordinal. ... |
hash2 13054 | Size of a finite ordinal. ... |
hash3 13055 | Size of a finite ordinal. ... |
hash4 13056 | Size of a finite ordinal. ... |
pr0hash2ex 13057 | There is (at least) one se... |
hashss 13058 | The size of a subset is le... |
prsshashgt1 13059 | The size of a superset of ... |
hashin 13060 | The size of the intersecti... |
hashssdif 13061 | The size of the difference... |
hashdif 13062 | The size of the difference... |
hashdifsn 13063 | The size of the difference... |
hashdifpr 13064 | The size of the difference... |
hashsn01 13065 | The size of a singleton is... |
hashsnle1 13066 | The size of a singleton is... |
hashsnlei 13067 | Get an upper bound on a co... |
hash1snb 13068 | The size of a set is 1 if ... |
euhash1 13069 | The size of a set is 1 in ... |
hashgt12el 13070 | In a set with more than on... |
hashgt12el2 13071 | In a set with more than on... |
hashunlei 13072 | Get an upper bound on a co... |
hashsslei 13073 | Get an upper bound on a co... |
hashfz 13074 | Value of the numeric cardi... |
fzsdom2 13075 | Condition for finite range... |
hashfzo 13076 | Cardinality of a half-open... |
hashfzo0 13077 | Cardinality of a half-open... |
hashfzp1 13078 | Value of the numeric cardi... |
hashfz0 13079 | Value of the numeric cardi... |
hashxplem 13080 | Lemma for ~ hashxp . (Con... |
hashxp 13081 | The size of the Cartesian ... |
hashmap 13082 | The size of the set expone... |
hashpw 13083 | The size of the power set ... |
hashfun 13084 | A finite set is a function... |
hashimarn 13085 | The size of the image of a... |
hashimarni 13086 | If the size of the image o... |
fnfz0hash 13087 | The size of a function on ... |
ffz0hash 13088 | The size of a function on ... |
fnfz0hashnn0 13089 | The size of a function on ... |
ffzo0hash 13090 | The size of a function on ... |
fnfzo0hash 13091 | The size of a function on ... |
fnfzo0hashnn0 13092 | The value of the size func... |
hashbclem 13093 | Lemma for ~ hashbc : induc... |
hashbc 13094 | The binomial coefficient c... |
hashfacen 13095 | The number of bijections b... |
hashf1lem1 13096 | Lemma for ~ hashf1 . (Con... |
hashf1lem2 13097 | Lemma for ~ hashf1 . (Con... |
hashf1 13098 | The permutation number ` |... |
hashfac 13099 | A factorial counts the num... |
leiso 13100 | Two ways to write a strict... |
leisorel 13101 | Version of ~ isorel for st... |
fz1isolem 13102 | Lemma for ~ fz1iso . (Con... |
fz1iso 13103 | Any finite ordered set has... |
ishashinf 13104 | Any set that is not finite... |
seqcoll 13105 | The function ` F ` contain... |
seqcoll2 13106 | The function ` F ` contain... |
hashprlei 13107 | An unordered pair has at m... |
hash2pr 13108 | A set of size two is an un... |
hash2prde 13109 | A set of size two is an un... |
hash2exprb 13110 | A set of size two is an un... |
hash2prb 13111 | A set of size two is a pro... |
prprrab 13112 | The set of proper pairs of... |
nehash2 13113 | The cardinality of a set w... |
hash2prd 13114 | A set of size two is an un... |
hash2pwpr 13115 | If the size of a subset of... |
pr2pwpr 13116 | The set of subsets of a pa... |
hashge2el2dif 13117 | A set with size at least 2... |
hashge2el2difr 13118 | A set with at least 2 diff... |
hashge2el2difb 13119 | A set has size at least 2 ... |
hashtplei 13120 | An unordered triple has at... |
hashtpg 13121 | The size of an unordered t... |
hashge3el3dif 13122 | A set with size at least 3... |
elss2prb 13123 | An element of the set of s... |
hash2sspr 13124 | A subset of size two is an... |
elss2prOLD 13125 | An element of the set of s... |
exprelprel 13126 | If there is an element of ... |
hash3tr 13127 | A set of size three is an ... |
hash1to3 13128 | If the size of a set is be... |
fundmge2nop0 13129 | A function with a domain c... |
fundmge2nop 13130 | A function with a domain c... |
fun2dmnop0 13131 | A function with a domain c... |
fun2dmnop 13132 | A function with a domain c... |
brfi1indlem 13133 | Lemma for ~ brfi1ind : Th... |
fi1uzind 13134 | Properties of an ordered p... |
brfi1uzind 13135 | Properties of a binary rel... |
brfi1ind 13136 | Properties of a binary rel... |
brfi1indALT 13137 | Alternate proof of ~ brfi1... |
opfi1uzind 13138 | Properties of an ordered p... |
opfi1ind 13139 | Properties of an ordered p... |
fi1uzindOLD 13140 | Obsolete version of ~ fi1u... |
brfi1uzindOLD 13141 | Obsolete version of ~ brfi... |
brfi1indOLD 13142 | Obsolete version of ~ brfi... |
brfi1indALTOLD 13143 | Obsolete version of ~ brfi... |
opfi1uzindOLD 13144 | Obsolete version of ~ opfi... |
opfi1indOLD 13145 | Obsolete version of ~ opfi... |
iswrd 13162 | Property of being a word o... |
wrdval 13163 | Value of the set of words ... |
iswrdi 13164 | A zero-based sequence is a... |
wrdf 13165 | A word is a zero-based seq... |
iswrdb 13166 | A word over an alphabet is... |
wrddm 13167 | The indices of a word (i.e... |
sswrd 13168 | The set of words respects ... |
snopiswrd 13169 | A singleton of an ordered ... |
wrdexg 13170 | The set of words over a se... |
wrdexb 13171 | The set of words over a se... |
wrdexi 13172 | The set of words over a se... |
wrdsymbcl 13173 | A symbol within a word ove... |
wrdfn 13174 | A word is a function with ... |
wrdv 13175 | A word over an alphabet is... |
wrdlndm 13176 | The length of a word is no... |
iswrdsymb 13177 | An arbitrary word is a wor... |
wrdfin 13178 | A word is a finite set. (... |
lencl 13179 | The length of a word is a ... |
lennncl 13180 | The length of a nonempty w... |
wrdffz 13181 | A word is a function from ... |
wrdeq 13182 | Equality theorem for the s... |
wrdeqi 13183 | Equality theorem for the s... |
iswrddm0 13184 | A function with empty doma... |
wrd0 13185 | The empty set is a word (t... |
0wrd0 13186 | The empty word is the only... |
ffz0iswrd 13187 | A sequence with zero-based... |
nfwrd 13188 | Hypothesis builder for ` W... |
csbwrdg 13189 | Class substitution for the... |
wrdnval 13190 | Words of a fixed length ar... |
wrdmap 13191 | Words as a mapping. (Cont... |
hashwrdn 13192 | If there is only a finite ... |
wrdnfi 13193 | If there is only a finite ... |
wrdsymb0 13194 | A symbol at a position "ou... |
wrdlenge1n0 13195 | A word with length at leas... |
wrdlenge2n0 13196 | A word with length at leas... |
wrdsymb1 13197 | The first symbol of a none... |
wrdlen1 13198 | A word of length 1 starts ... |
fstwrdne 13199 | The first symbol of a none... |
fstwrdne0 13200 | The first symbol of a none... |
eqwrd 13201 | Two words are equal iff th... |
elovmpt2wrd 13202 | Implications for the value... |
elovmptnn0wrd 13203 | Implications for the value... |
lsw 13204 | Extract the last symbol of... |
lsw0 13205 | The last symbol of an empt... |
lsw0g 13206 | The last symbol of an empt... |
lsw1 13207 | The last symbol of a word ... |
lswcl 13208 | Closure of the last symbol... |
lswlgt0cl 13209 | The last symbol of a nonem... |
ccatfn 13210 | The concatenation operator... |
ccatfval 13211 | Value of the concatenation... |
ccatcl 13212 | The concatenation of two w... |
ccatlen 13213 | The length of a concatenat... |
ccatval1 13214 | Value of a symbol in the l... |
ccatval2 13215 | Value of a symbol in the r... |
ccatval3 13216 | Value of a symbol in the r... |
elfzelfzccat 13217 | An element of a finite set... |
ccatvalfn 13218 | The concatenation of two w... |
ccatsymb 13219 | The symbol at a given posi... |
ccatfv0 13220 | The first symbol of a conc... |
ccatval1lsw 13221 | The last symbol of the lef... |
ccatlid 13222 | Concatenation of a word by... |
ccatrid 13223 | Concatenation of a word by... |
ccatass 13224 | Associative law for concat... |
ccatrn 13225 | The range of a concatenate... |
lswccatn0lsw 13226 | The last symbol of a word ... |
lswccat0lsw 13227 | The last symbol of a word ... |
ccatalpha 13228 | A concatenation of two arb... |
ccatrcl1 13229 | Reverse closure of a conca... |
ids1 13230 | Identity function protecti... |
s1val 13231 | Value of a single-symbol w... |
s1rn 13232 | The range of a single-symb... |
s1eq 13233 | Equality theorem for a sin... |
s1eqd 13234 | Equality theorem for a sin... |
s1cl 13235 | A singleton word is a word... |
s1cld 13236 | A singleton word is a word... |
s1cli 13237 | A singleton word is a word... |
s1len 13238 | Length of a singleton word... |
s1nz 13239 | A singleton word is not th... |
s1nzOLD 13240 | Obsolete proof of ~ s1nz a... |
s1dm 13241 | The domain of a singleton ... |
s1dmALT 13242 | Alternate version of ~ s1d... |
s1fv 13243 | Sole symbol of a singleton... |
lsws1 13244 | The last symbol of a singl... |
eqs1 13245 | A word of length 1 is a si... |
wrdl1exs1 13246 | A word of length 1 is a si... |
wrdl1s1 13247 | A word of length 1 is a si... |
s111 13248 | The singleton word functio... |
ccatws1cl 13249 | The concatenation of a wor... |
ccat2s1cl 13250 | The concatenation of two s... |
ccatws1len 13251 | The length of the concaten... |
wrdlenccats1lenm1 13252 | The length of a word is th... |
ccat2s1len 13253 | The length of the concaten... |
ccatw2s1len 13254 | The length of the concaten... |
ccats1val1 13255 | Value of a symbol in the l... |
ccats1val2 13256 | Value of the symbol concat... |
ccat2s1p1 13257 | Extract the first of two c... |
ccat2s1p2 13258 | Extract the second of two ... |
ccatw2s1ass 13259 | Associative law for a conc... |
ccatws1lenrev 13260 | The length of a word conca... |
ccatws1n0 13261 | The concatenation of a wor... |
ccatws1ls 13262 | The last symbol of the con... |
lswccats1 13263 | The last symbol of a word ... |
lswccats1fst 13264 | The last symbol of a nonem... |
ccatw2s1p1 13265 | Extract the symbol of the ... |
ccatw2s1p2 13266 | Extract the second of two ... |
ccat2s1fvw 13267 | Extract a symbol of a word... |
ccat2s1fst 13268 | The first symbol of the co... |
swrdval 13269 | Value of a subword. (Cont... |
swrd00 13270 | A zero length substring. ... |
swrdcl 13271 | Closure of the subword ext... |
swrdval2 13272 | Value of the subword extra... |
swrd0val 13273 | Value of the subword extra... |
swrd0len 13274 | Length of a left-anchored ... |
swrdlen 13275 | Length of an extracted sub... |
swrdfv 13276 | A symbol in an extracted s... |
swrdf 13277 | A subword of a word is a f... |
swrdvalfn 13278 | Value of the subword extra... |
swrd0f 13279 | A left-anchored subword of... |
swrdid 13280 | A word is a subword of its... |
swrdrn 13281 | The range of a subword of ... |
swrdn0 13282 | A prefixing subword consis... |
swrdlend 13283 | The value of the subword e... |
swrdnd 13284 | The value of the subword e... |
swrdnd2 13285 | Value of the subword extra... |
swrd0 13286 | A subword of an empty set ... |
swrdrlen 13287 | Length of a right-anchored... |
swrd0len0 13288 | Length of a prefix of a wo... |
addlenrevswrd 13289 | The sum of the lengths of ... |
addlenswrd 13290 | The sum of the lengths of ... |
swrd0fv 13291 | A symbol in an left-anchor... |
swrd0fv0 13292 | The first symbol in a left... |
swrdtrcfv 13293 | A symbol in a word truncat... |
swrdtrcfv0 13294 | The first symbol in a word... |
swrd0fvlsw 13295 | The last symbol in a left-... |
swrdeq 13296 | Two subwords of words are ... |
swrdlen2 13297 | Length of an extracted sub... |
swrdfv2 13298 | A symbol in an extracted s... |
swrdsb0eq 13299 | Two subwords with the same... |
swrdsbslen 13300 | Two subwords with the same... |
swrdspsleq 13301 | Two words have a common su... |
swrdtrcfvl 13302 | The last symbol in a word ... |
swrds1 13303 | Extract a single symbol fr... |
swrdlsw 13304 | Extract the last single sy... |
2swrdeqwrdeq 13305 | Two words are equal if and... |
2swrd1eqwrdeq 13306 | Two (nonempty) words are e... |
disjxwrd 13307 | Sets of words are disjoint... |
ccatswrd 13308 | Joining two adjacent subwo... |
swrdccat1 13309 | Recover the left half of a... |
swrdccat2 13310 | Recover the right half of ... |
swrdswrdlem 13311 | Lemma for ~ swrdswrd . (C... |
swrdswrd 13312 | A subword of a subword. (... |
swrd0swrd 13313 | A prefix of a subword. (C... |
swrdswrd0 13314 | A subword of a prefix. (C... |
swrd0swrd0 13315 | A prefix of a prefix. (Co... |
swrd0swrdid 13316 | A prefix of a prefix with ... |
wrdcctswrd 13317 | The concatenation of two p... |
lencctswrd 13318 | The length of two concaten... |
lenrevcctswrd 13319 | The length of two reversel... |
swrdccatwrd 13320 | Reconstruct a nonempty wor... |
ccats1swrdeq 13321 | The last symbol of a word ... |
ccatopth 13322 | An ~ opth -like theorem fo... |
ccatopth2 13323 | An ~ opth -like theorem fo... |
ccatlcan 13324 | Concatenation of words is ... |
ccatrcan 13325 | Concatenation of words is ... |
wrdeqs1cat 13326 | Decompose a nonempty word ... |
cats1un 13327 | Express a word with an ext... |
wrdind 13328 | Perform induction over the... |
wrd2ind 13329 | Perform induction over the... |
ccats1swrdeqrex 13330 | There exists a symbol such... |
reuccats1lem 13331 | Lemma for ~ reuccats1 . (... |
reuccats1 13332 | A set of words having the ... |
swrdccatfn 13333 | The subword of a concatena... |
swrdccatin1 13334 | The subword of a concatena... |
swrdccatin12lem1 13335 | Lemma 1 for ~ swrdccatin12... |
swrdccatin12lem2a 13336 | Lemma 1 for ~ swrdccatin12... |
swrdccatin12lem2b 13337 | Lemma 2 for ~ swrdccatin12... |
swrdccatin2 13338 | The subword of a concatena... |
swrdccatin12lem2c 13339 | Lemma for ~ swrdccatin12le... |
swrdccatin12lem2 13340 | Lemma 2 for ~ swrdccatin12... |
swrdccatin12lem3 13341 | Lemma 3 for ~ swrdccatin12... |
swrdccatin12 13342 | The subword of a concatena... |
swrdccat3 13343 | The subword of a concatena... |
swrdccat 13344 | The subword of a concatena... |
swrdccat3a 13345 | A prefix of a concatenatio... |
swrdccat3blem 13346 | Lemma for ~ swrdccat3b . ... |
swrdccat3b 13347 | A suffix of a concatenatio... |
swrdccatid 13348 | A prefix of a concatenatio... |
ccats1swrdeqbi 13349 | A word is a prefix of a wo... |
swrdccatin1d 13350 | The subword of a concatena... |
swrdccatin2d 13351 | The subword of a concatena... |
swrdccatin12d 13352 | The subword of a concatena... |
splval 13353 | Value of the substring rep... |
splcl 13354 | Closure of the substring r... |
splid 13355 | Splicing a subword for the... |
spllen 13356 | The length of a splice. (... |
splfv1 13357 | Symbols to the left of a s... |
splfv2a 13358 | Symbols within the replace... |
splval2 13359 | Value of a splice, assumin... |
revval 13360 | Value of the word reversin... |
revcl 13361 | The reverse of a word is a... |
revlen 13362 | The reverse of a word has ... |
revfv 13363 | Reverse of a word at a poi... |
rev0 13364 | The empty word is its own ... |
revs1 13365 | Singleton words are their ... |
revccat 13366 | Antiautomorphic property o... |
revrev 13367 | Reversion is an involution... |
reps 13368 | Construct a function mappi... |
repsundef 13369 | A function mapping a half-... |
repsconst 13370 | Construct a function mappi... |
repsf 13371 | The constructed function m... |
repswsymb 13372 | The symbols of a "repeated... |
repsw 13373 | A function mapping a half-... |
repswlen 13374 | The length of a "repeated ... |
repsw0 13375 | The "repeated symbol word"... |
repsdf2 13376 | Alternative definition of ... |
repswsymball 13377 | All the symbols of a "repe... |
repswsymballbi 13378 | A word is a "repeated symb... |
repswfsts 13379 | The first symbol of a none... |
repswlsw 13380 | The last symbol of a nonem... |
repsw1 13381 | The "repeated symbol word"... |
repswswrd 13382 | A subword of a "repeated s... |
repswccat 13383 | The concatenation of two "... |
repswrevw 13384 | The reverse of a "repeated... |
cshfn 13387 | Perform a cyclical shift f... |
cshword 13388 | Perform a cyclical shift f... |
cshnz 13389 | A cyclical shift is the em... |
0csh0 13390 | Cyclically shifting an emp... |
cshw0 13391 | A word cyclically shifted ... |
cshwmodn 13392 | Cyclically shifting a word... |
cshwsublen 13393 | Cyclically shifting a word... |
cshwn 13394 | A word cyclically shifted ... |
cshwcl 13395 | A cyclically shifted word ... |
cshwlen 13396 | The length of a cyclically... |
cshwf 13397 | A cyclically shifted word ... |
cshwfn 13398 | A cyclically shifted word ... |
cshwrn 13399 | The range of a cyclically ... |
cshwidxmod 13400 | The symbol at a given inde... |
cshwidxmodr 13401 | The symbol at a given inde... |
cshwidx0mod 13402 | The symbol at index 0 of a... |
cshwidx0 13403 | The symbol at index 0 of a... |
cshwidxm1 13404 | The symbol at index ((n-N)... |
cshwidxm 13405 | The symbol at index (n-N) ... |
cshwidxn 13406 | The symbol at index (n-1) ... |
cshf1 13407 | Cyclically shifting a word... |
cshinj 13408 | If a word is injectiv (reg... |
repswcshw 13409 | A cyclically shifted "repe... |
2cshw 13410 | Cyclically shifting a word... |
2cshwid 13411 | Cyclically shifting a word... |
lswcshw 13412 | The last symbol of a word ... |
2cshwcom 13413 | Cyclically shifting a word... |
cshwleneq 13414 | If the results of cyclical... |
3cshw 13415 | Cyclically shifting a word... |
cshweqdif2 13416 | If cyclically shifting two... |
cshweqdifid 13417 | If cyclically shifting a w... |
cshweqrep 13418 | If cyclically shifting a w... |
cshw1 13419 | If cyclically shifting a w... |
cshw1repsw 13420 | If cyclically shifting a w... |
cshwsexa 13421 | The class of (different!) ... |
2cshwcshw 13422 | If a word is a cyclically ... |
scshwfzeqfzo 13423 | For a nonempty word the se... |
cshwcshid 13424 | A cyclically shifted word ... |
cshwcsh2id 13425 | A cyclically shifted word ... |
cshimadifsn 13426 | The image of a cyclically ... |
cshimadifsn0 13427 | The image of a cyclically ... |
wrdco 13428 | Mapping a word by a functi... |
lenco 13429 | Length of a mapped word is... |
s1co 13430 | Mapping of a singleton wor... |
revco 13431 | Mapping of words commutes ... |
ccatco 13432 | Mapping of words commutes ... |
cshco 13433 | Mapping of words commutes ... |
swrdco 13434 | Mapping of words commutes ... |
lswco 13435 | Mapping of (nonempty) word... |
repsco 13436 | Mapping of words commutes ... |
cats1cld 13451 | Closure of concatenation w... |
cats1co 13452 | Closure of concatenation w... |
cats1cli 13453 | Closure of concatenation w... |
cats1fvn 13454 | The last symbol of a conca... |
cats1fv 13455 | A symbol other than the la... |
cats1len 13456 | The length of concatenatio... |
cats1cat 13457 | Closure of concatenation w... |
cats2cat 13458 | Closure of concatenation o... |
s2eqd 13459 | Equality theorem for a dou... |
s3eqd 13460 | Equality theorem for a len... |
s4eqd 13461 | Equality theorem for a len... |
s5eqd 13462 | Equality theorem for a len... |
s6eqd 13463 | Equality theorem for a len... |
s7eqd 13464 | Equality theorem for a len... |
s8eqd 13465 | Equality theorem for a len... |
s2cld 13466 | A doubleton word is a word... |
s3cld 13467 | A length 3 string is a wor... |
s4cld 13468 | A length 4 string is a wor... |
s5cld 13469 | A length 5 string is a wor... |
s6cld 13470 | A length 6 string is a wor... |
s7cld 13471 | A length 7 string is a wor... |
s8cld 13472 | A length 7 string is a wor... |
s2cl 13473 | A doubleton word is a word... |
s3cl 13474 | A length 3 string is a wor... |
s2cli 13475 | A doubleton word is a word... |
s3cli 13476 | A length 3 string is a wor... |
s4cli 13477 | A length 4 string is a wor... |
s5cli 13478 | A length 5 string is a wor... |
s6cli 13479 | A length 6 string is a wor... |
s7cli 13480 | A length 7 string is a wor... |
s8cli 13481 | A length 8 string is a wor... |
s2fv0 13482 | Extract the first symbol f... |
s2fv1 13483 | Extract the second symbol ... |
s2len 13484 | The length of a doubleton ... |
s2dm 13485 | The domain of a doubleton ... |
s3fv0 13486 | Extract the first symbol f... |
s3fv1 13487 | Extract the second symbol ... |
s3fv2 13488 | Extract the third symbol f... |
s3len 13489 | The length of a length 3 s... |
s4fv0 13490 | Extract the first symbol f... |
s4fv1 13491 | Extract the second symbol ... |
s4fv2 13492 | Extract the third symbol f... |
s4fv3 13493 | Extract the fourth symbol ... |
s4len 13494 | The length of a length 4 s... |
s5len 13495 | The length of a length 5 s... |
s6len 13496 | The length of a length 6 s... |
s7len 13497 | The length of a length 7 s... |
s8len 13498 | The length of a length 8 s... |
lsws2 13499 | The last symbol of a doubl... |
lsws3 13500 | The last symbol of a 3 let... |
lsws4 13501 | The last symbol of a 4 let... |
s2prop 13502 | A length 2 word is an unor... |
s2dmALT 13503 | Alternate version of ~ s2d... |
s3tpop 13504 | A length 3 word is an unor... |
s4prop 13505 | A length 4 word is a union... |
s3fn 13506 | A length 3 word is a funct... |
funcnvs1 13507 | The converse of a singleto... |
funcnvs2 13508 | The converse of a length 2... |
funcnvs3 13509 | The converse of a length 3... |
funcnvs4 13510 | The converse of a length 4... |
s2f1o 13511 | A length 2 word with mutua... |
f1oun2prg 13512 | A union of unordered pairs... |
s4f1o 13513 | A length 4 word with mutua... |
s4dom 13514 | The domain of a length 4 w... |
s2co 13515 | Mapping a doubleton word b... |
s3co 13516 | Mapping a length 3 string ... |
s0s1 13517 | Concatenation of fixed len... |
s1s2 13518 | Concatenation of fixed len... |
s1s3 13519 | Concatenation of fixed len... |
s1s4 13520 | Concatenation of fixed len... |
s1s5 13521 | Concatenation of fixed len... |
s1s6 13522 | Concatenation of fixed len... |
s1s7 13523 | Concatenation of fixed len... |
s2s2 13524 | Concatenation of fixed len... |
s4s2 13525 | Concatenation of fixed len... |
s4s3 13526 | Concatenation of fixed len... |
s4s4 13527 | Concatenation of fixed len... |
s3s4 13528 | Concatenation of fixed len... |
s2s5 13529 | Concatenation of fixed len... |
s5s2 13530 | Concatenation of fixed len... |
s2eq2s1eq 13531 | Two length 2 words are equ... |
s2eq2seq 13532 | Two length 2 words are equ... |
swrds2 13533 | Extract two adjacent symbo... |
wrdlen2i 13534 | Implications of a word of ... |
wrd2pr2op 13535 | A word of length 2 represe... |
wrdlen2 13536 | A word of length 2. (Cont... |
wrdlen2s2 13537 | A word of length 2 as doub... |
wrdl2exs2 13538 | A word of length 2 is a do... |
wrd3tpop 13539 | A word of length 3 represe... |
wrdlen3s3 13540 | A word of length 3 as leng... |
repsw2 13541 | The "repeated symbol word"... |
repsw3 13542 | The "repeated symbol word"... |
swrd2lsw 13543 | Extract the last two symbo... |
2swrd2eqwrdeq 13544 | Two words of length at lea... |
ccatw2s1ccatws2 13545 | The concatenation of a wor... |
ccat2s1fvwALT 13546 | Alternate proof of ~ ccat2... |
wwlktovf 13547 | Lemma 1 for ~ wrd2f1tovbij... |
wwlktovf1 13548 | Lemma 2 for ~ wrd2f1tovbij... |
wwlktovfo 13549 | Lemma 3 for ~ wrd2f1tovbij... |
wwlktovf1o 13550 | Lemma 4 for ~ wrd2f1tovbij... |
wrd2f1tovbij 13551 | There is a bijection betwe... |
eqwrds3 13552 | A word is equal with a len... |
wrdl3s3 13553 | A word of length 3 is a le... |
s3sndisj 13554 | The singletons consisting ... |
s3iunsndisj 13555 | The union of singletons co... |
ofccat 13556 | Letterwise operations on w... |
ofs1 13557 | Letterwise operations on a... |
ofs2 13558 | Letterwise operations on a... |
coss12d 13559 | Subset deduction for compo... |
trrelssd 13560 | The composition of subclas... |
xpcogend 13561 | The most interesting case ... |
xpcoidgend 13562 | If two classes are not dis... |
cotr2g 13563 | Two ways of saying that th... |
cotr2 13564 | Two ways of saying a relat... |
cotr3 13565 | Two ways of saying a relat... |
coemptyd 13566 | Deduction about compositio... |
xptrrel 13567 | The cross product is alway... |
0trrel 13568 | The empty class is a trans... |
cleq1lem 13569 | Equality implies bijection... |
cleq1 13570 | Equality of relations impl... |
clsslem 13571 | The closure of a subclass ... |
trcleq1 13576 | Equality of relations impl... |
trclsslem 13577 | The transitive closure (as... |
trcleq2lem 13578 | Equality implies bijection... |
cvbtrcl 13579 | Change of bound variable i... |
trcleq12lem 13580 | Equality implies bijection... |
trclexlem 13581 | Existence of relation impl... |
trclublem 13582 | If a relation exists then ... |
trclubi 13583 | The Cartesian product of t... |
trclubiOLD 13584 | Obsolete version of ~ trcl... |
trclubgi 13585 | The union with the Cartesi... |
trclubgiOLD 13586 | Obsolete version of ~ trcl... |
trclub 13587 | The Cartesian product of t... |
trclubg 13588 | The union with the Cartesi... |
trclfv 13589 | The transitive closure of ... |
brintclab 13590 | Two ways to express a bina... |
brtrclfv 13591 | Two ways of expressing the... |
brcnvtrclfv 13592 | Two ways of expressing the... |
brtrclfvcnv 13593 | Two ways of expressing the... |
brcnvtrclfvcnv 13594 | Two ways of expressing the... |
trclfvss 13595 | The transitive closure (as... |
trclfvub 13596 | The transitive closure of ... |
trclfvlb 13597 | The transitive closure of ... |
trclfvcotr 13598 | The transitive closure of ... |
trclfvlb2 13599 | The transitive closure of ... |
trclfvlb3 13600 | The transitive closure of ... |
cotrtrclfv 13601 | The transitive closure of ... |
trclidm 13602 | The transitive closure of ... |
trclun 13603 | Transitive closure of a un... |
trclfvg 13604 | The value of the transitiv... |
trclfvcotrg 13605 | The value of the transitiv... |
reltrclfv 13606 | The transitive closure of ... |
dmtrclfv 13607 | The domain of the transiti... |
relexp0g 13610 | A relation composed zero t... |
relexp0 13611 | A relation composed zero t... |
relexp0d 13612 | A relation composed zero t... |
relexpsucnnr 13613 | A reduction for relation e... |
relexp1g 13614 | A relation composed once i... |
dfid5 13615 | Identity relation is equal... |
dfid6 13616 | Identity relation expresse... |
relexpsucr 13617 | A reduction for relation e... |
relexpsucrd 13618 | A reduction for relation e... |
relexp1d 13619 | A relation composed once i... |
relexpsucnnl 13620 | A reduction for relation e... |
relexpsucl 13621 | A reduction for relation e... |
relexpsucld 13622 | A reduction for relation e... |
relexpcnv 13623 | Commutation of converse an... |
relexpcnvd 13624 | Commutation of converse an... |
relexp0rel 13625 | The exponentiation of a cl... |
relexprelg 13626 | The exponentiation of a cl... |
relexprel 13627 | The exponentiation of a re... |
relexpreld 13628 | The exponentiation of a re... |
relexpnndm 13629 | The domain of an exponenti... |
relexpdmg 13630 | The domain of an exponenti... |
relexpdm 13631 | The domain of an exponenti... |
relexpdmd 13632 | The domain of an exponenti... |
relexpnnrn 13633 | The range of an exponentia... |
relexprng 13634 | The range of an exponentia... |
relexprn 13635 | The range of an exponentia... |
relexprnd 13636 | The range of an exponentia... |
relexpfld 13637 | The field of an exponentia... |
relexpfldd 13638 | The field of an exponentia... |
relexpaddnn 13639 | Relation composition becom... |
relexpuzrel 13640 | The exponentiation of a cl... |
relexpaddg 13641 | Relation composition becom... |
relexpaddd 13642 | Relation composition becom... |
dfrtrclrec2 13645 | If two elements are connec... |
rtrclreclem1 13646 | The reflexive, transitive ... |
rtrclreclem2 13647 | The reflexive, transitive ... |
rtrclreclem3 13648 | The reflexive, transitive ... |
rtrclreclem4 13649 | The reflexive, transitive ... |
dfrtrcl2 13650 | The two definitions ` t* `... |
relexpindlem 13651 | Principle of transitive in... |
relexpind 13652 | Principle of transitive in... |
rtrclind 13653 | Principle of transitive in... |
shftlem 13656 | Two ways to write a shifte... |
shftuz 13657 | A shift of the upper integ... |
shftfval 13658 | The value of the sequence ... |
shftdm 13659 | Domain of a relation shift... |
shftfib 13660 | Value of a fiber of the re... |
shftfn 13661 | Functionality and domain o... |
shftval 13662 | Value of a sequence shifte... |
shftval2 13663 | Value of a sequence shifte... |
shftval3 13664 | Value of a sequence shifte... |
shftval4 13665 | Value of a sequence shifte... |
shftval5 13666 | Value of a shifted sequenc... |
shftf 13667 | Functionality of a shifted... |
2shfti 13668 | Composite shift operations... |
shftidt2 13669 | Identity law for the shift... |
shftidt 13670 | Identity law for the shift... |
shftcan1 13671 | Cancellation law for the s... |
shftcan2 13672 | Cancellation law for the s... |
seqshft 13673 | Shifting the index set of ... |
sgnval 13676 | Value of Signum function. ... |
sgn0 13677 | Proof that signum of 0 is ... |
sgnp 13678 | Proof that signum of posit... |
sgnrrp 13679 | Proof that signum of posit... |
sgn1 13680 | Proof that the signum of 1... |
sgnpnf 13681 | Proof that the signum of `... |
sgnn 13682 | Proof that signum of negat... |
sgnmnf 13683 | Proof that the signum of `... |
cjval 13690 | The value of the conjugate... |
cjth 13691 | The defining property of t... |
cjf 13692 | Domain and codomain of the... |
cjcl 13693 | The conjugate of a complex... |
reval 13694 | The value of the real part... |
imval 13695 | The value of the imaginary... |
imre 13696 | The imaginary part of a co... |
reim 13697 | The real part of a complex... |
recl 13698 | The real part of a complex... |
imcl 13699 | The imaginary part of a co... |
ref 13700 | Domain and codomain of the... |
imf 13701 | Domain and codomain of the... |
crre 13702 | The real part of a complex... |
crim 13703 | The real part of a complex... |
replim 13704 | Reconstruct a complex numb... |
remim 13705 | Value of the conjugate of ... |
reim0 13706 | The imaginary part of a re... |
reim0b 13707 | A number is real iff its i... |
rereb 13708 | A number is real iff it eq... |
mulre 13709 | A product with a nonzero r... |
rere 13710 | A real number equals its r... |
cjreb 13711 | A number is real iff it eq... |
recj 13712 | Real part of a complex con... |
reneg 13713 | Real part of negative. (C... |
readd 13714 | Real part distributes over... |
resub 13715 | Real part distributes over... |
remullem 13716 | Lemma for ~ remul , ~ immu... |
remul 13717 | Real part of a product. (... |
remul2 13718 | Real part of a product. (... |
rediv 13719 | Real part of a division. ... |
imcj 13720 | Imaginary part of a comple... |
imneg 13721 | The imaginary part of a ne... |
imadd 13722 | Imaginary part distributes... |
imsub 13723 | Imaginary part distributes... |
immul 13724 | Imaginary part of a produc... |
immul2 13725 | Imaginary part of a produc... |
imdiv 13726 | Imaginary part of a divisi... |
cjre 13727 | A real number equals its c... |
cjcj 13728 | The conjugate of the conju... |
cjadd 13729 | Complex conjugate distribu... |
cjmul 13730 | Complex conjugate distribu... |
ipcnval 13731 | Standard inner product on ... |
cjmulrcl 13732 | A complex number times its... |
cjmulval 13733 | A complex number times its... |
cjmulge0 13734 | A complex number times its... |
cjneg 13735 | Complex conjugate of negat... |
addcj 13736 | A number plus its conjugat... |
cjsub 13737 | Complex conjugate distribu... |
cjexp 13738 | Complex conjugate of posit... |
imval2 13739 | The imaginary part of a nu... |
re0 13740 | The real part of zero. (C... |
im0 13741 | The imaginary part of zero... |
re1 13742 | The real part of one. (Co... |
im1 13743 | The imaginary part of one.... |
rei 13744 | The real part of ` _i ` . ... |
imi 13745 | The imaginary part of ` _i... |
cj0 13746 | The conjugate of zero. (C... |
cji 13747 | The complex conjugate of t... |
cjreim 13748 | The conjugate of a represe... |
cjreim2 13749 | The conjugate of the repre... |
cj11 13750 | Complex conjugate is a one... |
cjne0 13751 | A number is nonzero iff it... |
cjdiv 13752 | Complex conjugate distribu... |
cnrecnv 13753 | The inverse to the canonic... |
sqeqd 13754 | A deduction for showing tw... |
recli 13755 | The real part of a complex... |
imcli 13756 | The imaginary part of a co... |
cjcli 13757 | Closure law for complex co... |
replimi 13758 | Construct a complex number... |
cjcji 13759 | The conjugate of the conju... |
reim0bi 13760 | A number is real iff its i... |
rerebi 13761 | A real number equals its r... |
cjrebi 13762 | A number is real iff it eq... |
recji 13763 | Real part of a complex con... |
imcji 13764 | Imaginary part of a comple... |
cjmulrcli 13765 | A complex number times its... |
cjmulvali 13766 | A complex number times its... |
cjmulge0i 13767 | A complex number times its... |
renegi 13768 | Real part of negative. (C... |
imnegi 13769 | Imaginary part of negative... |
cjnegi 13770 | Complex conjugate of negat... |
addcji 13771 | A number plus its conjugat... |
readdi 13772 | Real part distributes over... |
imaddi 13773 | Imaginary part distributes... |
remuli 13774 | Real part of a product. (... |
immuli 13775 | Imaginary part of a produc... |
cjaddi 13776 | Complex conjugate distribu... |
cjmuli 13777 | Complex conjugate distribu... |
ipcni 13778 | Standard inner product on ... |
cjdivi 13779 | Complex conjugate distribu... |
crrei 13780 | The real part of a complex... |
crimi 13781 | The imaginary part of a co... |
recld 13782 | The real part of a complex... |
imcld 13783 | The imaginary part of a co... |
cjcld 13784 | Closure law for complex co... |
replimd 13785 | Construct a complex number... |
remimd 13786 | Value of the conjugate of ... |
cjcjd 13787 | The conjugate of the conju... |
reim0bd 13788 | A number is real iff its i... |
rerebd 13789 | A real number equals its r... |
cjrebd 13790 | A number is real iff it eq... |
cjne0d 13791 | A number is nonzero iff it... |
recjd 13792 | Real part of a complex con... |
imcjd 13793 | Imaginary part of a comple... |
cjmulrcld 13794 | A complex number times its... |
cjmulvald 13795 | A complex number times its... |
cjmulge0d 13796 | A complex number times its... |
renegd 13797 | Real part of negative. (C... |
imnegd 13798 | Imaginary part of negative... |
cjnegd 13799 | Complex conjugate of negat... |
addcjd 13800 | A number plus its conjugat... |
cjexpd 13801 | Complex conjugate of posit... |
readdd 13802 | Real part distributes over... |
imaddd 13803 | Imaginary part distributes... |
resubd 13804 | Real part distributes over... |
imsubd 13805 | Imaginary part distributes... |
remuld 13806 | Real part of a product. (... |
immuld 13807 | Imaginary part of a produc... |
cjaddd 13808 | Complex conjugate distribu... |
cjmuld 13809 | Complex conjugate distribu... |
ipcnd 13810 | Standard inner product on ... |
cjdivd 13811 | Complex conjugate distribu... |
rered 13812 | A real number equals its r... |
reim0d 13813 | The imaginary part of a re... |
cjred 13814 | A real number equals its c... |
remul2d 13815 | Real part of a product. (... |
immul2d 13816 | Imaginary part of a produc... |
redivd 13817 | Real part of a division. ... |
imdivd 13818 | Imaginary part of a divisi... |
crred 13819 | The real part of a complex... |
crimd 13820 | The imaginary part of a co... |
sqrtval 13825 | Value of square root funct... |
absval 13826 | The absolute value (modulu... |
rennim 13827 | A real number does not lie... |
cnpart 13828 | The specification of restr... |
sqr0lem 13829 | Square root of zero. (Con... |
sqrt0 13830 | Square root of zero. (Con... |
sqrlem1 13831 | Lemma for ~ 01sqrex . (Co... |
sqrlem2 13832 | Lemma for ~ 01sqrex . (Co... |
sqrlem3 13833 | Lemma for ~ 01sqrex . (Co... |
sqrlem4 13834 | Lemma for ~ 01sqrex . (Co... |
sqrlem5 13835 | Lemma for ~ 01sqrex . (Co... |
sqrlem6 13836 | Lemma for ~ 01sqrex . (Co... |
sqrlem7 13837 | Lemma for ~ 01sqrex . (Co... |
01sqrex 13838 | Existence of a square root... |
resqrex 13839 | Existence of a square root... |
sqrmo 13840 | Uniqueness for the square ... |
resqreu 13841 | Existence and uniqueness f... |
resqrtcl 13842 | Closure of the square root... |
resqrtthlem 13843 | Lemma for ~ resqrtth . (C... |
resqrtth 13844 | Square root theorem over t... |
remsqsqrt 13845 | Square of square root. (C... |
sqrtge0 13846 | The square root function i... |
sqrtgt0 13847 | The square root function i... |
sqrtmul 13848 | Square root distributes ov... |
sqrtle 13849 | Square root is monotonic. ... |
sqrtlt 13850 | Square root is strictly mo... |
sqrt11 13851 | The square root function i... |
sqrt00 13852 | A square root is zero iff ... |
rpsqrtcl 13853 | The square root of a posit... |
sqrtdiv 13854 | Square root distributes ov... |
sqrtneglem 13855 | The square root of a negat... |
sqrtneg 13856 | The square root of a negat... |
sqrtsq2 13857 | Relationship between squar... |
sqrtsq 13858 | Square root of square. (C... |
sqrtmsq 13859 | Square root of square. (C... |
sqrt1 13860 | The square root of 1 is 1.... |
sqrt4 13861 | The square root of 4 is 2.... |
sqrt9 13862 | The square root of 9 is 3.... |
sqrt2gt1lt2 13863 | The square root of 2 is bo... |
sqrtm1 13864 | The imaginary unit is the ... |
absneg 13865 | Absolute value of negative... |
abscl 13866 | Real closure of absolute v... |
abscj 13867 | The absolute value of a nu... |
absvalsq 13868 | Square of value of absolut... |
absvalsq2 13869 | Square of value of absolut... |
sqabsadd 13870 | Square of absolute value o... |
sqabssub 13871 | Square of absolute value o... |
absval2 13872 | Value of absolute value fu... |
abs0 13873 | The absolute value of 0. ... |
absi 13874 | The absolute value of the ... |
absge0 13875 | Absolute value is nonnegat... |
absrpcl 13876 | The absolute value of a no... |
abs00 13877 | The absolute value of a nu... |
abs00ad 13878 | A complex number is zero i... |
abs00bd 13879 | If a complex number is zer... |
absreimsq 13880 | Square of the absolute val... |
absreim 13881 | Absolute value of a number... |
absmul 13882 | Absolute value distributes... |
absdiv 13883 | Absolute value distributes... |
absid 13884 | A nonnegative number is it... |
abs1 13885 | The absolute value of 1. ... |
absnid 13886 | A negative number is the n... |
leabs 13887 | A real number is less than... |
absor 13888 | The absolute value of a re... |
absre 13889 | Absolute value of a real n... |
absresq 13890 | Square of the absolute val... |
absmod0 13891 | ` A ` is divisible by ` B ... |
absexp 13892 | Absolute value of positive... |
absexpz 13893 | Absolute value of integer ... |
abssq 13894 | Square can be moved in and... |
sqabs 13895 | The squares of two reals a... |
absrele 13896 | The absolute value of a co... |
absimle 13897 | The absolute value of a co... |
max0add 13898 | The sum of the positive an... |
absz 13899 | A real number is an intege... |
nn0abscl 13900 | The absolute value of an i... |
zabscl 13901 | The absolute value of an i... |
abslt 13902 | Absolute value and 'less t... |
absle 13903 | Absolute value and 'less t... |
abssubne0 13904 | If the absolute value of a... |
absdiflt 13905 | The absolute value of a di... |
absdifle 13906 | The absolute value of a di... |
elicc4abs 13907 | Membership in a symmetric ... |
lenegsq 13908 | Comparison to a nonnegativ... |
releabs 13909 | The real part of a number ... |
recval 13910 | Reciprocal expressed with ... |
absidm 13911 | The absolute value functio... |
absgt0 13912 | The absolute value of a no... |
nnabscl 13913 | The absolute value of a no... |
abssub 13914 | Swapping order of subtract... |
abssubge0 13915 | Absolute value of a nonneg... |
abssuble0 13916 | Absolute value of a nonpos... |
absmax 13917 | The maximum of two numbers... |
abstri 13918 | Triangle inequality for ab... |
abs3dif 13919 | Absolute value of differen... |
abs2dif 13920 | Difference of absolute val... |
abs2dif2 13921 | Difference of absolute val... |
abs2difabs 13922 | Absolute value of differen... |
abs1m 13923 | For any complex number, th... |
recan 13924 | Cancellation law involving... |
absf 13925 | Mapping domain and codomai... |
abs3lem 13926 | Lemma involving absolute v... |
abslem2 13927 | Lemma involving absolute v... |
rddif 13928 | The difference between a r... |
absrdbnd 13929 | Bound on the absolute valu... |
fzomaxdiflem 13930 | Lemma for ~ fzomaxdif . (... |
fzomaxdif 13931 | A bound on the separation ... |
uzin2 13932 | The upper integers are clo... |
rexanuz 13933 | Combine two different uppe... |
rexanre 13934 | Combine two different uppe... |
rexfiuz 13935 | Combine finitely many diff... |
rexuz3 13936 | Restrict the base of the u... |
rexanuz2 13937 | Combine two different uppe... |
r19.29uz 13938 | A version of ~ 19.29 for u... |
r19.2uz 13939 | A version of ~ r19.2z for ... |
rexuzre 13940 | Convert an upper real quan... |
rexico 13941 | Restrict the base of an up... |
cau3lem 13942 | Lemma for ~ cau3 . (Contr... |
cau3 13943 | Convert between three-quan... |
cau4 13944 | Change the base of a Cauch... |
caubnd2 13945 | A Cauchy sequence of compl... |
caubnd 13946 | A Cauchy sequence of compl... |
sqreulem 13947 | Lemma for ~ sqreu : write ... |
sqreu 13948 | Existence and uniqueness f... |
sqrtcl 13949 | Closure of the square root... |
sqrtthlem 13950 | Lemma for ~ sqrtth . (Con... |
sqrtf 13951 | Mapping domain and codomai... |
sqrtth 13952 | Square root theorem over t... |
sqrtrege0 13953 | The square root function m... |
eqsqrtor 13954 | Solve an equation containi... |
eqsqrtd 13955 | A deduction for showing th... |
eqsqrt2d 13956 | A deduction for showing th... |
amgm2 13957 | Arithmetic-geometric mean ... |
sqrtthi 13958 | Square root theorem. Theo... |
sqrtcli 13959 | The square root of a nonne... |
sqrtgt0i 13960 | The square root of a posit... |
sqrtmsqi 13961 | Square root of square. (C... |
sqrtsqi 13962 | Square root of square. (C... |
sqsqrti 13963 | Square of square root. (C... |
sqrtge0i 13964 | The square root of a nonne... |
absidi 13965 | A nonnegative number is it... |
absnidi 13966 | A negative number is the n... |
leabsi 13967 | A real number is less than... |
absori 13968 | The absolute value of a re... |
absrei 13969 | Absolute value of a real n... |
sqrtpclii 13970 | The square root of a posit... |
sqrtgt0ii 13971 | The square root of a posit... |
sqrt11i 13972 | The square root function i... |
sqrtmuli 13973 | Square root distributes ov... |
sqrtmulii 13974 | Square root distributes ov... |
sqrtmsq2i 13975 | Relationship between squar... |
sqrtlei 13976 | Square root is monotonic. ... |
sqrtlti 13977 | Square root is strictly mo... |
abslti 13978 | Absolute value and 'less t... |
abslei 13979 | Absolute value and 'less t... |
absvalsqi 13980 | Square of value of absolut... |
absvalsq2i 13981 | Square of value of absolut... |
abscli 13982 | Real closure of absolute v... |
absge0i 13983 | Absolute value is nonnegat... |
absval2i 13984 | Value of absolute value fu... |
abs00i 13985 | The absolute value of a nu... |
absgt0i 13986 | The absolute value of a no... |
absnegi 13987 | Absolute value of negative... |
abscji 13988 | The absolute value of a nu... |
releabsi 13989 | The real part of a number ... |
abssubi 13990 | Swapping order of subtract... |
absmuli 13991 | Absolute value distributes... |
sqabsaddi 13992 | Square of absolute value o... |
sqabssubi 13993 | Square of absolute value o... |
absdivzi 13994 | Absolute value distributes... |
abstrii 13995 | Triangle inequality for ab... |
abs3difi 13996 | Absolute value of differen... |
abs3lemi 13997 | Lemma involving absolute v... |
rpsqrtcld 13998 | The square root of a posit... |
sqrtgt0d 13999 | The square root of a posit... |
absnidd 14000 | A negative number is the n... |
leabsd 14001 | A real number is less than... |
absord 14002 | The absolute value of a re... |
absred 14003 | Absolute value of a real n... |
resqrtcld 14004 | The square root of a nonne... |
sqrtmsqd 14005 | Square root of square. (C... |
sqrtsqd 14006 | Square root of square. (C... |
sqrtge0d 14007 | The square root of a nonne... |
sqrtnegd 14008 | The square root of a negat... |
absidd 14009 | A nonnegative number is it... |
sqrtdivd 14010 | Square root distributes ov... |
sqrtmuld 14011 | Square root distributes ov... |
sqrtsq2d 14012 | Relationship between squar... |
sqrtled 14013 | Square root is monotonic. ... |
sqrtltd 14014 | Square root is strictly mo... |
sqr11d 14015 | The square root function i... |
absltd 14016 | Absolute value and 'less t... |
absled 14017 | Absolute value and 'less t... |
abssubge0d 14018 | Absolute value of a nonneg... |
abssuble0d 14019 | Absolute value of a nonpos... |
absdifltd 14020 | The absolute value of a di... |
absdifled 14021 | The absolute value of a di... |
icodiamlt 14022 | Two elements in a half-ope... |
abscld 14023 | Real closure of absolute v... |
sqrtcld 14024 | Closure of the square root... |
sqrtrege0d 14025 | The real part of the squar... |
sqsqrtd 14026 | Square root theorem. Theo... |
msqsqrtd 14027 | Square root theorem. Theo... |
sqr00d 14028 | A square root is zero iff ... |
absvalsqd 14029 | Square of value of absolut... |
absvalsq2d 14030 | Square of value of absolut... |
absge0d 14031 | Absolute value is nonnegat... |
absval2d 14032 | Value of absolute value fu... |
abs00d 14033 | The absolute value of a nu... |
absne0d 14034 | The absolute value of a nu... |
absrpcld 14035 | The absolute value of a no... |
absnegd 14036 | Absolute value of negative... |
abscjd 14037 | The absolute value of a nu... |
releabsd 14038 | The real part of a number ... |
absexpd 14039 | Absolute value of positive... |
abssubd 14040 | Swapping order of subtract... |
absmuld 14041 | Absolute value distributes... |
absdivd 14042 | Absolute value distributes... |
abstrid 14043 | Triangle inequality for ab... |
abs2difd 14044 | Difference of absolute val... |
abs2dif2d 14045 | Difference of absolute val... |
abs2difabsd 14046 | Absolute value of differen... |
abs3difd 14047 | Absolute value of differen... |
abs3lemd 14048 | Lemma involving absolute v... |
limsupgord 14051 | Ordering property of the s... |
limsupcl 14052 | Closure of the superior li... |
limsupval 14053 | The superior limit of an i... |
limsupgf 14054 | Closure of the superior li... |
limsupgval 14055 | Value of the superior limi... |
limsupgle 14056 | The defining property of t... |
limsuple 14057 | The defining property of t... |
limsuplt 14058 | The defining property of t... |
limsupval2 14059 | The superior limit, relati... |
limsupgre 14060 | If a sequence of real numb... |
limsupbnd1 14061 | If a sequence is eventuall... |
limsupbnd2 14062 | If a sequence is eventuall... |
climrel 14071 | The limit relation is a re... |
rlimrel 14072 | The limit relation is a re... |
clim 14073 | Express the predicate: Th... |
rlim 14074 | Express the predicate: Th... |
rlim2 14075 | Rewrite ~ rlim for a mappi... |
rlim2lt 14076 | Use strictly less-than in ... |
rlim3 14077 | Restrict the range of the ... |
climcl 14078 | Closure of the limit of a ... |
rlimpm 14079 | Closure of a function with... |
rlimf 14080 | Closure of a function with... |
rlimss 14081 | Domain closure of a functi... |
rlimcl 14082 | Closure of the limit of a ... |
clim2 14083 | Express the predicate: Th... |
clim2c 14084 | Express the predicate ` F ... |
clim0 14085 | Express the predicate ` F ... |
clim0c 14086 | Express the predicate ` F ... |
rlim0 14087 | Express the predicate ` B ... |
rlim0lt 14088 | Use strictly less-than in ... |
climi 14089 | Convergence of a sequence ... |
climi2 14090 | Convergence of a sequence ... |
climi0 14091 | Convergence of a sequence ... |
rlimi 14092 | Convergence at infinity of... |
rlimi2 14093 | Convergence at infinity of... |
ello1 14094 | Elementhood in the set of ... |
ello12 14095 | Elementhood in the set of ... |
ello12r 14096 | Sufficient condition for e... |
lo1f 14097 | An eventually upper bounde... |
lo1dm 14098 | An eventually upper bounde... |
lo1bdd 14099 | The defining property of a... |
ello1mpt 14100 | Elementhood in the set of ... |
ello1mpt2 14101 | Elementhood in the set of ... |
ello1d 14102 | Sufficient condition for e... |
lo1bdd2 14103 | If an eventually bounded f... |
lo1bddrp 14104 | Refine ~ o1bdd2 to give a ... |
elo1 14105 | Elementhood in the set of ... |
elo12 14106 | Elementhood in the set of ... |
elo12r 14107 | Sufficient condition for e... |
o1f 14108 | An eventually bounded func... |
o1dm 14109 | An eventually bounded func... |
o1bdd 14110 | The defining property of a... |
lo1o1 14111 | A function is eventually b... |
lo1o12 14112 | A function is eventually b... |
elo1mpt 14113 | Elementhood in the set of ... |
elo1mpt2 14114 | Elementhood in the set of ... |
elo1d 14115 | Sufficient condition for e... |
o1lo1 14116 | A real function is eventua... |
o1lo12 14117 | A lower bounded real funct... |
o1lo1d 14118 | A real eventually bounded ... |
icco1 14119 | Derive eventual boundednes... |
o1bdd2 14120 | If an eventually bounded f... |
o1bddrp 14121 | Refine ~ o1bdd2 to give a ... |
climconst 14122 | An (eventually) constant s... |
rlimconst 14123 | A constant sequence conver... |
rlimclim1 14124 | Forward direction of ~ rli... |
rlimclim 14125 | A sequence on an upper int... |
climrlim2 14126 | Produce a real limit from ... |
climconst2 14127 | A constant sequence conver... |
climz 14128 | The zero sequence converge... |
rlimuni 14129 | A real function whose doma... |
rlimdm 14130 | Two ways to express that a... |
climuni 14131 | An infinite sequence of co... |
fclim 14132 | The limit relation is func... |
climdm 14133 | Two ways to express that a... |
climeu 14134 | An infinite sequence of co... |
climreu 14135 | An infinite sequence of co... |
climmo 14136 | An infinite sequence of co... |
rlimres 14137 | The restriction of a funct... |
lo1res 14138 | The restriction of an even... |
o1res 14139 | The restriction of an even... |
rlimres2 14140 | The restriction of a funct... |
lo1res2 14141 | The restriction of a funct... |
o1res2 14142 | The restriction of a funct... |
lo1resb 14143 | The restriction of a funct... |
rlimresb 14144 | The restriction of a funct... |
o1resb 14145 | The restriction of a funct... |
climeq 14146 | Two functions that are eve... |
lo1eq 14147 | Two functions that are eve... |
rlimeq 14148 | Two functions that are eve... |
o1eq 14149 | Two functions that are eve... |
climmpt 14150 | Exhibit a function ` G ` w... |
2clim 14151 | If two sequences converge ... |
climmpt2 14152 | Relate an integer limit on... |
climshftlem 14153 | A shifted function converg... |
climres 14154 | A function restricted to u... |
climshft 14155 | A shifted function converg... |
serclim0 14156 | The zero series converges ... |
rlimcld2 14157 | If ` D ` is a closed set i... |
rlimrege0 14158 | The limit of a sequence of... |
rlimrecl 14159 | The limit of a real sequen... |
rlimge0 14160 | The limit of a sequence of... |
climshft2 14161 | A shifted function converg... |
climrecl 14162 | The limit of a convergent ... |
climge0 14163 | A nonnegative sequence con... |
climabs0 14164 | Convergence to zero of the... |
o1co 14165 | Sufficient condition for t... |
o1compt 14166 | Sufficient condition for t... |
rlimcn1 14167 | Image of a limit under a c... |
rlimcn1b 14168 | Image of a limit under a c... |
rlimcn2 14169 | Image of a limit under a c... |
climcn1 14170 | Image of a limit under a c... |
climcn2 14171 | Image of a limit under a c... |
addcn2 14172 | Complex number addition is... |
subcn2 14173 | Complex number subtraction... |
mulcn2 14174 | Complex number multiplicat... |
reccn2 14175 | The reciprocal function is... |
cn1lem 14176 | A sufficient condition for... |
abscn2 14177 | The absolute value functio... |
cjcn2 14178 | The complex conjugate func... |
recn2 14179 | The real part function is ... |
imcn2 14180 | The imaginary part functio... |
climcn1lem 14181 | The limit of a continuous ... |
climabs 14182 | Limit of the absolute valu... |
climcj 14183 | Limit of the complex conju... |
climre 14184 | Limit of the real part of ... |
climim 14185 | Limit of the imaginary par... |
rlimmptrcl 14186 | Reverse closure for a real... |
rlimabs 14187 | Limit of the absolute valu... |
rlimcj 14188 | Limit of the complex conju... |
rlimre 14189 | Limit of the real part of ... |
rlimim 14190 | Limit of the imaginary par... |
o1of2 14191 | Show that a binary operati... |
o1add 14192 | The sum of two eventually ... |
o1mul 14193 | The product of two eventua... |
o1sub 14194 | The difference of two even... |
rlimo1 14195 | Any function with a finite... |
rlimdmo1 14196 | A convergent function is e... |
o1rlimmul 14197 | The product of an eventual... |
o1const 14198 | A constant function is eve... |
lo1const 14199 | A constant function is eve... |
lo1mptrcl 14200 | Reverse closure for an eve... |
o1mptrcl 14201 | Reverse closure for an eve... |
o1add2 14202 | The sum of two eventually ... |
o1mul2 14203 | The product of two eventua... |
o1sub2 14204 | The product of two eventua... |
lo1add 14205 | The sum of two eventually ... |
lo1mul 14206 | The product of an eventual... |
lo1mul2 14207 | The product of an eventual... |
o1dif 14208 | If the difference of two f... |
lo1sub 14209 | The difference of an event... |
climadd 14210 | Limit of the sum of two co... |
climmul 14211 | Limit of the product of tw... |
climsub 14212 | Limit of the difference of... |
climaddc1 14213 | Limit of a constant ` C ` ... |
climaddc2 14214 | Limit of a constant ` C ` ... |
climmulc2 14215 | Limit of a sequence multip... |
climsubc1 14216 | Limit of a constant ` C ` ... |
climsubc2 14217 | Limit of a constant ` C ` ... |
climle 14218 | Comparison of the limits o... |
climsqz 14219 | Convergence of a sequence ... |
climsqz2 14220 | Convergence of a sequence ... |
rlimadd 14221 | Limit of the sum of two co... |
rlimsub 14222 | Limit of the difference of... |
rlimmul 14223 | Limit of the product of tw... |
rlimdiv 14224 | Limit of the quotient of t... |
rlimneg 14225 | Limit of the negative of a... |
rlimle 14226 | Comparison of the limits o... |
rlimsqzlem 14227 | Lemma for ~ rlimsqz and ~ ... |
rlimsqz 14228 | Convergence of a sequence ... |
rlimsqz2 14229 | Convergence of a sequence ... |
lo1le 14230 | Transfer eventual upper bo... |
o1le 14231 | Transfer eventual boundedn... |
rlimno1 14232 | A function whose inverse c... |
clim2ser 14233 | The limit of an infinite s... |
clim2ser2 14234 | The limit of an infinite s... |
iserex 14235 | An infinite series converg... |
isermulc2 14236 | Multiplication of an infin... |
climlec2 14237 | Comparison of a constant t... |
iserle 14238 | Comparison of the limits o... |
iserge0 14239 | The limit of an infinite s... |
climub 14240 | The limit of a monotonic s... |
climserle 14241 | The partial sums of a conv... |
isershft 14242 | Index shift of the limit o... |
isercolllem1 14243 | Lemma for ~ isercoll . (C... |
isercolllem2 14244 | Lemma for ~ isercoll . (C... |
isercolllem3 14245 | Lemma for ~ isercoll . (C... |
isercoll 14246 | Rearrange an infinite seri... |
isercoll2 14247 | Generalize ~ isercoll so t... |
climsup 14248 | A bounded monotonic sequen... |
climcau 14249 | A converging sequence of c... |
climbdd 14250 | A converging sequence of c... |
caucvgrlem 14251 | Lemma for ~ caurcvgr . (C... |
caurcvgr 14252 | A Cauchy sequence of real ... |
caucvgrlem2 14253 | Lemma for ~ caucvgr . (Co... |
caucvgr 14254 | A Cauchy sequence of compl... |
caurcvg 14255 | A Cauchy sequence of real ... |
caurcvg2 14256 | A Cauchy sequence of real ... |
caucvg 14257 | A Cauchy sequence of compl... |
caucvgb 14258 | A function is convergent i... |
serf0 14259 | If an infinite series conv... |
iseraltlem1 14260 | Lemma for ~ iseralt . A d... |
iseraltlem2 14261 | Lemma for ~ iseralt . The... |
iseraltlem3 14262 | Lemma for ~ iseralt . Fro... |
iseralt 14263 | The alternating series tes... |
sumex 14266 | A sum is a set. (Contribu... |
sumeq1 14267 | Equality theorem for a sum... |
nfsum1 14268 | Bound-variable hypothesis ... |
nfsum 14269 | Bound-variable hypothesis ... |
sumeq2w 14270 | Equality theorem for sum, ... |
sumeq2ii 14271 | Equality theorem for sum, ... |
sumeq2 14272 | Equality theorem for sum. ... |
cbvsum 14273 | Change bound variable in a... |
cbvsumv 14274 | Change bound variable in a... |
cbvsumi 14275 | Change bound variable in a... |
sumeq1i 14276 | Equality inference for sum... |
sumeq2i 14277 | Equality inference for sum... |
sumeq12i 14278 | Equality inference for sum... |
sumeq1d 14279 | Equality deduction for sum... |
sumeq2d 14280 | Equality deduction for sum... |
sumeq2dv 14281 | Equality deduction for sum... |
sumeq2sdv 14282 | Equality deduction for sum... |
2sumeq2dv 14283 | Equality deduction for dou... |
sumeq12dv 14284 | Equality deduction for sum... |
sumeq12rdv 14285 | Equality deduction for sum... |
sum2id 14286 | The second class argument ... |
sumfc 14287 | A lemma to facilitate conv... |
fz1f1o 14288 | A lemma for working with f... |
sumrblem 14289 | Lemma for ~ sumrb . (Cont... |
fsumcvg 14290 | The sequence of partial su... |
sumrb 14291 | Rebase the starting point ... |
summolem3 14292 | Lemma for ~ summo . (Cont... |
summolem2a 14293 | Lemma for ~ summo . (Cont... |
summolem2 14294 | Lemma for ~ summo . (Cont... |
summo 14295 | A sum has at most one limi... |
zsum 14296 | Series sum with index set ... |
isum 14297 | Series sum with an upper i... |
fsum 14298 | The value of a sum over a ... |
sum0 14299 | Any sum over the empty set... |
sumz 14300 | Any sum of zero over a sum... |
fsumf1o 14301 | Re-index a finite sum usin... |
sumss 14302 | Change the index set to a ... |
fsumss 14303 | Change the index set to a ... |
sumss2 14304 | Change the index set of a ... |
fsumcvg2 14305 | The sequence of partial su... |
fsumsers 14306 | Special case of series sum... |
fsumcvg3 14307 | A finite sum is convergent... |
fsumser 14308 | A finite sum expressed in ... |
fsumcl2lem 14309 | - Lemma for finite sum clo... |
fsumcllem 14310 | - Lemma for finite sum clo... |
fsumcl 14311 | Closure of a finite sum of... |
fsumrecl 14312 | Closure of a finite sum of... |
fsumzcl 14313 | Closure of a finite sum of... |
fsumnn0cl 14314 | Closure of a finite sum of... |
fsumrpcl 14315 | Closure of a finite sum of... |
fsumzcl2 14316 | A finite sum with integer ... |
fsumadd 14317 | The sum of two finite sums... |
fsumsplit 14318 | Split a sum into two parts... |
sumsn 14319 | A sum of a singleton is th... |
fsum1 14320 | The finite sum of ` A ( k ... |
sumpr 14321 | A sum over a pair is the s... |
sumtp 14322 | A sum over a triple is the... |
sumsns 14323 | A sum of a singleton is th... |
fsumm1 14324 | Separate out the last term... |
fzosump1 14325 | Separate out the last term... |
fsum1p 14326 | Separate out the first ter... |
fsummsnunz 14327 | A finite sum with an addit... |
fsumsplitsnun 14328 | Separate out a term in a f... |
fsump1 14329 | The addition of the next t... |
isumclim 14330 | An infinite sum equals the... |
isumclim2 14331 | A converging series conver... |
isumclim3 14332 | The sequence of partial fi... |
sumnul 14333 | The sum of a non-convergen... |
isumcl 14334 | The sum of a converging in... |
isummulc2 14335 | An infinite sum multiplied... |
isummulc1 14336 | An infinite sum multiplied... |
isumdivc 14337 | An infinite sum divided by... |
isumrecl 14338 | The sum of a converging in... |
isumge0 14339 | An infinite sum of nonnega... |
isumadd 14340 | Addition of infinite sums.... |
sumsplit 14341 | Split a sum into two parts... |
fsump1i 14342 | Optimized version of ~ fsu... |
fsum2dlem 14343 | Lemma for ~ fsum2d - induc... |
fsum2d 14344 | Write a double sum as a su... |
fsumxp 14345 | Combine two sums into a si... |
fsumcnv 14346 | Transform a region of summ... |
fsumcom2 14347 | Interchange order of summa... |
fsumcom2OLD 14348 | Obsolete proof of ~ fsumco... |
fsumcom 14349 | Interchange order of summa... |
fsum0diaglem 14350 | Lemma for ~ fsum0diag . (... |
fsum0diag 14351 | Two ways to express "the s... |
mptfzshft 14352 | 1-1 onto function in maps-... |
fsumrev 14353 | Reversal of a finite sum. ... |
fsumshft 14354 | Index shift of a finite su... |
fsumshftm 14355 | Negative index shift of a ... |
fsumrev2 14356 | Reversal of a finite sum. ... |
fsum0diag2 14357 | Two ways to express "the s... |
fsummulc2 14358 | A finite sum multiplied by... |
fsummulc1 14359 | A finite sum multiplied by... |
fsumdivc 14360 | A finite sum divided by a ... |
fsumneg 14361 | Negation of a finite sum. ... |
fsumsub 14362 | Split a finite sum over a ... |
fsum2mul 14363 | Separate the nested sum of... |
fsumconst 14364 | The sum of constant terms ... |
modfsummodslem1 14365 | Lemma 1 for ~ modfsummods ... |
modfsummods 14366 | Induction step for ~ modfs... |
modfsummod 14367 | A finite sum modulo a posi... |
fsumge0 14368 | If all of the terms of a f... |
fsumless 14369 | A shorter sum of nonnegati... |
fsumge1 14370 | A sum of nonnegative numbe... |
fsum00 14371 | A sum of nonnegative numbe... |
fsumle 14372 | If all of the terms of fin... |
fsumlt 14373 | If every term in one finit... |
fsumabs 14374 | Generalized triangle inequ... |
telfsumo 14375 | Sum of a telescoping serie... |
telfsumo2 14376 | Sum of a telescoping serie... |
telfsum 14377 | Sum of a telescoping serie... |
telfsum2 14378 | Sum of a telescoping serie... |
fsumparts 14379 | Summation by parts. (Cont... |
fsumrelem 14380 | Lemma for ~ fsumre , ~ fsu... |
fsumre 14381 | The real part of a sum. (... |
fsumim 14382 | The imaginary part of a su... |
fsumcj 14383 | The complex conjugate of a... |
fsumrlim 14384 | Limit of a finite sum of c... |
fsumo1 14385 | The finite sum of eventual... |
o1fsum 14386 | If ` A ( k ) ` is O(1), th... |
seqabs 14387 | Generalized triangle inequ... |
iserabs 14388 | Generalized triangle inequ... |
cvgcmp 14389 | A comparison test for conv... |
cvgcmpub 14390 | An upper bound for the lim... |
cvgcmpce 14391 | A comparison test for conv... |
abscvgcvg 14392 | An absolutely convergent s... |
climfsum 14393 | Limit of a finite sum of c... |
fsumiun 14394 | Sum over a disjoint indexe... |
hashiun 14395 | The cardinality of a disjo... |
hashrabrex 14396 | The number of elements in ... |
hashuni 14397 | The cardinality of a disjo... |
qshash 14398 | The cardinality of a set w... |
ackbijnn 14399 | Translate the Ackermann bi... |
binomlem 14400 | Lemma for ~ binom (binomia... |
binom 14401 | The binomial theorem: ` ( ... |
binom1p 14402 | Special case of the binomi... |
binom11 14403 | Special case of the binomi... |
binom1dif 14404 | A summation for the differ... |
bcxmaslem1 14405 | Lemma for ~ bcxmas . (Con... |
bcxmas 14406 | Parallel summation (Christ... |
incexclem 14407 | Lemma for ~ incexc . (Con... |
incexc 14408 | The inclusion/exclusion pr... |
incexc2 14409 | The inclusion/exclusion pr... |
isumshft 14410 | Index shift of an infinite... |
isumsplit 14411 | Split off the first ` N ` ... |
isum1p 14412 | The infinite sum of a conv... |
isumnn0nn 14413 | Sum from 0 to infinity in ... |
isumrpcl 14414 | The infinite sum of positi... |
isumle 14415 | Comparison of two infinite... |
isumless 14416 | A finite sum of nonnegativ... |
isumsup2 14417 | An infinite sum of nonnega... |
isumsup 14418 | An infinite sum of nonnega... |
isumltss 14419 | A partial sum of a series ... |
climcndslem1 14420 | Lemma for ~ climcnds : bou... |
climcndslem2 14421 | Lemma for ~ climcnds : bou... |
climcnds 14422 | The Cauchy condensation te... |
divrcnv 14423 | The sequence of reciprocal... |
divcnv 14424 | The sequence of reciprocal... |
flo1 14425 | The floor function satisfi... |
divcnvshft 14426 | Limit of a ratio function.... |
supcvg 14427 | Extract a sequence ` f ` i... |
infcvgaux1i 14428 | Auxiliary theorem for appl... |
infcvgaux2i 14429 | Auxiliary theorem for appl... |
harmonic 14430 | The harmonic series ` H ` ... |
arisum 14431 | Arithmetic series sum of t... |
arisum2 14432 | Arithmetic series sum of t... |
trireciplem 14433 | Lemma for ~ trirecip . Sh... |
trirecip 14434 | The sum of the reciprocals... |
expcnv 14435 | A sequence of powers of a ... |
explecnv 14436 | A sequence of terms conver... |
geoserg 14437 | The value of the finite ge... |
geoser 14438 | The value of the finite ge... |
pwm1geoser 14439 | The n-th power of a number... |
geolim 14440 | The partial sums in the in... |
geolim2 14441 | The partial sums in the ge... |
georeclim 14442 | The limit of a geometric s... |
geo2sum 14443 | The value of the finite ge... |
geo2sum2 14444 | The value of the finite ge... |
geo2lim 14445 | The value of the infinite ... |
geomulcvg 14446 | The geometric series conve... |
geoisum 14447 | The infinite sum of ` 1 + ... |
geoisumr 14448 | The infinite sum of recipr... |
geoisum1 14449 | The infinite sum of ` A ^ ... |
geoisum1c 14450 | The infinite sum of ` A x.... |
0.999... 14451 | The recurring decimal 0.99... |
0.999...OLD 14452 | Obsolete version of ~ 0.99... |
geoihalfsum 14453 | Prove that the infinite ge... |
cvgrat 14454 | Ratio test for convergence... |
mertenslem1 14455 | Lemma for ~ mertens . (Co... |
mertenslem2 14456 | Lemma for ~ mertens . (Co... |
mertens 14457 | Mertens' theorem. If ` A ... |
prodf 14458 | An infinite product of com... |
clim2prod 14459 | The limit of an infinite p... |
clim2div 14460 | The limit of an infinite p... |
prodfmul 14461 | The product of two infinit... |
prodf1 14462 | The value of the partial p... |
prodf1f 14463 | A one-valued infinite prod... |
prodfclim1 14464 | The constant one product c... |
prodfn0 14465 | No term of a nonzero infin... |
prodfrec 14466 | The reciprocal of an infin... |
prodfdiv 14467 | The quotient of two infini... |
ntrivcvg 14468 | A non-trivially converging... |
ntrivcvgn0 14469 | A product that converges t... |
ntrivcvgfvn0 14470 | Any value of a product seq... |
ntrivcvgtail 14471 | A tail of a non-trivially ... |
ntrivcvgmullem 14472 | Lemma for ~ ntrivcvgmul . ... |
ntrivcvgmul 14473 | The product of two non-tri... |
prodex 14476 | A product is a set. (Cont... |
prodeq1f 14477 | Equality theorem for a pro... |
prodeq1 14478 | Equality theorem for a pro... |
nfcprod1 14479 | Bound-variable hypothesis ... |
nfcprod 14480 | Bound-variable hypothesis ... |
prodeq2w 14481 | Equality theorem for produ... |
prodeq2ii 14482 | Equality theorem for produ... |
prodeq2 14483 | Equality theorem for produ... |
cbvprod 14484 | Change bound variable in a... |
cbvprodv 14485 | Change bound variable in a... |
cbvprodi 14486 | Change bound variable in a... |
prodeq1i 14487 | Equality inference for pro... |
prodeq2i 14488 | Equality inference for pro... |
prodeq12i 14489 | Equality inference for pro... |
prodeq1d 14490 | Equality deduction for pro... |
prodeq2d 14491 | Equality deduction for pro... |
prodeq2dv 14492 | Equality deduction for pro... |
prodeq2sdv 14493 | Equality deduction for pro... |
2cprodeq2dv 14494 | Equality deduction for dou... |
prodeq12dv 14495 | Equality deduction for pro... |
prodeq12rdv 14496 | Equality deduction for pro... |
prod2id 14497 | The second class argument ... |
prodrblem 14498 | Lemma for ~ prodrb . (Con... |
fprodcvg 14499 | The sequence of partial pr... |
prodrblem2 14500 | Lemma for ~ prodrb . (Con... |
prodrb 14501 | Rebase the starting point ... |
prodmolem3 14502 | Lemma for ~ prodmo . (Con... |
prodmolem2a 14503 | Lemma for ~ prodmo . (Con... |
prodmolem2 14504 | Lemma for ~ prodmo . (Con... |
prodmo 14505 | A product has at most one ... |
zprod 14506 | Series product with index ... |
iprod 14507 | Series product with an upp... |
zprodn0 14508 | Nonzero series product wit... |
iprodn0 14509 | Nonzero series product wit... |
fprod 14510 | The value of a product ove... |
fprodntriv 14511 | A non-triviality lemma for... |
prod0 14512 | A product over the empty s... |
prod1 14513 | Any product of one over a ... |
prodfc 14514 | A lemma to facilitate conv... |
fprodf1o 14515 | Re-index a finite product ... |
prodss 14516 | Change the index set to a ... |
fprodss 14517 | Change the index set to a ... |
fprodser 14518 | A finite product expressed... |
fprodcl2lem 14519 | Finite product closure lem... |
fprodcllem 14520 | Finite product closure lem... |
fprodcl 14521 | Closure of a finite produc... |
fprodrecl 14522 | Closure of a finite produc... |
fprodzcl 14523 | Closure of a finite produc... |
fprodnncl 14524 | Closure of a finite produc... |
fprodrpcl 14525 | Closure of a finite produc... |
fprodnn0cl 14526 | Closure of a finite produc... |
fprodcllemf 14527 | Finite product closure lem... |
fprodreclf 14528 | Closure of a finite produc... |
fprodmul 14529 | The product of two finite ... |
fproddiv 14530 | The quotient of two finite... |
prodsn 14531 | A product of a singleton i... |
fprod1 14532 | A finite product of only o... |
prodsnf 14533 | A product of a singleton i... |
climprod1 14534 | The limit of a product ove... |
fprodsplit 14535 | Split a finite product int... |
fprodm1 14536 | Separate out the last term... |
fprod1p 14537 | Separate out the first ter... |
fprodp1 14538 | Multiply in the last term ... |
fprodm1s 14539 | Separate out the last term... |
fprodp1s 14540 | Multiply in the last term ... |
prodsns 14541 | A product of the singleton... |
fprodfac 14542 | Factorial using product no... |
fprodabs 14543 | The absolute value of a fi... |
fprodeq0 14544 | Anything finite product co... |
fprodshft 14545 | Shift the index of a finit... |
fprodrev 14546 | Reversal of a finite produ... |
fprodconst 14547 | The product of constant te... |
fprodn0 14548 | A finite product of nonzer... |
fprod2dlem 14549 | Lemma for ~ fprod2d - indu... |
fprod2d 14550 | Write a double product as ... |
fprodxp 14551 | Combine two products into ... |
fprodcnv 14552 | Transform a product region... |
fprodcom2 14553 | Interchange order of multi... |
fprodcom2OLD 14554 | Obsolete proof of ~ fprodc... |
fprodcom 14555 | Interchange product order.... |
fprod0diag 14556 | Two ways to express "the p... |
fproddivf 14557 | The quotient of two finite... |
fprodsplitf 14558 | Split a finite product int... |
fprodsplitsn 14559 | Separate out a term in a f... |
fprodsplit1f 14560 | Separate out a term in a f... |
fprodn0f 14561 | A finite product of nonzer... |
fprodclf 14562 | Closure of a finite produc... |
fprodge0 14563 | If all the terms of a fini... |
fprodeq0g 14564 | Any finite product contain... |
fprodge1 14565 | If all of the terms of a f... |
fprodle 14566 | If all the terms of two fi... |
fprodmodd 14567 | If all factors of two fini... |
iprodclim 14568 | An infinite product equals... |
iprodclim2 14569 | A converging product conve... |
iprodclim3 14570 | The sequence of partial fi... |
iprodcl 14571 | The product of a non-trivi... |
iprodrecl 14572 | The product of a non-trivi... |
iprodmul 14573 | Multiplication of infinite... |
risefacval 14578 | The value of the rising fa... |
fallfacval 14579 | The value of the falling f... |
risefacval2 14580 | One-based value of rising ... |
fallfacval2 14581 | One-based value of falling... |
fallfacval3 14582 | A product representation o... |
risefaccllem 14583 | Lemma for rising factorial... |
fallfaccllem 14584 | Lemma for falling factoria... |
risefaccl 14585 | Closure law for rising fac... |
fallfaccl 14586 | Closure law for falling fa... |
rerisefaccl 14587 | Closure law for rising fac... |
refallfaccl 14588 | Closure law for falling fa... |
nnrisefaccl 14589 | Closure law for rising fac... |
zrisefaccl 14590 | Closure law for rising fac... |
zfallfaccl 14591 | Closure law for falling fa... |
nn0risefaccl 14592 | Closure law for rising fac... |
rprisefaccl 14593 | Closure law for rising fac... |
risefallfac 14594 | A relationship between ris... |
fallrisefac 14595 | A relationship between fal... |
risefall0lem 14596 | Lemma for ~ risefac0 and ~... |
risefac0 14597 | The value of the rising fa... |
fallfac0 14598 | The value of the falling f... |
risefacp1 14599 | The value of the rising fa... |
fallfacp1 14600 | The value of the falling f... |
risefacp1d 14601 | The value of the rising fa... |
fallfacp1d 14602 | The value of the falling f... |
risefac1 14603 | The value of rising factor... |
fallfac1 14604 | The value of falling facto... |
risefacfac 14605 | Relate rising factorial to... |
fallfacfwd 14606 | The forward difference of ... |
0fallfac 14607 | The value of the zero fall... |
0risefac 14608 | The value of the zero risi... |
binomfallfaclem1 14609 | Lemma for ~ binomfallfac .... |
binomfallfaclem2 14610 | Lemma for ~ binomfallfac .... |
binomfallfac 14611 | A version of the binomial ... |
binomrisefac 14612 | A version of the binomial ... |
fallfacval4 14613 | Represent the falling fact... |
bcfallfac 14614 | Binomial coefficient in te... |
fallfacfac 14615 | Relate falling factorial t... |
bpolylem 14618 | Lemma for ~ bpolyval . (C... |
bpolyval 14619 | The value of the Bernoulli... |
bpoly0 14620 | The value of the Bernoulli... |
bpoly1 14621 | The value of the Bernoulli... |
bpolycl 14622 | Closure law for Bernoulli ... |
bpolysum 14623 | A sum for Bernoulli polyno... |
bpolydiflem 14624 | Lemma for ~ bpolydif . (C... |
bpolydif 14625 | Calculate the difference b... |
fsumkthpow 14626 | A closed-form expression f... |
bpoly2 14627 | The Bernoulli polynomials ... |
bpoly3 14628 | The Bernoulli polynomials ... |
bpoly4 14629 | The Bernoulli polynomials ... |
fsumcube 14630 | Express the sum of cubes i... |
eftcl 14643 | Closure of a term in the s... |
reeftcl 14644 | The terms of the series ex... |
eftabs 14645 | The absolute value of a te... |
eftval 14646 | The value of a term in the... |
efcllem 14647 | Lemma for ~ efcl . The se... |
ef0lem 14648 | The series defining the ex... |
efval 14649 | Value of the exponential f... |
esum 14650 | Value of Euler's constant ... |
eff 14651 | Domain and codomain of the... |
efcl 14652 | Closure law for the expone... |
efval2 14653 | Value of the exponential f... |
efcvg 14654 | The series that defines th... |
efcvgfsum 14655 | Exponential function conve... |
reefcl 14656 | The exponential function i... |
reefcld 14657 | The exponential function i... |
ere 14658 | Euler's constant ` _e ` = ... |
ege2le3 14659 | Lemma for ~ egt2lt3 . (Co... |
ef0 14660 | Value of the exponential f... |
efcj 14661 | Exponential function of a ... |
efaddlem 14662 | Lemma for ~ efadd (exponen... |
efadd 14663 | Sum of exponents law for e... |
fprodefsum 14664 | Move the exponential funct... |
efcan 14665 | Cancellation of law for ex... |
efne0 14666 | The exponential function n... |
efneg 14667 | Exponent of a negative num... |
eff2 14668 | The exponential function m... |
efsub 14669 | Difference of exponents la... |
efexp 14670 | Exponential function to an... |
efzval 14671 | Value of the exponential f... |
efgt0 14672 | The exponential function o... |
rpefcl 14673 | The exponential function o... |
rpefcld 14674 | The exponential function o... |
eftlcvg 14675 | The tail series of the exp... |
eftlcl 14676 | Closure of the sum of an i... |
reeftlcl 14677 | Closure of the sum of an i... |
eftlub 14678 | An upper bound on the abso... |
efsep 14679 | Separate out the next term... |
effsumlt 14680 | The partial sums of the se... |
eft0val 14681 | The value of the first ter... |
ef4p 14682 | Separate out the first fou... |
efgt1p2 14683 | The exponential function o... |
efgt1p 14684 | The exponential function o... |
efgt1 14685 | The exponential function o... |
eflt 14686 | The exponential function o... |
efle 14687 | The exponential function o... |
reef11 14688 | The exponential function o... |
reeff1 14689 | The exponential function m... |
eflegeo 14690 | The exponential function o... |
sinval 14691 | Value of the sine function... |
cosval 14692 | Value of the cosine functi... |
sinf 14693 | Domain and codomain of the... |
cosf 14694 | Domain and codomain of the... |
sincl 14695 | Closure of the sine functi... |
coscl 14696 | Closure of the cosine func... |
tanval 14697 | Value of the tangent funct... |
tancl 14698 | The closure of the tangent... |
sincld 14699 | Closure of the sine functi... |
coscld 14700 | Closure of the cosine func... |
tancld 14701 | Closure of the tangent fun... |
tanval2 14702 | Express the tangent functi... |
tanval3 14703 | Express the tangent functi... |
resinval 14704 | The sine of a real number ... |
recosval 14705 | The cosine of a real numbe... |
efi4p 14706 | Separate out the first fou... |
resin4p 14707 | Separate out the first fou... |
recos4p 14708 | Separate out the first fou... |
resincl 14709 | The sine of a real number ... |
recoscl 14710 | The cosine of a real numbe... |
retancl 14711 | The closure of the tangent... |
resincld 14712 | Closure of the sine functi... |
recoscld 14713 | Closure of the cosine func... |
retancld 14714 | Closure of the tangent fun... |
sinneg 14715 | The sine of a negative is ... |
cosneg 14716 | The cosines of a number an... |
tanneg 14717 | The tangent of a negative ... |
sin0 14718 | Value of the sine function... |
cos0 14719 | Value of the cosine functi... |
tan0 14720 | The value of the tangent f... |
efival 14721 | The exponential function i... |
efmival 14722 | The exponential function i... |
sinhval 14723 | Value of the hyperbolic si... |
coshval 14724 | Value of the hyperbolic co... |
resinhcl 14725 | The hyperbolic sine of a r... |
rpcoshcl 14726 | The hyperbolic cosine of a... |
recoshcl 14727 | The hyperbolic cosine of a... |
retanhcl 14728 | The hyperbolic tangent of ... |
tanhlt1 14729 | The hyperbolic tangent of ... |
tanhbnd 14730 | The hyperbolic tangent of ... |
efeul 14731 | Eulerian representation of... |
efieq 14732 | The exponentials of two im... |
sinadd 14733 | Addition formula for sine.... |
cosadd 14734 | Addition formula for cosin... |
tanaddlem 14735 | A useful intermediate step... |
tanadd 14736 | Addition formula for tange... |
sinsub 14737 | Sine of difference. (Cont... |
cossub 14738 | Cosine of difference. (Co... |
addsin 14739 | Sum of sines. (Contribute... |
subsin 14740 | Difference of sines. (Con... |
sinmul 14741 | Product of sines can be re... |
cosmul 14742 | Product of cosines can be ... |
addcos 14743 | Sum of cosines. (Contribu... |
subcos 14744 | Difference of cosines. (C... |
sincossq 14745 | Sine squared plus cosine s... |
sin2t 14746 | Double-angle formula for s... |
cos2t 14747 | Double-angle formula for c... |
cos2tsin 14748 | Double-angle formula for c... |
sinbnd 14749 | The sine of a real number ... |
cosbnd 14750 | The cosine of a real numbe... |
sinbnd2 14751 | The sine of a real number ... |
cosbnd2 14752 | The cosine of a real numbe... |
ef01bndlem 14753 | Lemma for ~ sin01bnd and ~... |
sin01bnd 14754 | Bounds on the sine of a po... |
cos01bnd 14755 | Bounds on the cosine of a ... |
cos1bnd 14756 | Bounds on the cosine of 1.... |
cos2bnd 14757 | Bounds on the cosine of 2.... |
sinltx 14758 | The sine of a positive rea... |
sin01gt0 14759 | The sine of a positive rea... |
cos01gt0 14760 | The cosine of a positive r... |
sin02gt0 14761 | The sine of a positive rea... |
sincos1sgn 14762 | The signs of the sine and ... |
sincos2sgn 14763 | The signs of the sine and ... |
sin4lt0 14764 | The sine of 4 is negative.... |
absefi 14765 | The absolute value of the ... |
absef 14766 | The absolute value of the ... |
absefib 14767 | A number is real iff its i... |
efieq1re 14768 | A number whose imaginary e... |
demoivre 14769 | De Moivre's Formula. Proo... |
demoivreALT 14770 | Alternate proof of ~ demoi... |
eirrlem 14771 | Lemma for ~ eirr . (Contr... |
eirr 14772 | ` _e ` is irrational. (Co... |
egt2lt3 14773 | Euler's constant ` _e ` = ... |
epos 14774 | Euler's constant ` _e ` is... |
epr 14775 | Euler's constant ` _e ` is... |
ene0 14776 | ` _e ` is not 0. (Contrib... |
ene1 14777 | ` _e ` is not 1. (Contrib... |
xpnnen 14778 | The Cartesian product of t... |
znnenlem 14779 | Lemma for ~ znnen . (Cont... |
znnen 14780 | The set of integers and th... |
qnnen 14781 | The rational numbers are c... |
rpnnen2lem1 14782 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem2 14783 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem3 14784 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem4 14785 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem5 14786 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem6 14787 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem7 14788 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem8 14789 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem9 14790 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem10 14791 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem11 14792 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2lem12 14793 | Lemma for ~ rpnnen2 . (Co... |
rpnnen2 14794 | The other half of ~ rpnnen... |
rpnnen 14795 | The cardinality of the con... |
rexpen 14796 | The real numbers are equin... |
cpnnen 14797 | The complex numbers are eq... |
rucALT 14798 | Alternate proof of ~ ruc .... |
ruclem1 14799 | Lemma for ~ ruc (the reals... |
ruclem2 14800 | Lemma for ~ ruc . Orderin... |
ruclem3 14801 | Lemma for ~ ruc . The con... |
ruclem4 14802 | Lemma for ~ ruc . Initial... |
ruclem6 14803 | Lemma for ~ ruc . Domain ... |
ruclem7 14804 | Lemma for ~ ruc . Success... |
ruclem8 14805 | Lemma for ~ ruc . The int... |
ruclem9 14806 | Lemma for ~ ruc . The fir... |
ruclem10 14807 | Lemma for ~ ruc . Every f... |
ruclem11 14808 | Lemma for ~ ruc . Closure... |
ruclem12 14809 | Lemma for ~ ruc . The sup... |
ruclem13 14810 | Lemma for ~ ruc . There i... |
ruc 14811 | The set of positive intege... |
resdomq 14812 | The set of rationals is st... |
aleph1re 14813 | There are at least aleph-o... |
aleph1irr 14814 | There are at least aleph-o... |
cnso 14815 | The complex numbers can be... |
sqr2irrlem 14816 | Lemma for irrationality of... |
sqrt2irr 14817 | The square root of 2 is ir... |
sqrt2re 14818 | The square root of 2 exist... |
nthruc 14819 | The sequence ` NN ` , ` ZZ... |
nthruz 14820 | The sequence ` NN ` , ` NN... |
divides 14823 | Define the divides relatio... |
dvdsval2 14824 | One nonzero integer divide... |
dvdsval3 14825 | One nonzero integer divide... |
dvdszrcl 14826 | Reverse closure for the di... |
nndivdvds 14827 | Strong form of ~ dvdsval2 ... |
nndivides 14828 | Definition of the divides ... |
moddvds 14829 | Two ways to say ` A == B `... |
dvds0lem 14830 | A lemma to assist theorems... |
dvds1lem 14831 | A lemma to assist theorems... |
dvds2lem 14832 | A lemma to assist theorems... |
iddvds 14833 | An integer divides itself.... |
1dvds 14834 | 1 divides any integer. Th... |
dvds0 14835 | Any integer divides 0. Th... |
negdvdsb 14836 | An integer divides another... |
dvdsnegb 14837 | An integer divides another... |
absdvdsb 14838 | An integer divides another... |
dvdsabsb 14839 | An integer divides another... |
0dvds 14840 | Only 0 is divisible by 0. ... |
dvdsmul1 14841 | An integer divides a multi... |
dvdsmul2 14842 | An integer divides a multi... |
iddvdsexp 14843 | An integer divides a posit... |
muldvds1 14844 | If a product divides an in... |
muldvds2 14845 | If a product divides an in... |
dvdscmul 14846 | Multiplication by a consta... |
dvdsmulc 14847 | Multiplication by a consta... |
dvdscmulr 14848 | Cancellation law for the d... |
dvdsmulcr 14849 | Cancellation law for the d... |
summodnegmod 14850 | The sum of two integers mo... |
modmulconst 14851 | Constant multiplication in... |
dvds2ln 14852 | If an integer divides each... |
dvds2add 14853 | If an integer divides each... |
dvds2sub 14854 | If an integer divides each... |
dvds2subd 14855 | Natural deduction form of ... |
dvdstr 14856 | The divides relation is tr... |
dvdsmultr1 14857 | If an integer divides anot... |
dvdsmultr1d 14858 | Natural deduction form of ... |
dvdsmultr2 14859 | If an integer divides anot... |
ordvdsmul 14860 | If an integer divides eith... |
dvdssub2 14861 | If an integer divides a di... |
dvdsadd 14862 | An integer divides another... |
dvdsaddr 14863 | An integer divides another... |
dvdssub 14864 | An integer divides another... |
dvdssubr 14865 | An integer divides another... |
dvdsadd2b 14866 | Adding a multiple of the b... |
dvdsaddre2b 14867 | Adding a multiple of the b... |
fsumdvds 14868 | If every term in a sum is ... |
dvdslelem 14869 | Lemma for ~ dvdsle . (Con... |
dvdsle 14870 | The divisors of a positive... |
dvdsleabs 14871 | The divisors of a nonzero ... |
dvdsleabs2 14872 | Transfer divisibility to a... |
dvdsabseq 14873 | If two integers divide eac... |
dvdseq 14874 | If two nonnegative integer... |
divconjdvds 14875 | If a nonzero integer ` M `... |
dvdsdivcl 14876 | The complement of a diviso... |
dvdsflip 14877 | An involution of the divis... |
dvdsssfz1 14878 | The set of divisors of a n... |
dvds1 14879 | The only nonnegative integ... |
alzdvds 14880 | Only 0 is divisible by all... |
dvdsext 14881 | Poset extensionality for d... |
fzm1ndvds 14882 | No number between ` 1 ` an... |
fzo0dvdseq 14883 | Zero is the only one of th... |
fzocongeq 14884 | Two different elements of ... |
addmodlteqALT 14885 | Two nonnegative integers l... |
dvdsfac 14886 | A positive integer divides... |
dvdsexp 14887 | A power divides a power wi... |
dvdsmod 14888 | Any number ` K ` whose mod... |
mulmoddvds 14889 | If an integer is divisible... |
3dvds 14890 | A rule for divisibility by... |
3dvdsOLD 14891 | Obsolete version of ~ 3dvd... |
3dvdsdec 14892 | A decimal number is divisi... |
3dvdsdecOLD 14893 | Obsolete proof of ~ 3dvdsd... |
3dvds2dec 14894 | A decimal number is divisi... |
3dvds2decOLD 14895 | Old version of ~ 3dvds2dec... |
fprodfvdvdsd 14896 | A finite product of intege... |
fproddvdsd 14897 | A finite product of intege... |
evenelz 14898 | An even number is an integ... |
zeo3 14899 | An integer is even or odd.... |
zeo4 14900 | An integer is even or odd ... |
zeneo 14901 | No even integer equals an ... |
odd2np1lem 14902 | Lemma for ~ odd2np1 . (Co... |
odd2np1 14903 | An integer is odd iff it i... |
even2n 14904 | An integer is even iff it ... |
oddm1even 14905 | An integer is odd iff its ... |
oddp1even 14906 | An integer is odd iff its ... |
oexpneg 14907 | The exponential of the neg... |
mod2eq0even 14908 | An integer is 0 modulo 2 i... |
mod2eq1n2dvds 14909 | An integer is 1 modulo 2 i... |
oddnn02np1 14910 | A nonnegative integer is o... |
oddge22np1 14911 | An integer greater than on... |
evennn02n 14912 | A nonnegative integer is e... |
evennn2n 14913 | A positive integer is even... |
2tp1odd 14914 | A number which is twice an... |
mulsucdiv2z 14915 | An integer multiplied with... |
sqoddm1div8z 14916 | A squared odd number minus... |
2teven 14917 | A number which is twice an... |
zeo5 14918 | An integer is either even ... |
evend2 14919 | An integer is even iff its... |
oddp1d2 14920 | An integer is odd iff its ... |
zob 14921 | Alternate characterization... |
oddm1d2 14922 | An integer is odd iff its ... |
ltoddhalfle 14923 | An integer is less than ha... |
halfleoddlt 14924 | An integer is greater than... |
opoe 14925 | The sum of two odds is eve... |
omoe 14926 | The difference of two odds... |
opeo 14927 | The sum of an odd and an e... |
omeo 14928 | The difference of an odd a... |
m1expe 14929 | Exponentiation of -1 by an... |
m1expo 14930 | Exponentiation of -1 by an... |
m1exp1 14931 | Exponentiation of negative... |
nn0enne 14932 | A positive integer is an e... |
nn0ehalf 14933 | The half of an even nonneg... |
nnehalf 14934 | The half of an even positi... |
nn0o1gt2 14935 | An odd nonnegative integer... |
nno 14936 | An alternate characterizat... |
nn0o 14937 | An alternate characterizat... |
nn0ob 14938 | Alternate characterization... |
nn0oddm1d2 14939 | A positive integer is odd ... |
nnoddm1d2 14940 | A positive integer is odd ... |
z0even 14941 | 0 is even. (Contributed b... |
n2dvds1 14942 | 2 does not divide 1 (commo... |
n2dvdsm1 14943 | 2 does not divide -1. Tha... |
z2even 14944 | 2 is even. (Contributed b... |
n2dvds3 14945 | 2 does not divide 3, i.e. ... |
z4even 14946 | 4 is an even number. (Con... |
4dvdseven 14947 | An integer which is divisi... |
sumeven 14948 | If every term in a sum is ... |
sumodd 14949 | If every term in a sum is ... |
evensumodd 14950 | If every term in a sum wit... |
oddsumodd 14951 | If every term in a sum wit... |
pwp1fsum 14952 | The n-th power of a number... |
oddpwp1fsum 14953 | An odd power of a number i... |
divalglem0 14954 | Lemma for ~ divalg . (Con... |
divalglem1 14955 | Lemma for ~ divalg . (Con... |
divalglem2 14956 | Lemma for ~ divalg . (Con... |
divalglem4 14957 | Lemma for ~ divalg . (Con... |
divalglem5 14958 | Lemma for ~ divalg . (Con... |
divalglem6 14959 | Lemma for ~ divalg . (Con... |
divalglem7 14960 | Lemma for ~ divalg . (Con... |
divalglem8 14961 | Lemma for ~ divalg . (Con... |
divalglem9 14962 | Lemma for ~ divalg . (Con... |
divalglem10 14963 | Lemma for ~ divalg . (Con... |
divalg 14964 | The division algorithm (th... |
divalgb 14965 | Express the division algor... |
divalg2 14966 | The division algorithm (th... |
divalgmod 14967 | The result of the ` mod ` ... |
divalgmodOLD 14968 | Obsolete proof of ~ divalg... |
divalgmodcl 14969 | The result of the ` mod ` ... |
modremain 14970 | The result of the modulo o... |
ndvdssub 14971 | Corollary of the division ... |
ndvdsadd 14972 | Corollary of the division ... |
ndvdsp1 14973 | Special case of ~ ndvdsadd... |
ndvdsi 14974 | A quick test for non-divis... |
flodddiv4 14975 | The floor of an odd intege... |
fldivndvdslt 14976 | The floor of an integer di... |
flodddiv4lt 14977 | The floor of an odd number... |
flodddiv4t2lthalf 14978 | The floor of an odd number... |
bitsfval 14983 | Expand the definition of t... |
bitsval 14984 | Expand the definition of t... |
bitsval2 14985 | Expand the definition of t... |
bitsss 14986 | The set of bits of an inte... |
bitsf 14987 | The ` bits ` function is a... |
bits0 14988 | Value of the zeroth bit. ... |
bits0e 14989 | The zeroth bit of an even ... |
bits0o 14990 | The zeroth bit of an odd n... |
bitsp1 14991 | The ` M + 1 ` -th bit of `... |
bitsp1e 14992 | The ` M + 1 ` -th bit of `... |
bitsp1o 14993 | The ` M + 1 ` -th bit of `... |
bitsfzolem 14994 | Lemma for ~ bitsfzo . (Co... |
bitsfzo 14995 | The bits of a number are a... |
bitsmod 14996 | Truncating the bit sequenc... |
bitsfi 14997 | Every number is associated... |
bitscmp 14998 | The bit complement of ` N ... |
0bits 14999 | The bits of zero. (Contri... |
m1bits 15000 | The bits of negative one. ... |
bitsinv1lem 15001 | Lemma for ~ bitsinv1 . (C... |
bitsinv1 15002 | There is an explicit inver... |
bitsinv2 15003 | There is an explicit inver... |
bitsf1ocnv 15004 | The ` bits ` function rest... |
bitsf1o 15005 | The ` bits ` function rest... |
bitsf1 15006 | The ` bits ` function is a... |
2ebits 15007 | The bits of a power of two... |
bitsinv 15008 | The inverse of the ` bits ... |
bitsinvp1 15009 | Recursive definition of th... |
sadadd2lem2 15010 | The core of the proof of ~... |
sadfval 15012 | Define the addition of two... |
sadcf 15013 | The carry sequence is a se... |
sadc0 15014 | The initial element of the... |
sadcp1 15015 | The carry sequence (which ... |
sadval 15016 | The full adder sequence is... |
sadcaddlem 15017 | Lemma for ~ sadcadd . (Co... |
sadcadd 15018 | Non-recursive definition o... |
sadadd2lem 15019 | Lemma for ~ sadadd2 . (Co... |
sadadd2 15020 | Sum of initial segments of... |
sadadd3 15021 | Sum of initial segments of... |
sadcl 15022 | The sum of two sequences i... |
sadcom 15023 | The adder sequence functio... |
saddisjlem 15024 | Lemma for ~ sadadd . (Con... |
saddisj 15025 | The sum of disjoint sequen... |
sadaddlem 15026 | Lemma for ~ sadadd . (Con... |
sadadd 15027 | For sequences that corresp... |
sadid1 15028 | The adder sequence functio... |
sadid2 15029 | The adder sequence functio... |
sadasslem 15030 | Lemma for ~ sadass . (Con... |
sadass 15031 | Sequence addition is assoc... |
sadeq 15032 | Any element of a sequence ... |
bitsres 15033 | Restrict the bits of a num... |
bitsuz 15034 | The bits of a number are a... |
bitsshft 15035 | Shifting a bit sequence to... |
smufval 15037 | The multiplication of two ... |
smupf 15038 | The sequence of partial su... |
smup0 15039 | The initial element of the... |
smupp1 15040 | The initial element of the... |
smuval 15041 | Define the addition of two... |
smuval2 15042 | The partial sum sequence s... |
smupvallem 15043 | If ` A ` only has elements... |
smucl 15044 | The product of two sequenc... |
smu01lem 15045 | Lemma for ~ smu01 and ~ sm... |
smu01 15046 | Multiplication of a sequen... |
smu02 15047 | Multiplication of a sequen... |
smupval 15048 | Rewrite the elements of th... |
smup1 15049 | Rewrite ~ smupp1 using onl... |
smueqlem 15050 | Any element of a sequence ... |
smueq 15051 | Any element of a sequence ... |
smumullem 15052 | Lemma for ~ smumul . (Con... |
smumul 15053 | For sequences that corresp... |
gcdval 15056 | The value of the ` gcd ` o... |
gcd0val 15057 | The value, by convention, ... |
gcdn0val 15058 | The value of the ` gcd ` o... |
gcdcllem1 15059 | Lemma for ~ gcdn0cl , ~ gc... |
gcdcllem2 15060 | Lemma for ~ gcdn0cl , ~ gc... |
gcdcllem3 15061 | Lemma for ~ gcdn0cl , ~ gc... |
gcdn0cl 15062 | Closure of the ` gcd ` ope... |
gcddvds 15063 | The gcd of two integers di... |
dvdslegcd 15064 | An integer which divides b... |
nndvdslegcd 15065 | A positive integer which d... |
gcdcl 15066 | Closure of the ` gcd ` ope... |
gcdnncl 15067 | Closure of the ` gcd ` ope... |
gcdcld 15068 | Closure of the ` gcd ` ope... |
gcd2n0cl 15069 | Closure of the ` gcd ` ope... |
zeqzmulgcd 15070 | An integer is the product ... |
divgcdz 15071 | An integer divided by the ... |
gcdf 15072 | Domain and codomain of the... |
gcdcom 15073 | The ` gcd ` operator is co... |
divgcdnn 15074 | A positive integer divided... |
divgcdnnr 15075 | A positive integer divided... |
gcdeq0 15076 | The gcd of two integers is... |
gcdn0gt0 15077 | The gcd of two integers is... |
gcd0id 15078 | The gcd of 0 and an intege... |
gcdid0 15079 | The gcd of an integer and ... |
nn0gcdid0 15080 | The gcd of a nonnegative i... |
gcdneg 15081 | Negating one operand of th... |
neggcd 15082 | Negating one operand of th... |
gcdaddmlem 15083 | Lemma for ~ gcdaddm . (Co... |
gcdaddm 15084 | Adding a multiple of one o... |
gcdadd 15085 | The GCD of two numbers is ... |
gcdid 15086 | The gcd of a number and it... |
gcd1 15087 | The gcd of a number with 1... |
gcdabs 15088 | The gcd of two integers is... |
gcdabs1 15089 | ` gcd ` of the absolute va... |
gcdabs2 15090 | ` gcd ` of the absolute va... |
modgcd 15091 | The gcd remains unchanged ... |
1gcd 15092 | The GCD of one and an inte... |
6gcd4e2 15093 | The greatest common diviso... |
bezoutlem1 15094 | Lemma for ~ bezout . (Con... |
bezoutlem2 15095 | Lemma for ~ bezout . (Con... |
bezoutlem3 15096 | Lemma for ~ bezout . (Con... |
bezoutlem4 15097 | Lemma for ~ bezout . (Con... |
bezout 15098 | Bézout's identity: ... |
dvdsgcd 15099 | An integer which divides e... |
dvdsgcdb 15100 | Biconditional form of ~ dv... |
dfgcd2 15101 | Alternate definition of th... |
gcdass 15102 | Associative law for ` gcd ... |
mulgcd 15103 | Distribute multiplication ... |
absmulgcd 15104 | Distribute absolute value ... |
mulgcdr 15105 | Reverse distribution law f... |
gcddiv 15106 | Division law for GCD. (Con... |
gcdmultiple 15107 | The GCD of a multiple of a... |
gcdmultiplez 15108 | Extend ~ gcdmultiple so ` ... |
gcdzeq 15109 | A positive integer ` A ` i... |
gcdeq 15110 | ` A ` is equal to its gcd ... |
dvdssqim 15111 | Unidirectional form of ~ d... |
dvdsmulgcd 15112 | A divisibility equivalent ... |
rpmulgcd 15113 | If ` K ` and ` M ` are rel... |
rplpwr 15114 | If ` A ` and ` B ` are rel... |
rppwr 15115 | If ` A ` and ` B ` are rel... |
sqgcd 15116 | Square distributes over GC... |
dvdssqlem 15117 | Lemma for ~ dvdssq . (Con... |
dvdssq 15118 | Two numbers are divisible ... |
bezoutr 15119 | Partial converse to ~ bezo... |
bezoutr1 15120 | Converse of ~ bezout for t... |
nn0seqcvgd 15121 | A strictly-decreasing nonn... |
seq1st 15122 | A sequence whose iteration... |
algr0 15123 | The value of the algorithm... |
algrf 15124 | An algorithm is a step fun... |
algrp1 15125 | The value of the algorithm... |
alginv 15126 | If ` I ` is an invariant o... |
algcvg 15127 | One way to prove that an a... |
algcvgblem 15128 | Lemma for ~ algcvgb . (Co... |
algcvgb 15129 | Two ways of expressing tha... |
algcvga 15130 | The countdown function ` C... |
algfx 15131 | If ` F ` reaches a fixed p... |
eucalgval2 15132 | The value of the step func... |
eucalgval 15133 | Euclid's Algorithm ~ eucal... |
eucalgf 15134 | Domain and codomain of the... |
eucalginv 15135 | The invariant of the step ... |
eucalglt 15136 | The second member of the s... |
eucalgcvga 15137 | Once Euclid's Algorithm ha... |
eucalg 15138 | Euclid's Algorithm compute... |
lcmval 15143 | Value of the ` lcm ` opera... |
lcmcom 15144 | The ` lcm ` operator is co... |
lcm0val 15145 | The value, by convention, ... |
lcmn0val 15146 | The value of the ` lcm ` o... |
lcmcllem 15147 | Lemma for ~ lcmn0cl and ~ ... |
lcmn0cl 15148 | Closure of the ` lcm ` ope... |
dvdslcm 15149 | The lcm of two integers is... |
lcmledvds 15150 | A positive integer which b... |
lcmeq0 15151 | The lcm of two integers is... |
lcmcl 15152 | Closure of the ` lcm ` ope... |
gcddvdslcm 15153 | The greatest common diviso... |
lcmneg 15154 | Negating one operand of th... |
neglcm 15155 | Negating one operand of th... |
lcmabs 15156 | The lcm of two integers is... |
lcmgcdlem 15157 | Lemma for ~ lcmgcd and ~ l... |
lcmgcd 15158 | The product of two numbers... |
lcmdvds 15159 | The lcm of two integers di... |
lcmid 15160 | The lcm of an integer and ... |
lcm1 15161 | The lcm of an integer and ... |
lcmgcdnn 15162 | The product of two positiv... |
lcmgcdeq 15163 | Two integers' absolute val... |
lcmdvdsb 15164 | Biconditional form of ~ lc... |
lcmass 15165 | Associative law for ` lcm ... |
3lcm2e6woprm 15166 | The least common multiple ... |
6lcm4e12 15167 | The least common multiple ... |
absproddvds 15168 | The absolute value of the ... |
absprodnn 15169 | The absolute value of the ... |
fissn0dvds 15170 | For each finite subset of ... |
fissn0dvdsn0 15171 | For each finite subset of ... |
lcmfval 15172 | Value of the ` _lcm ` func... |
lcmf0val 15173 | The value, by convention, ... |
lcmfn0val 15174 | The value of the ` _lcm ` ... |
lcmfnnval 15175 | The value of the ` _lcm ` ... |
lcmfcllem 15176 | Lemma for ~ lcmfn0cl and ~... |
lcmfn0cl 15177 | Closure of the ` _lcm ` fu... |
lcmfpr 15178 | The value of the ` _lcm ` ... |
lcmfcl 15179 | Closure of the ` _lcm ` fu... |
lcmfnncl 15180 | Closure of the ` _lcm ` fu... |
lcmfeq0b 15181 | The least common multiple ... |
dvdslcmf 15182 | The least common multiple ... |
lcmfledvds 15183 | A positive integer which i... |
lcmf 15184 | Characterization of the le... |
lcmf0 15185 | The least common multiple ... |
lcmfsn 15186 | The least common multiple ... |
lcmftp 15187 | The least common multiple ... |
lcmfunsnlem1 15188 | Lemma for ~ lcmfdvds and ~... |
lcmfunsnlem2lem1 15189 | Lemma 1 for ~ lcmfunsnlem2... |
lcmfunsnlem2lem2 15190 | Lemma 2 for ~ lcmfunsnlem2... |
lcmfunsnlem2 15191 | Lemma for ~ lcmfunsn and ~... |
lcmfunsnlem 15192 | Lemma for ~ lcmfdvds and ~... |
lcmfdvds 15193 | The least common multiple ... |
lcmfdvdsb 15194 | Biconditional form of ~ lc... |
lcmfunsn 15195 | The ` _lcm ` function for ... |
lcmfun 15196 | The ` _lcm ` function for ... |
lcmfass 15197 | Associative law for the ` ... |
lcmf2a3a4e12 15198 | The least common multiple ... |
lcmflefac 15199 | The least common multiple ... |
coprmgcdb 15200 | Two positive integers are ... |
ncoprmgcdne1b 15201 | Two positive integers are ... |
ncoprmgcdgt1b 15202 | Two positive integers are ... |
coprmdvds1 15203 | If two positive integers a... |
coprmdvds 15204 | Euclid's Lemma (see ProofW... |
coprmdvdsOLD 15205 | If an integer divides the ... |
coprmdvds2 15206 | If an integer is divisible... |
mulgcddvds 15207 | One half of ~ rpmulgcd2 , ... |
rpmulgcd2 15208 | If ` M ` is relatively pri... |
qredeq 15209 | Two equal reduced fraction... |
qredeu 15210 | Every rational number has ... |
rpmul 15211 | If ` K ` is relatively pri... |
rpdvds 15212 | If ` K ` is relatively pri... |
coprmprod 15213 | The product of the element... |
coprmproddvdslem 15214 | Lemma for ~ coprmproddvds ... |
coprmproddvds 15215 | If a positive integer is d... |
congr 15216 | Definition of congruence b... |
divgcdcoprm0 15217 | Integers divided by gcd ar... |
divgcdcoprmex 15218 | Integers divided by gcd ar... |
cncongr1 15219 | One direction of the bicon... |
cncongr2 15220 | The other direction of the... |
cncongr 15221 | Cancellability of Congruen... |
cncongrcoprm 15222 | Corollary 1 of Cancellabil... |
isprm 15225 | The predicate "is a prime ... |
prmnn 15226 | A prime number is a positi... |
prmz 15227 | A prime number is an integ... |
prmssnn 15228 | The prime numbers are a su... |
prmex 15229 | The set of prime numbers e... |
1nprm 15230 | 1 is not a prime number. ... |
1idssfct 15231 | The positive divisors of a... |
isprm2lem 15232 | Lemma for ~ isprm2 . (Con... |
isprm2 15233 | The predicate "is a prime ... |
isprm3 15234 | The predicate "is a prime ... |
isprm4 15235 | The predicate "is a prime ... |
prmind2 15236 | A variation on ~ prmind as... |
prmind 15237 | Perform induction over the... |
dvdsprime 15238 | If ` M ` divides a prime, ... |
nprm 15239 | A product of two integers ... |
nprmi 15240 | An inference for composite... |
dvdsnprmd 15241 | If a number is divisible b... |
prm2orodd 15242 | A prime number is either 2... |
2prm 15243 | 2 is a prime number. (Con... |
3prm 15244 | 3 is a prime number. (Con... |
4nprm 15245 | 4 is not a prime number. ... |
prmuz2 15246 | A prime number is an integ... |
prmgt1 15247 | A prime number is an integ... |
prmm2nn0 15248 | Subtracting 2 from a prime... |
oddprmgt2 15249 | An odd prime is greater th... |
oddprmge3 15250 | An odd prime is greater th... |
prmn2uzge3OLD 15251 | Obsolete version of ~ oddp... |
sqnprm 15252 | A square is never prime. ... |
dvdsprm 15253 | An integer greater than or... |
exprmfct 15254 | Every integer greater than... |
prmdvdsfz 15255 | Each integer greater than ... |
nprmdvds1 15256 | No prime number divides 1.... |
isprm5 15257 | One need only check prime ... |
isprm7 15258 | One need only check prime ... |
maxprmfct 15259 | The set of prime factors o... |
divgcdodd 15260 | Either ` A / ( A gcd B ) `... |
coprm 15261 | A prime number either divi... |
prmrp 15262 | Unequal prime numbers are ... |
euclemma 15263 | Euclid's lemma. A prime n... |
isprm6 15264 | A number is prime iff it s... |
prmdvdsexp 15265 | A prime divides a positive... |
prmdvdsexpb 15266 | A prime divides a positive... |
prmdvdsexpr 15267 | If a prime divides a nonne... |
prmexpb 15268 | Two positive prime powers ... |
prmfac1 15269 | The factorial of a number ... |
rpexp 15270 | If two numbers ` A ` and `... |
rpexp1i 15271 | Relative primality passes ... |
rpexp12i 15272 | Relative primality passes ... |
prmndvdsfaclt 15273 | A prime number does not di... |
ncoprmlnprm 15274 | If two positive integers a... |
cncongrprm 15275 | Corollary 2 of Cancellabil... |
isevengcd2 15276 | The predicate "is an even ... |
isoddgcd1 15277 | The predicate "is an odd n... |
3lcm2e6 15278 | The least common multiple ... |
qnumval 15283 | Value of the canonical num... |
qdenval 15284 | Value of the canonical den... |
qnumdencl 15285 | Lemma for ~ qnumcl and ~ q... |
qnumcl 15286 | The canonical numerator of... |
qdencl 15287 | The canonical denominator ... |
fnum 15288 | Canonical numerator define... |
fden 15289 | Canonical denominator defi... |
qnumdenbi 15290 | Two numbers are the canoni... |
qnumdencoprm 15291 | The canonical representati... |
qeqnumdivden 15292 | Recover a rational number ... |
qmuldeneqnum 15293 | Multiplying a rational by ... |
divnumden 15294 | Calculate the reduced form... |
divdenle 15295 | Reducing a quotient never ... |
qnumgt0 15296 | A rational is positive iff... |
qgt0numnn 15297 | A rational is positive iff... |
nn0gcdsq 15298 | Squaring commutes with GCD... |
zgcdsq 15299 | ~ nn0gcdsq extended to int... |
numdensq 15300 | Squaring a rational square... |
numsq 15301 | Square commutes with canon... |
densq 15302 | Square commutes with canon... |
qden1elz 15303 | A rational is an integer i... |
zsqrtelqelz 15304 | If an integer has a ration... |
nonsq 15305 | Any integer strictly betwe... |
phival 15310 | Value of the Euler ` phi `... |
phicl2 15311 | Bounds and closure for the... |
phicl 15312 | Closure for the value of t... |
phibndlem 15313 | Lemma for ~ phibnd . (Con... |
phibnd 15314 | A slightly tighter bound o... |
phicld 15315 | Closure for the value of t... |
phi1 15316 | Value of the Euler ` phi `... |
dfphi2 15317 | Alternate definition of th... |
hashdvds 15318 | The number of numbers in a... |
phiprmpw 15319 | Value of the Euler ` phi `... |
phiprm 15320 | Value of the Euler ` phi `... |
crth 15321 | The Chinese Remainder Theo... |
phimullem 15322 | Lemma for ~ phimul . (Con... |
phimul 15323 | The Euler ` phi ` function... |
eulerthlem1 15324 | Lemma for ~ eulerth . (Co... |
eulerthlem2 15325 | Lemma for ~ eulerth . (Co... |
eulerth 15326 | Euler's theorem, a general... |
fermltl 15327 | Fermat's little theorem. ... |
prmdiv 15328 | Show an explicit expressio... |
prmdiveq 15329 | The modular inverse of ` A... |
prmdivdiv 15330 | The (modular) inverse of t... |
hashgcdlem 15331 | A correspondence between e... |
hashgcdeq 15332 | Number of initial positive... |
phisum 15333 | The divisor sum identity o... |
odzval 15334 | Value of the order functio... |
odzcllem 15335 | - Lemma for ~ odzcl , show... |
odzcl 15336 | The order of a group eleme... |
odzid 15337 | Any element raised to the ... |
odzdvds 15338 | The only powers of ` A ` t... |
odzphi 15339 | The order of any group ele... |
modprm1div 15340 | A prime number divides an ... |
m1dvdsndvds 15341 | If an integer minus 1 is d... |
modprminv 15342 | Show an explicit expressio... |
modprminveq 15343 | The modular inverse of ` A... |
vfermltl 15344 | Variant of Fermat's little... |
vfermltlALT 15345 | Alternate proof of ~ vferm... |
powm2modprm 15346 | If an integer minus 1 is d... |
reumodprminv 15347 | For any prime number and f... |
modprm0 15348 | For two positive integers ... |
nnnn0modprm0 15349 | For a positive integer and... |
modprmn0modprm0 15350 | For an integer not being 0... |
coprimeprodsq 15351 | If three numbers are copri... |
coprimeprodsq2 15352 | If three numbers are copri... |
oddprm 15353 | A prime not equal to ` 2 `... |
nnoddn2prm 15354 | A prime not equal to ` 2 `... |
oddn2prm 15355 | A prime not equal to ` 2 `... |
nnoddn2prmb 15356 | A number is a prime number... |
prm23lt5 15357 | A prime less than 5 is eit... |
prm23ge5 15358 | A prime is either 2 or 3 o... |
pythagtriplem1 15359 | Lemma for ~ pythagtrip . ... |
pythagtriplem2 15360 | Lemma for ~ pythagtrip . ... |
pythagtriplem3 15361 | Lemma for ~ pythagtrip . ... |
pythagtriplem4 15362 | Lemma for ~ pythagtrip . ... |
pythagtriplem10 15363 | Lemma for ~ pythagtrip . ... |
pythagtriplem6 15364 | Lemma for ~ pythagtrip . ... |
pythagtriplem7 15365 | Lemma for ~ pythagtrip . ... |
pythagtriplem8 15366 | Lemma for ~ pythagtrip . ... |
pythagtriplem9 15367 | Lemma for ~ pythagtrip . ... |
pythagtriplem11 15368 | Lemma for ~ pythagtrip . ... |
pythagtriplem12 15369 | Lemma for ~ pythagtrip . ... |
pythagtriplem13 15370 | Lemma for ~ pythagtrip . ... |
pythagtriplem14 15371 | Lemma for ~ pythagtrip . ... |
pythagtriplem15 15372 | Lemma for ~ pythagtrip . ... |
pythagtriplem16 15373 | Lemma for ~ pythagtrip . ... |
pythagtriplem17 15374 | Lemma for ~ pythagtrip . ... |
pythagtriplem18 15375 | Lemma for ~ pythagtrip . ... |
pythagtriplem19 15376 | Lemma for ~ pythagtrip . ... |
pythagtrip 15377 | Parameterize the Pythagore... |
iserodd 15378 | Collect the odd terms in a... |
pclem 15381 | - Lemma for the prime powe... |
pcprecl 15382 | Closure of the prime power... |
pcprendvds 15383 | Non-divisibility property ... |
pcprendvds2 15384 | Non-divisibility property ... |
pcpre1 15385 | Value of the prime power p... |
pcpremul 15386 | Multiplicative property of... |
pcval 15387 | The value of the prime pow... |
pceulem 15388 | Lemma for ~ pceu . (Contr... |
pceu 15389 | Uniqueness for the prime p... |
pczpre 15390 | Connect the prime count pr... |
pczcl 15391 | Closure of the prime power... |
pccl 15392 | Closure of the prime power... |
pccld 15393 | Closure of the prime power... |
pcmul 15394 | Multiplication property of... |
pcdiv 15395 | Division property of the p... |
pcqmul 15396 | Multiplication property of... |
pc0 15397 | The value of the prime pow... |
pc1 15398 | Value of the prime count f... |
pcqcl 15399 | Closure of the general pri... |
pcqdiv 15400 | Division property of the p... |
pcrec 15401 | Prime power of a reciproca... |
pcexp 15402 | Prime power of an exponent... |
pcxcl 15403 | Extended real closure of t... |
pcge0 15404 | The prime count of an inte... |
pczdvds 15405 | Defining property of the p... |
pcdvds 15406 | Defining property of the p... |
pczndvds 15407 | Defining property of the p... |
pcndvds 15408 | Defining property of the p... |
pczndvds2 15409 | The remainder after dividi... |
pcndvds2 15410 | The remainder after dividi... |
pcdvdsb 15411 | ` P ^ A ` divides ` N ` if... |
pcelnn 15412 | There are a positive numbe... |
pceq0 15413 | There are zero powers of a... |
pcidlem 15414 | The prime count of a prime... |
pcid 15415 | The prime count of a prime... |
pcneg 15416 | The prime count of a negat... |
pcabs 15417 | The prime count of an abso... |
pcdvdstr 15418 | The prime count increases ... |
pcgcd1 15419 | The prime count of a GCD i... |
pcgcd 15420 | The prime count of a GCD i... |
pc2dvds 15421 | A characterization of divi... |
pc11 15422 | The prime count function, ... |
pcz 15423 | The prime count function c... |
pcprmpw2 15424 | Self-referential expressio... |
pcprmpw 15425 | Self-referential expressio... |
dvdsprmpweq 15426 | If a positive integer divi... |
dvdsprmpweqnn 15427 | If an integer greater than... |
dvdsprmpweqle 15428 | If a positive integer divi... |
difsqpwdvds 15429 | If the difference of two s... |
pcaddlem 15430 | Lemma for ~ pcadd . The o... |
pcadd 15431 | An inequality for the prim... |
pcadd2 15432 | The inequality of ~ pcadd ... |
pcmptcl 15433 | Closure for the prime powe... |
pcmpt 15434 | Construct a function with ... |
pcmpt2 15435 | Dividing two prime count m... |
pcmptdvds 15436 | The partial products of th... |
pcprod 15437 | The product of the primes ... |
sumhash 15438 | The sum of 1 over a set is... |
fldivp1 15439 | The difference between the... |
pcfaclem 15440 | Lemma for ~ pcfac . (Cont... |
pcfac 15441 | Calculate the prime count ... |
pcbc 15442 | Calculate the prime count ... |
qexpz 15443 | If a power of a rational n... |
expnprm 15444 | A second or higher power o... |
oddprmdvds 15445 | Every positive integer whi... |
prmpwdvds 15446 | A relation involving divis... |
pockthlem 15447 | Lemma for ~ pockthg . (Co... |
pockthg 15448 | The generalized Pocklingto... |
pockthi 15449 | Pocklington's theorem, whi... |
unbenlem 15450 | Lemma for ~ unben . (Cont... |
unben 15451 | An unbounded set of positi... |
infpnlem1 15452 | Lemma for ~ infpn . The s... |
infpnlem2 15453 | Lemma for ~ infpn . For a... |
infpn 15454 | There exist infinitely man... |
infpn2 15455 | There exist infinitely man... |
prmunb 15456 | The primes are unbounded. ... |
prminf 15457 | There are an infinite numb... |
prmreclem1 15458 | Lemma for ~ prmrec . Prop... |
prmreclem2 15459 | Lemma for ~ prmrec . Ther... |
prmreclem3 15460 | Lemma for ~ prmrec . The ... |
prmreclem4 15461 | Lemma for ~ prmrec . Show... |
prmreclem5 15462 | Lemma for ~ prmrec . Here... |
prmreclem6 15463 | Lemma for ~ prmrec . If t... |
prmrec 15464 | The sum of the reciprocals... |
1arithlem1 15465 | Lemma for ~ 1arith . (Con... |
1arithlem2 15466 | Lemma for ~ 1arith . (Con... |
1arithlem3 15467 | Lemma for ~ 1arith . (Con... |
1arithlem4 15468 | Lemma for ~ 1arith . (Con... |
1arith 15469 | Fundamental theorem of ari... |
1arith2 15470 | Fundamental theorem of ari... |
elgz 15473 | Elementhood in the gaussia... |
gzcn 15474 | A gaussian integer is a co... |
zgz 15475 | An integer is a gaussian i... |
igz 15476 | ` _i ` is a gaussian integ... |
gznegcl 15477 | The gaussian integers are ... |
gzcjcl 15478 | The gaussian integers are ... |
gzaddcl 15479 | The gaussian integers are ... |
gzmulcl 15480 | The gaussian integers are ... |
gzreim 15481 | Construct a gaussian integ... |
gzsubcl 15482 | The gaussian integers are ... |
gzabssqcl 15483 | The squared norm of a gaus... |
4sqlem5 15484 | Lemma for ~ 4sq . (Contri... |
4sqlem6 15485 | Lemma for ~ 4sq . (Contri... |
4sqlem7 15486 | Lemma for ~ 4sq . (Contri... |
4sqlem8 15487 | Lemma for ~ 4sq . (Contri... |
4sqlem9 15488 | Lemma for ~ 4sq . (Contri... |
4sqlem10 15489 | Lemma for ~ 4sq . (Contri... |
4sqlem1 15490 | Lemma for ~ 4sq . The set... |
4sqlem2 15491 | Lemma for ~ 4sq . Change ... |
4sqlem3 15492 | Lemma for ~ 4sq . Suffici... |
4sqlem4a 15493 | Lemma for ~ 4sqlem4 . (Co... |
4sqlem4 15494 | Lemma for ~ 4sq . We can ... |
mul4sqlem 15495 | Lemma for ~ mul4sq : algeb... |
mul4sq 15496 | Euler's four-square identi... |
4sqlem11 15497 | Lemma for ~ 4sq . Use the... |
4sqlem12 15498 | Lemma for ~ 4sq . For any... |
4sqlem13 15499 | Lemma for ~ 4sq . (Contri... |
4sqlem14 15500 | Lemma for ~ 4sq . (Contri... |
4sqlem15 15501 | Lemma for ~ 4sq . (Contri... |
4sqlem16 15502 | Lemma for ~ 4sq . (Contri... |
4sqlem17 15503 | Lemma for ~ 4sq . (Contri... |
4sqlem18 15504 | Lemma for ~ 4sq . Inducti... |
4sqlem19 15505 | Lemma for ~ 4sq . The pro... |
4sq 15506 | Lagrange's four-square the... |
vdwapfval 15513 | Define the arithmetic prog... |
vdwapf 15514 | The arithmetic progression... |
vdwapval 15515 | Value of the arithmetic pr... |
vdwapun 15516 | Remove the first element o... |
vdwapid1 15517 | The first element of an ar... |
vdwap0 15518 | Value of a length-1 arithm... |
vdwap1 15519 | Value of a length-1 arithm... |
vdwmc 15520 | The predicate " The ` <. R... |
vdwmc2 15521 | Expand out the definition ... |
vdwpc 15522 | The predicate " The colori... |
vdwlem1 15523 | Lemma for ~ vdw . (Contri... |
vdwlem2 15524 | Lemma for ~ vdw . (Contri... |
vdwlem3 15525 | Lemma for ~ vdw . (Contri... |
vdwlem4 15526 | Lemma for ~ vdw . (Contri... |
vdwlem5 15527 | Lemma for ~ vdw . (Contri... |
vdwlem6 15528 | Lemma for ~ vdw . (Contri... |
vdwlem7 15529 | Lemma for ~ vdw . (Contri... |
vdwlem8 15530 | Lemma for ~ vdw . (Contri... |
vdwlem9 15531 | Lemma for ~ vdw . (Contri... |
vdwlem10 15532 | Lemma for ~ vdw . Set up ... |
vdwlem11 15533 | Lemma for ~ vdw . (Contri... |
vdwlem12 15534 | Lemma for ~ vdw . ` K = 2 ... |
vdwlem13 15535 | Lemma for ~ vdw . Main in... |
vdw 15536 | Van der Waerden's theorem.... |
vdwnnlem1 15537 | Corollary of ~ vdw , and l... |
vdwnnlem2 15538 | Lemma for ~ vdwnn . The s... |
vdwnnlem3 15539 | Lemma for ~ vdwnn . (Cont... |
vdwnn 15540 | Van der Waerden's theorem,... |
ramtlecl 15542 | The set ` T ` of numbers w... |
hashbcval 15544 | Value of the "binomial set... |
hashbccl 15545 | The binomial set is a fini... |
hashbcss 15546 | Subset relation for the bi... |
hashbc0 15547 | The set of subsets of size... |
hashbc2 15548 | The size of the binomial s... |
0hashbc 15549 | There are no subsets of th... |
ramval 15550 | The value of the Ramsey nu... |
ramcl2lem 15551 | Lemma for extended real cl... |
ramtcl 15552 | The Ramsey number has the ... |
ramtcl2 15553 | The Ramsey number is an in... |
ramtub 15554 | The Ramsey number is a low... |
ramub 15555 | The Ramsey number is a low... |
ramub2 15556 | It is sufficient to check ... |
rami 15557 | The defining property of a... |
ramcl2 15558 | The Ramsey number is eithe... |
ramxrcl 15559 | The Ramsey number is an ex... |
ramubcl 15560 | If the Ramsey number is up... |
ramlb 15561 | Establish a lower bound on... |
0ram 15562 | The Ramsey number when ` M... |
0ram2 15563 | The Ramsey number when ` M... |
ram0 15564 | The Ramsey number when ` R... |
0ramcl 15565 | Lemma for ~ ramcl : Exist... |
ramz2 15566 | The Ramsey number when ` F... |
ramz 15567 | The Ramsey number when ` F... |
ramub1lem1 15568 | Lemma for ~ ramub1 . (Con... |
ramub1lem2 15569 | Lemma for ~ ramub1 . (Con... |
ramub1 15570 | Inductive step for Ramsey'... |
ramcl 15571 | Ramsey's theorem: the Rams... |
ramsey 15572 | Ramsey's theorem with the ... |
prmoval 15575 | Value of the primorial fun... |
prmocl 15576 | Closure of the primorial f... |
prmone0 15577 | The primorial function is ... |
prmo0 15578 | The primorial of 0. (Cont... |
prmo1 15579 | The primorial of 1. (Cont... |
prmop1 15580 | The primorial of a success... |
prmonn2 15581 | Value of the primorial fun... |
prmo2 15582 | The primorial of 2. (Cont... |
prmo3 15583 | The primorial of 3. (Cont... |
prmdvdsprmo 15584 | The primorial of a number ... |
prmdvdsprmop 15585 | The primorial of a number ... |
fvprmselelfz 15586 | The value of the prime sel... |
fvprmselgcd1 15587 | The greatest common diviso... |
prmolefac 15588 | The primorial of a positiv... |
prmodvdslcmf 15589 | The primorial of a nonnega... |
prmolelcmf 15590 | The primorial of a positiv... |
prmgaplem1 15591 | Lemma for ~ prmgap : The ... |
prmgaplem2 15592 | Lemma for ~ prmgap : The ... |
prmgaplcmlem1 15593 | Lemma for ~ prmgaplcm : T... |
prmgaplcmlem2 15594 | Lemma for ~ prmgaplcm : T... |
prmgaplem3 15595 | Lemma for ~ prmgap . (Con... |
prmgaplem4 15596 | Lemma for ~ prmgap . (Con... |
prmgaplem5 15597 | Lemma for ~ prmgap : for e... |
prmgaplem6 15598 | Lemma for ~ prmgap : for e... |
prmgaplem7 15599 | Lemma for ~ prmgap . (Con... |
prmgaplem8 15600 | Lemma for ~ prmgap . (Con... |
prmgap 15601 | The prime gap theorem: for... |
prmgaplcm 15602 | Alternate proof of ~ prmga... |
prmgapprmolem 15603 | Lemma for ~ prmgapprmo : ... |
prmgapprmo 15604 | Alternate proof of ~ prmga... |
dec2dvds 15605 | Divisibility by two is obv... |
dec5dvds 15606 | Divisibility by five is ob... |
dec5dvds2 15607 | Divisibility by five is ob... |
dec5nprm 15608 | Divisibility by five is ob... |
dec2nprm 15609 | Divisibility by two is obv... |
modxai 15610 | Add exponents in a power m... |
mod2xi 15611 | Double exponents in a powe... |
modxp1i 15612 | Add one to an exponent in ... |
mod2xnegi 15613 | Version of ~ mod2xi with a... |
modsubi 15614 | Subtract from within a mod... |
gcdi 15615 | Calculate a GCD via Euclid... |
gcdmodi 15616 | Calculate a GCD via Euclid... |
decexp2 15617 | Calculate a power of two. ... |
numexp0 15618 | Calculate an integer power... |
numexp1 15619 | Calculate an integer power... |
numexpp1 15620 | Calculate an integer power... |
numexp2x 15621 | Double an integer power. ... |
decsplit0b 15622 | Split a decimal number int... |
decsplit0 15623 | Split a decimal number int... |
decsplit1 15624 | Split a decimal number int... |
decsplit 15625 | Split a decimal number int... |
decsplit0bOLD 15626 | Obsolete version of ~ decs... |
decsplit0OLD 15627 | Obsolete version of ~ decs... |
decsplit1OLD 15628 | Obsolete version of ~ decs... |
decsplitOLD 15629 | Obsolete version of ~ decs... |
karatsuba 15630 | The Karatsuba multiplicati... |
karatsubaOLD 15631 | Obsolete version of ~ kara... |
2exp4 15632 | Two to the fourth power is... |
2exp6 15633 | Two to the sixth power is ... |
2exp8 15634 | Two to the eighth power is... |
2exp16 15635 | Two to the sixteenth power... |
3exp3 15636 | Three to the third power i... |
2expltfac 15637 | The factorial grows faster... |
cshwsidrepsw 15638 | If cyclically shifting a w... |
cshwsidrepswmod0 15639 | If cyclically shifting a w... |
cshwshashlem1 15640 | If cyclically shifting a w... |
cshwshashlem2 15641 | If cyclically shifting a w... |
cshwshashlem3 15642 | If cyclically shifting a w... |
cshwsdisj 15643 | The singletons resulting b... |
cshwsiun 15644 | The set of (different!) wo... |
cshwsex 15645 | The class of (different!) ... |
cshws0 15646 | The size of the set of (di... |
cshwrepswhash1 15647 | The size of the set of (di... |
cshwshashnsame 15648 | If a word (not consisting ... |
cshwshash 15649 | If a word has a length bei... |
prmlem0 15650 | Lemma for ~ prmlem1 and ~ ... |
prmlem1a 15651 | A quick proof skeleton to ... |
prmlem1 15652 | A quick proof skeleton to ... |
5prm 15653 | 5 is a prime number. (Con... |
6nprm 15654 | 6 is not a prime number. ... |
7prm 15655 | 7 is a prime number. (Con... |
8nprm 15656 | 8 is not a prime number. ... |
9nprm 15657 | 9 is not a prime number. ... |
10nprm 15658 | 10 is not a prime number. ... |
10nprmOLD 15659 | Obsolete version of ~ 10np... |
11prm 15660 | 11 is a prime number. (Co... |
13prm 15661 | 13 is a prime number. (Co... |
17prm 15662 | 17 is a prime number. (Co... |
19prm 15663 | 19 is a prime number. (Co... |
23prm 15664 | 23 is a prime number. (Co... |
prmlem2 15665 | Our last proving session g... |
37prm 15666 | 37 is a prime number. (Co... |
43prm 15667 | 43 is a prime number. (Co... |
83prm 15668 | 83 is a prime number. (Co... |
139prm 15669 | 139 is a prime number. (C... |
163prm 15670 | 163 is a prime number. (C... |
317prm 15671 | 317 is a prime number. (C... |
631prm 15672 | 631 is a prime number. (C... |
prmo4 15673 | The primorial of 4. (Cont... |
prmo5 15674 | The primorial of 5. (Cont... |
prmo6 15675 | The primorial of 6. (Cont... |
1259lem1 15676 | Lemma for ~ 1259prm . Cal... |
1259lem2 15677 | Lemma for ~ 1259prm . Cal... |
1259lem3 15678 | Lemma for ~ 1259prm . Cal... |
1259lem4 15679 | Lemma for ~ 1259prm . Cal... |
1259lem5 15680 | Lemma for ~ 1259prm . Cal... |
1259prm 15681 | 1259 is a prime number. (... |
2503lem1 15682 | Lemma for ~ 2503prm . Cal... |
2503lem2 15683 | Lemma for ~ 2503prm . Cal... |
2503lem3 15684 | Lemma for ~ 2503prm . Cal... |
2503prm 15685 | 2503 is a prime number. (... |
4001lem1 15686 | Lemma for ~ 4001prm . Cal... |
4001lem2 15687 | Lemma for ~ 4001prm . Cal... |
4001lem3 15688 | Lemma for ~ 4001prm . Cal... |
4001lem4 15689 | Lemma for ~ 4001prm . Cal... |
4001prm 15690 | 4001 is a prime number. (... |
brstruct 15703 | The structure relation is ... |
isstruct2 15704 | The property of being a st... |
isstruct 15705 | The property of being a st... |
structcnvcnv 15706 | Two ways to express the re... |
structfun 15707 | Convert between two kinds ... |
structfn 15708 | Convert between two kinds ... |
slotfn 15709 | A slot is a function on se... |
strfvnd 15710 | Deduction version of ~ str... |
wunndx 15711 | Closure of the index extra... |
strfvn 15712 | Value of a structure compo... |
strfvss 15713 | A structure component extr... |
wunstr 15714 | Closure of a structure ind... |
ndxarg 15715 | Get the numeric argument f... |
ndxid 15716 | A structure component extr... |
strndxid 15717 | The value of a structure c... |
reldmsets 15718 | The structure override ope... |
setsvalg 15719 | Value of the structure rep... |
setsval 15720 | Value of the structure rep... |
setsidvald 15721 | Value of the structure rep... |
fvsetsid 15722 | The value of the structure... |
fsets 15723 | The structure replacement ... |
setsdm 15724 | The domain of a structure ... |
setsfun 15725 | A structure with replaceme... |
setsfun0 15726 | A structure with replaceme... |
setsstruct 15727 | An extensible structure wi... |
wunsets 15728 | Closure of structure repla... |
setsres 15729 | The structure replacement ... |
setsabs 15730 | Replacing the same compone... |
setscom 15731 | Component-setting is commu... |
strfvd 15732 | Deduction version of ~ str... |
strfv2d 15733 | Deduction version of ~ str... |
strfv2 15734 | A variation on ~ strfv to ... |
strfv 15735 | Extract a structure compon... |
strfv3 15736 | Variant on ~ strfv for lar... |
strssd 15737 | Deduction version of ~ str... |
strss 15738 | Propagate component extrac... |
str0 15739 | All components of the empt... |
base0 15740 | The base set of the empty ... |
strfvi 15741 | Structure slot extractors ... |
setsid 15742 | Value of the structure rep... |
setsnid 15743 | Value of the structure rep... |
sbcie2s 15744 | A special version of class... |
sbcie3s 15745 | A special version of class... |
baseval 15746 | Value of the base set extr... |
baseid 15747 | Utility theorem: index-ind... |
elbasfv 15748 | Utility theorem: reverse c... |
elbasov 15749 | Utility theorem: reverse c... |
strov2rcl 15750 | Partial reverse closure fo... |
basendx 15751 | Index value of the base se... |
basendxnn 15752 | The index value of the bas... |
reldmress 15753 | The structure restriction ... |
ressval 15754 | Value of structure restric... |
ressid2 15755 | General behavior of trivia... |
ressval2 15756 | Value of nontrivial struct... |
ressbas 15757 | Base set of a structure re... |
ressbas2 15758 | Base set of a structure re... |
ressbasss 15759 | The base set of a restrict... |
resslem 15760 | Other elements of a struct... |
ress0 15761 | All restrictions of the nu... |
ressid 15762 | Behavior of trivial restri... |
ressinbas 15763 | Restriction only cares abo... |
ressval3d 15764 | Value of structure restric... |
ressress 15765 | Restriction composition la... |
ressabs 15766 | Restriction absorption law... |
wunress 15767 | Closure of structure restr... |
dfpleOLD 15789 | Obsolete version of ~ df-p... |
strlemor0 15795 | Structure definition utili... |
strlemor1 15796 | Add one element to the end... |
strlemor2 15797 | Add two elements to the en... |
strlemor3 15798 | Add three elements to the ... |
strleun 15799 | Combine two structures int... |
strle1 15800 | Make a structure from a si... |
strle2 15801 | Make a structure from a pa... |
strle3 15802 | Make a structure from a tr... |
plusgndx 15803 | Index value of the ~ df-pl... |
plusgid 15804 | Utility theorem: index-ind... |
1strstr 15805 | A constructed one-slot str... |
1strbas 15806 | The base set of a construc... |
1strwunbndx 15807 | A constructed one-slot str... |
1strwun 15808 | A constructed one-slot str... |
2strstr 15809 | A constructed two-slot str... |
2strbas 15810 | The base set of a construc... |
2strop 15811 | The other slot of a constr... |
2strstr1 15812 | A constructed two-slot str... |
2strbas1 15813 | The base set of a construc... |
2strop1 15814 | The other slot of a constr... |
grpstr 15815 | A constructed group is a s... |
grpbase 15816 | The base set of a construc... |
grpplusg 15817 | The operation of a constru... |
ressplusg 15818 | ` +g ` is unaffected by re... |
grpbasex 15819 | The base of an explicitly ... |
grpplusgx 15820 | The operation of an explic... |
mulrndx 15821 | Index value of the ~ df-mu... |
mulrid 15822 | Utility theorem: index-ind... |
rngstr 15823 | A constructed ring is a st... |
rngbase 15824 | The base set of a construc... |
rngplusg 15825 | The additive operation of ... |
rngmulr 15826 | The multiplicative operati... |
starvndx 15827 | Index value of the ~ df-st... |
starvid 15828 | Utility theorem: index-ind... |
ressmulr 15829 | ` .r ` is unaffected by re... |
ressstarv 15830 | ` *r ` is unaffected by re... |
srngfn 15831 | A constructed star ring is... |
srngbase 15832 | The base set of a construc... |
srngplusg 15833 | The addition operation of ... |
srngmulr 15834 | The multiplication operati... |
srnginvl 15835 | The involution function of... |
scandx 15836 | Index value of the ~ df-sc... |
scaid 15837 | Utility theorem: index-ind... |
vscandx 15838 | Index value of the ~ df-vs... |
vscaid 15839 | Utility theorem: index-ind... |
lmodstr 15840 | A constructed left module ... |
lmodbase 15841 | The base set of a construc... |
lmodplusg 15842 | The additive operation of ... |
lmodsca 15843 | The set of scalars of a co... |
lmodvsca 15844 | The scalar product operati... |
ipndx 15845 | Index value of the ~ df-ip... |
ipid 15846 | Utility theorem: index-ind... |
ipsstr 15847 | Lemma to shorten proofs of... |
ipsbase 15848 | The base set of a construc... |
ipsaddg 15849 | The additive operation of ... |
ipsmulr 15850 | The multiplicative operati... |
ipssca 15851 | The set of scalars of a co... |
ipsvsca 15852 | The scalar product operati... |
ipsip 15853 | The multiplicative operati... |
resssca 15854 | ` Scalar ` is unaffected b... |
ressvsca 15855 | ` .s ` is unaffected by re... |
ressip 15856 | The inner product is unaff... |
phlstr 15857 | A constructed pre-Hilbert ... |
phlbase 15858 | The base set of a construc... |
phlplusg 15859 | The additive operation of ... |
phlsca 15860 | The ring of scalars of a c... |
phlvsca 15861 | The scalar product operati... |
phlip 15862 | The inner product (Hermiti... |
tsetndx 15863 | Index value of the ~ df-ts... |
tsetid 15864 | Utility theorem: index-ind... |
topgrpstr 15865 | A constructed topological ... |
topgrpbas 15866 | The base set of a construc... |
topgrpplusg 15867 | The additive operation of ... |
topgrptset 15868 | The topology of a construc... |
resstset 15869 | ` TopSet ` is unaffected b... |
plendx 15870 | Index value of the ~ df-pl... |
plendxOLD 15871 | Obsolete version of ~ df-p... |
pleid 15872 | Utility theorem: self-refe... |
pleidOLD 15873 | Obsolete version of ~ otps... |
otpsstr 15874 | Functionality of a topolog... |
otpsbas 15875 | The base set of a topologi... |
otpstset 15876 | The open sets of a topolog... |
otpsle 15877 | The order of a topological... |
otpsstrOLD 15878 | Obsolete version of ~ otps... |
otpsbasOLD 15879 | Obsolete version of ~ otps... |
otpstsetOLD 15880 | Obsolete version of ~ otps... |
otpsleOLD 15881 | Obsolete version of ~ otps... |
ressle 15882 | ` le ` is unaffected by re... |
ocndx 15883 | Index value of the ~ df-oc... |
ocid 15884 | Utility theorem: index-ind... |
dsndx 15885 | Index value of the ~ df-ds... |
dsid 15886 | Utility theorem: index-ind... |
unifndx 15887 | Index value of the ~ df-un... |
unifid 15888 | Utility theorem: index-ind... |
odrngstr 15889 | Functionality of an ordere... |
odrngbas 15890 | The base set of an ordered... |
odrngplusg 15891 | The addition operation of ... |
odrngmulr 15892 | The multiplication operati... |
odrngtset 15893 | The open sets of an ordere... |
odrngle 15894 | The order of an ordered me... |
odrngds 15895 | The metric of an ordered m... |
ressds 15896 | ` dist ` is unaffected by ... |
homndx 15897 | Index value of the ~ df-ho... |
homid 15898 | Utility theorem: index-ind... |
ccondx 15899 | Index value of the ~ df-cc... |
ccoid 15900 | Utility theorem: index-ind... |
resshom 15901 | ` Hom ` is unaffected by r... |
ressco 15902 | ` comp ` is unaffected by ... |
slotsbhcdif 15903 | The slots ` Base ` , ` Hom... |
restfn 15908 | The subspace topology oper... |
topnfn 15909 | The topology extractor fun... |
restval 15910 | The subspace topology indu... |
elrest 15911 | The predicate "is an open ... |
elrestr 15912 | Sufficient condition for b... |
0rest 15913 | Value of the structure res... |
restid2 15914 | The subspace topology over... |
restsspw 15915 | The subspace topology is a... |
firest 15916 | The finite intersections o... |
restid 15917 | The subspace topology of t... |
topnval 15918 | Value of the topology extr... |
topnid 15919 | Value of the topology extr... |
topnpropd 15920 | The topology extractor fun... |
reldmprds 15932 | The structure product is a... |
prdsbasex 15934 | Lemma for structure produc... |
imasvalstr 15935 | Structure product value is... |
prdsvalstr 15936 | Structure product value is... |
prdsvallem 15937 | Lemma for ~ prdsbas and si... |
prdsval 15938 | Value of the structure pro... |
prdssca 15939 | Scalar ring of a structure... |
prdsbas 15940 | Base set of a structure pr... |
prdsplusg 15941 | Addition in a structure pr... |
prdsmulr 15942 | Multiplication in a struct... |
prdsvsca 15943 | Scalar multiplication in a... |
prdsip 15944 | Inner product in a structu... |
prdsle 15945 | Structure product weak ord... |
prdsless 15946 | Closure of the order relat... |
prdsds 15947 | Structure product distance... |
prdsdsfn 15948 | Structure product distance... |
prdstset 15949 | Structure product topology... |
prdshom 15950 | Structure product hom-sets... |
prdsco 15951 | Structure product composit... |
prdsbas2 15952 | The base set of a structur... |
prdsbasmpt 15953 | A constructed tuple is a p... |
prdsbasfn 15954 | Points in the structure pr... |
prdsbasprj 15955 | Each point in a structure ... |
prdsplusgval 15956 | Value of a componentwise s... |
prdsplusgfval 15957 | Value of a structure produ... |
prdsmulrval 15958 | Value of a componentwise r... |
prdsmulrfval 15959 | Value of a structure produ... |
prdsleval 15960 | Value of the product order... |
prdsdsval 15961 | Value of the metric in a s... |
prdsvscaval 15962 | Scalar multiplication in a... |
prdsvscafval 15963 | Scalar multiplication of a... |
prdsbas3 15964 | The base set of an indexed... |
prdsbasmpt2 15965 | A constructed tuple is a p... |
prdsbascl 15966 | An element of the base has... |
prdsdsval2 15967 | Value of the metric in a s... |
prdsdsval3 15968 | Value of the metric in a s... |
pwsval 15969 | Value of a structure power... |
pwsbas 15970 | Base set of a structure po... |
pwselbasb 15971 | Membership in the base set... |
pwselbas 15972 | An element of a structure ... |
pwsplusgval 15973 | Value of addition in a str... |
pwsmulrval 15974 | Value of multiplication in... |
pwsle 15975 | Ordering in a structure po... |
pwsleval 15976 | Ordering in a structure po... |
pwsvscafval 15977 | Scalar multiplication in a... |
pwsvscaval 15978 | Scalar multiplication of a... |
pwssca 15979 | The ring of scalars of a s... |
pwsdiagel 15980 | Membership of diagonal ele... |
pwssnf1o 15981 | Triviality of singleton po... |
imasval 15994 | Value of an image structur... |
imasbas 15995 | The base set of an image s... |
imasds 15996 | The distance function of a... |
imasdsfn 15997 | The distance function is a... |
imasdsval 15998 | The distance function of a... |
imasdsval2 15999 | The distance function of a... |
imasplusg 16000 | The group operation in an ... |
imasmulr 16001 | The ring multiplication in... |
imassca 16002 | The scalar field of an ima... |
imasvsca 16003 | The scalar multiplication ... |
imasip 16004 | The inner product of an im... |
imastset 16005 | The topology of an image s... |
imasle 16006 | The ordering of an image s... |
f1ocpbllem 16007 | Lemma for ~ f1ocpbl . (Co... |
f1ocpbl 16008 | An injection is compatible... |
f1ovscpbl 16009 | An injection is compatible... |
f1olecpbl 16010 | An injection is compatible... |
imasaddfnlem 16011 | The image structure operat... |
imasaddvallem 16012 | The operation of an image ... |
imasaddflem 16013 | The image set operations a... |
imasaddfn 16014 | The image structure's grou... |
imasaddval 16015 | The value of an image stru... |
imasaddf 16016 | The image structure's grou... |
imasmulfn 16017 | The image structure's ring... |
imasmulval 16018 | The value of an image stru... |
imasmulf 16019 | The image structure's ring... |
imasvscafn 16020 | The image structure's scal... |
imasvscaval 16021 | The value of an image stru... |
imasvscaf 16022 | The image structure's scal... |
imasless 16023 | The order relation defined... |
imasleval 16024 | The value of the image str... |
qusval 16025 | Value of a quotient struct... |
quslem 16026 | The function in ~ qusval i... |
qusin 16027 | Restrict the equivalence r... |
qusbas 16028 | Base set of a quotient str... |
quss 16029 | The scalar field of a quot... |
divsfval 16030 | Value of the function in ~... |
ercpbllem 16031 | Lemma for ~ ercpbl . (Con... |
ercpbl 16032 | Translate the function com... |
erlecpbl 16033 | Translate the relation com... |
qusaddvallem 16034 | Value of an operation defi... |
qusaddflem 16035 | The operation of a quotien... |
qusaddval 16036 | The base set of an image s... |
qusaddf 16037 | The base set of an image s... |
qusmulval 16038 | The base set of an image s... |
qusmulf 16039 | The base set of an image s... |
xpsc 16040 | A short expression for the... |
xpscg 16041 | A short expression for the... |
xpscfn 16042 | The pair function is a fun... |
xpsc0 16043 | The pair function maps ` 0... |
xpsc1 16044 | The pair function maps ` 1... |
xpscfv 16045 | The value of the pair func... |
xpsfrnel 16046 | Elementhood in the target ... |
xpsfeq 16047 | A function on ` 2o ` is de... |
xpsfrnel2 16048 | Elementhood in the target ... |
xpscf 16049 | Equivalent condition for t... |
xpsfval 16050 | The value of the function ... |
xpsff1o 16051 | The function appearing in ... |
xpsfrn 16052 | A short expression for the... |
xpsfrn2 16053 | A short expression for the... |
xpsff1o2 16054 | The function appearing in ... |
xpsval 16055 | Value of the binary struct... |
xpslem 16056 | The indexed structure prod... |
xpsbas 16057 | The base set of the binary... |
xpsaddlem 16058 | Lemma for ~ xpsadd and ~ x... |
xpsadd 16059 | Value of the addition oper... |
xpsmul 16060 | Value of the multiplicatio... |
xpssca 16061 | Value of the scalar field ... |
xpsvsca 16062 | Value of the scalar multip... |
xpsless 16063 | Closure of the ordering in... |
xpsle 16064 | Value of the ordering in a... |
ismre 16073 | Property of being a Moore ... |
fnmre 16074 | The Moore collection gener... |
mresspw 16075 | A Moore collection is a su... |
mress 16076 | A Moore-closed subset is a... |
mre1cl 16077 | In any Moore collection th... |
mreintcl 16078 | A nonempty collection of c... |
mreiincl 16079 | A nonempty indexed interse... |
mrerintcl 16080 | The relative intersection ... |
mreriincl 16081 | The relative intersection ... |
mreincl 16082 | Two closed sets have a clo... |
mreuni 16083 | Since the entire base set ... |
mreunirn 16084 | Two ways to express the no... |
ismred 16085 | Properties that determine ... |
ismred2 16086 | Properties that determine ... |
mremre 16087 | The Moore collections of s... |
submre 16088 | The subcollection of a clo... |
mrcflem 16089 | The domain and range of th... |
fnmrc 16090 | Moore-closure is a well-be... |
mrcfval 16091 | Value of the function expr... |
mrcf 16092 | The Moore closure is a fun... |
mrcval 16093 | Evaluation of the Moore cl... |
mrccl 16094 | The Moore closure of a set... |
mrcsncl 16095 | The Moore closure of a sin... |
mrcid 16096 | The closure of a closed se... |
mrcssv 16097 | The closure of a set is a ... |
mrcidb 16098 | A set is closed iff it is ... |
mrcss 16099 | Closure preserves subset o... |
mrcssid 16100 | The closure of a set is a ... |
mrcidb2 16101 | A set is closed iff it con... |
mrcidm 16102 | The closure operation is i... |
mrcsscl 16103 | The closure is the minimal... |
mrcuni 16104 | Idempotence of closure und... |
mrcun 16105 | Idempotence of closure und... |
mrcssvd 16106 | The Moore closure of a set... |
mrcssd 16107 | Moore closure preserves su... |
mrcssidd 16108 | A set is contained in its ... |
mrcidmd 16109 | Moore closure is idempoten... |
mressmrcd 16110 | In a Moore system, if a se... |
submrc 16111 | In a closure system which ... |
mrieqvlemd 16112 | In a Moore system, if ` Y ... |
mrisval 16113 | Value of the set of indepe... |
ismri 16114 | Criterion for a set to be ... |
ismri2 16115 | Criterion for a subset of ... |
ismri2d 16116 | Criterion for a subset of ... |
ismri2dd 16117 | Definition of independence... |
mriss 16118 | An independent set of a Mo... |
mrissd 16119 | An independent set of a Mo... |
ismri2dad 16120 | Consequence of a set in a ... |
mrieqvd 16121 | In a Moore system, a set i... |
mrieqv2d 16122 | In a Moore system, a set i... |
mrissmrcd 16123 | In a Moore system, if an i... |
mrissmrid 16124 | In a Moore system, subsets... |
mreexd 16125 | In a Moore system, the clo... |
mreexmrid 16126 | In a Moore system whose cl... |
mreexexlemd 16127 | This lemma is used to gene... |
mreexexlem2d 16128 | Used in ~ mreexexlem4d to ... |
mreexexlem3d 16129 | Base case of the induction... |
mreexexlem4d 16130 | Induction step of the indu... |
mreexexd 16131 | Exchange-type theorem. In... |
mreexexdOLD 16132 | Obsolete proof of ~ mreexe... |
mreexdomd 16133 | In a Moore system whose cl... |
mreexfidimd 16134 | In a Moore system whose cl... |
isacs 16135 | A set is an algebraic clos... |
acsmre 16136 | Algebraic closure systems ... |
isacs2 16137 | In the definition of an al... |
acsfiel 16138 | A set is closed in an alge... |
acsfiel2 16139 | A set is closed in an alge... |
acsmred 16140 | An algebraic closure syste... |
isacs1i 16141 | A closure system determine... |
mreacs 16142 | Algebraicity is a composab... |
acsfn 16143 | Algebraicity of a conditio... |
acsfn0 16144 | Algebraicity of a point cl... |
acsfn1 16145 | Algebraicity of a one-argu... |
acsfn1c 16146 | Algebraicity of a one-argu... |
acsfn2 16147 | Algebraicity of a two-argu... |
iscat 16156 | The predicate "is a catego... |
iscatd 16157 | Properties that determine ... |
catidex 16158 | Each object in a category ... |
catideu 16159 | Each object in a category ... |
cidfval 16160 | Each object in a category ... |
cidval 16161 | Each object in a category ... |
cidffn 16162 | The identity arrow constru... |
cidfn 16163 | The identity arrow operato... |
catidd 16164 | Deduce the identity arrow ... |
iscatd2 16165 | Version of ~ iscatd with a... |
catidcl 16166 | Each object in a category ... |
catlid 16167 | Left identity property of ... |
catrid 16168 | Right identity property of... |
catcocl 16169 | Closure of a composition a... |
catass 16170 | Associativity of compositi... |
0catg 16171 | Any structure with an empt... |
0cat 16172 | The empty set is a categor... |
homffval 16173 | Value of the functionalize... |
fnhomeqhomf 16174 | If the Hom-set operation i... |
homfval 16175 | Value of the functionalize... |
homffn 16176 | The functionalized Hom-set... |
homfeq 16177 | Condition for two categori... |
homfeqd 16178 | If two structures have the... |
homfeqbas 16179 | Deduce equality of base se... |
homfeqval 16180 | Value of the functionalize... |
comfffval 16181 | Value of the functionalize... |
comffval 16182 | Value of the functionalize... |
comfval 16183 | Value of the functionalize... |
comfffval2 16184 | Value of the functionalize... |
comffval2 16185 | Value of the functionalize... |
comfval2 16186 | Value of the functionalize... |
comfffn 16187 | The functionalized composi... |
comffn 16188 | The functionalized composi... |
comfeq 16189 | Condition for two categori... |
comfeqd 16190 | Condition for two categori... |
comfeqval 16191 | Equality of two compositio... |
catpropd 16192 | Two structures with the sa... |
cidpropd 16193 | Two structures with the sa... |
oppcval 16196 | Value of the opposite cate... |
oppchomfval 16197 | Hom-sets of the opposite c... |
oppchom 16198 | Hom-sets of the opposite c... |
oppccofval 16199 | Composition in the opposit... |
oppcco 16200 | Composition in the opposit... |
oppcbas 16201 | Base set of an opposite ca... |
oppccatid 16202 | Lemma for ~ oppccat . (Co... |
oppchomf 16203 | Hom-sets of the opposite c... |
oppcid 16204 | Identity function of an op... |
oppccat 16205 | An opposite category is a ... |
2oppcbas 16206 | The double opposite catego... |
2oppchomf 16207 | The double opposite catego... |
2oppccomf 16208 | The double opposite catego... |
oppchomfpropd 16209 | If two categories have the... |
oppccomfpropd 16210 | If two categories have the... |
monfval 16215 | Definition of a monomorphi... |
ismon 16216 | Definition of a monomorphi... |
ismon2 16217 | Write out the monomorphism... |
monhom 16218 | A monomorphism is a morphi... |
moni 16219 | Property of a monomorphism... |
monpropd 16220 | If two categories have the... |
oppcmon 16221 | A monomorphism in the oppo... |
oppcepi 16222 | An epimorphism in the oppo... |
isepi 16223 | Definition of an epimorphi... |
isepi2 16224 | Write out the epimorphism ... |
epihom 16225 | An epimorphism is a morphi... |
epii 16226 | Property of an epimorphism... |
sectffval 16233 | Value of the section opera... |
sectfval 16234 | Value of the section relat... |
sectss 16235 | The section relation is a ... |
issect 16236 | The property " ` F ` is a ... |
issect2 16237 | Property of being a sectio... |
sectcan 16238 | If ` G ` is a section of `... |
sectco 16239 | Composition of two section... |
isofval 16240 | Function value of the func... |
invffval 16241 | Value of the inverse relat... |
invfval 16242 | Value of the inverse relat... |
isinv 16243 | Value of the inverse relat... |
invss 16244 | The inverse relation is a ... |
invsym 16245 | The inverse relation is sy... |
invsym2 16246 | The inverse relation is sy... |
invfun 16247 | The inverse relation is a ... |
isoval 16248 | The isomorphisms are the d... |
inviso1 16249 | If ` G ` is an inverse to ... |
inviso2 16250 | If ` G ` is an inverse to ... |
invf 16251 | The inverse relation is a ... |
invf1o 16252 | The inverse relation is a ... |
invinv 16253 | The inverse of the inverse... |
invco 16254 | The composition of two iso... |
dfiso2 16255 | Alternate definition of an... |
dfiso3 16256 | Alternate definition of an... |
inveq 16257 | If there are two inverses ... |
isofn 16258 | The function value of the ... |
isohom 16259 | An isomorphism is a homomo... |
isoco 16260 | The composition of two iso... |
oppcsect 16261 | A section in the opposite ... |
oppcsect2 16262 | A section in the opposite ... |
oppcinv 16263 | An inverse in the opposite... |
oppciso 16264 | An isomorphism in the oppo... |
sectmon 16265 | If ` F ` is a section of `... |
monsect 16266 | If ` F ` is a monomorphism... |
sectepi 16267 | If ` F ` is a section of `... |
episect 16268 | If ` F ` is an epimorphism... |
sectid 16269 | The identity is a section ... |
invid 16270 | The inverse of the identit... |
idiso 16271 | The identity is an isomorp... |
idinv 16272 | The inverse of the identit... |
invisoinvl 16273 | The inverse of an isomorph... |
invisoinvr 16274 | The inverse of an isomorph... |
invcoisoid 16275 | The inverse of an isomorph... |
isocoinvid 16276 | The inverse of an isomorph... |
rcaninv 16277 | Right cancellation of an i... |
cicfval 16280 | The set of isomorphic obje... |
brcic 16281 | The relation "is isomorphi... |
cic 16282 | Objects ` X ` and ` Y ` in... |
brcici 16283 | Prove that two objects are... |
cicref 16284 | Isomorphism is reflexive. ... |
ciclcl 16285 | Isomorphism implies the le... |
cicrcl 16286 | Isomorphism implies the ri... |
cicsym 16287 | Isomorphism is symmetric. ... |
cictr 16288 | Isomorphism is transitive.... |
cicer 16289 | Isomorphism is an equivale... |
sscrel 16296 | The subcategory subset rel... |
brssc 16297 | The subcategory subset rel... |
sscpwex 16298 | An analogue of ~ pwex for ... |
subcrcl 16299 | Reverse closure for the su... |
sscfn1 16300 | The subcategory subset rel... |
sscfn2 16301 | The subcategory subset rel... |
ssclem 16302 | Lemma for ~ ssc1 and simil... |
isssc 16303 | Value of the subcategory s... |
ssc1 16304 | Infer subset relation on o... |
ssc2 16305 | Infer subset relation on m... |
sscres 16306 | Any function restricted to... |
sscid 16307 | The subcategory subset rel... |
ssctr 16308 | The subcategory subset rel... |
ssceq 16309 | The subcategory subset rel... |
rescval 16310 | Value of the category rest... |
rescval2 16311 | Value of the category rest... |
rescbas 16312 | Base set of the category r... |
reschom 16313 | Hom-sets of the category r... |
reschomf 16314 | Hom-sets of the category r... |
rescco 16315 | Composition in the categor... |
rescabs 16316 | Restriction absorption law... |
rescabs2 16317 | Restriction absorption law... |
issubc 16318 | Elementhood in the set of ... |
issubc2 16319 | Elementhood in the set of ... |
0ssc 16320 | For any category ` C ` , t... |
0subcat 16321 | For any category ` C ` , t... |
catsubcat 16322 | For any category ` C ` , `... |
subcssc 16323 | An element in the set of s... |
subcfn 16324 | An element in the set of s... |
subcss1 16325 | The objects of a subcatego... |
subcss2 16326 | The morphisms of a subcate... |
subcidcl 16327 | The identity of the origin... |
subccocl 16328 | A subcategory is closed un... |
subccatid 16329 | A subcategory is a categor... |
subcid 16330 | The identity in a subcateg... |
subccat 16331 | A subcategory is a categor... |
issubc3 16332 | Alternate definition of a ... |
fullsubc 16333 | The full subcategory gener... |
fullresc 16334 | The category formed by str... |
resscat 16335 | A category restricted to a... |
subsubc 16336 | A subcategory of a subcate... |
relfunc 16345 | The set of functors is a r... |
funcrcl 16346 | Reverse closure for a func... |
isfunc 16347 | Value of the set of functo... |
isfuncd 16348 | Deduce that an operation i... |
funcf1 16349 | The object part of a funct... |
funcixp 16350 | The morphism part of a fun... |
funcf2 16351 | The morphism part of a fun... |
funcfn2 16352 | The morphism part of a fun... |
funcid 16353 | A functor maps each identi... |
funcco 16354 | A functor maps composition... |
funcsect 16355 | The image of a section und... |
funcinv 16356 | The image of an inverse un... |
funciso 16357 | The image of an isomorphis... |
funcoppc 16358 | A functor on categories yi... |
idfuval 16359 | Value of the identity func... |
idfu2nd 16360 | Value of the morphism part... |
idfu2 16361 | Value of the morphism part... |
idfu1st 16362 | Value of the object part o... |
idfu1 16363 | Value of the object part o... |
idfucl 16364 | The identity functor is a ... |
cofuval 16365 | Value of the composition o... |
cofu1st 16366 | Value of the object part o... |
cofu1 16367 | Value of the object part o... |
cofu2nd 16368 | Value of the morphism part... |
cofu2 16369 | Value of the morphism part... |
cofuval2 16370 | Value of the composition o... |
cofucl 16371 | The composition of two fun... |
cofuass 16372 | Functor composition is ass... |
cofulid 16373 | The identity functor is a ... |
cofurid 16374 | The identity functor is a ... |
resfval 16375 | Value of the functor restr... |
resfval2 16376 | Value of the functor restr... |
resf1st 16377 | Value of the functor restr... |
resf2nd 16378 | Value of the functor restr... |
funcres 16379 | A functor restricted to a ... |
funcres2b 16380 | Condition for a functor to... |
funcres2 16381 | A functor into a restricte... |
wunfunc 16382 | A weak universe is closed ... |
funcpropd 16383 | If two categories have the... |
funcres2c 16384 | Condition for a functor to... |
fullfunc 16389 | A full functor is a functo... |
fthfunc 16390 | A faithful functor is a fu... |
relfull 16391 | The set of full functors i... |
relfth 16392 | The set of faithful functo... |
isfull 16393 | Value of the set of full f... |
isfull2 16394 | Equivalent condition for a... |
fullfo 16395 | The morphism map of a full... |
fulli 16396 | The morphism map of a full... |
isfth 16397 | Value of the set of faithf... |
isfth2 16398 | Equivalent condition for a... |
isffth2 16399 | A fully faithful functor i... |
fthf1 16400 | The morphism map of a fait... |
fthi 16401 | The morphism map of a fait... |
ffthf1o 16402 | The morphism map of a full... |
fullpropd 16403 | If two categories have the... |
fthpropd 16404 | If two categories have the... |
fulloppc 16405 | The opposite functor of a ... |
fthoppc 16406 | The opposite functor of a ... |
ffthoppc 16407 | The opposite functor of a ... |
fthsect 16408 | A faithful functor reflect... |
fthinv 16409 | A faithful functor reflect... |
fthmon 16410 | A faithful functor reflect... |
fthepi 16411 | A faithful functor reflect... |
ffthiso 16412 | A fully faithful functor r... |
fthres2b 16413 | Condition for a faithful f... |
fthres2c 16414 | Condition for a faithful f... |
fthres2 16415 | A faithful functor into a ... |
idffth 16416 | The identity functor is a ... |
cofull 16417 | The composition of two ful... |
cofth 16418 | The composition of two fai... |
coffth 16419 | The composition of two ful... |
rescfth 16420 | The inclusion functor from... |
ressffth 16421 | The inclusion functor from... |
fullres2c 16422 | Condition for a full funct... |
ffthres2c 16423 | Condition for a fully fait... |
fnfuc 16428 | The ` FuncCat ` operation ... |
natfval 16429 | Value of the function givi... |
isnat 16430 | Property of being a natura... |
isnat2 16431 | Property of being a natura... |
natffn 16432 | The natural transformation... |
natrcl 16433 | Reverse closure for a natu... |
nat1st2nd 16434 | Rewrite the natural transf... |
natixp 16435 | A natural transformation i... |
natcl 16436 | A component of a natural t... |
natfn 16437 | A natural transformation i... |
nati 16438 | Naturality property of a n... |
wunnat 16439 | A weak universe is closed ... |
catstr 16440 | A category structure is a ... |
fucval 16441 | Value of the functor categ... |
fuccofval 16442 | Value of the functor categ... |
fucbas 16443 | The objects of the functor... |
fuchom 16444 | The morphisms in the funct... |
fucco 16445 | Value of the composition o... |
fuccoval 16446 | Value of the functor categ... |
fuccocl 16447 | The composition of two nat... |
fucidcl 16448 | The identity natural trans... |
fuclid 16449 | Left identity of natural t... |
fucrid 16450 | Right identity of natural ... |
fucass 16451 | Associativity of natural t... |
fuccatid 16452 | The functor category is a ... |
fuccat 16453 | The functor category is a ... |
fucid 16454 | The identity morphism in t... |
fucsect 16455 | Two natural transformation... |
fucinv 16456 | Two natural transformation... |
invfuc 16457 | If ` V ( x ) ` is an inver... |
fuciso 16458 | A natural transformation i... |
natpropd 16459 | If two categories have the... |
fucpropd 16460 | If two categories have the... |
initorcl 16467 | Reverse closure for an ini... |
termorcl 16468 | Reverse closure for a term... |
zeroorcl 16469 | Reverse closure for a zero... |
initoval 16470 | The value of the initial o... |
termoval 16471 | The value of the terminal ... |
zerooval 16472 | The value of the zero obje... |
isinito 16473 | The predicate "is an initi... |
istermo 16474 | The predicate "is a termin... |
iszeroo 16475 | The predicate "is a zero o... |
isinitoi 16476 | Implication of a class bei... |
istermoi 16477 | Implication of a class bei... |
initoid 16478 | For an initial object, the... |
termoid 16479 | For a terminal object, the... |
initoo 16480 | An initial object is an ob... |
termoo 16481 | A terminal object is an ob... |
iszeroi 16482 | Implication of a class bei... |
2initoinv 16483 | Morphisms between two init... |
initoeu1 16484 | Initial objects are essent... |
initoeu1w 16485 | Initial objects are essent... |
initoeu2lem0 16486 | Lemma 0 for ~ initoeu2 . ... |
initoeu2lem1 16487 | Lemma 1 for ~ initoeu2 . ... |
initoeu2lem2 16488 | Lemma 2 for ~ initoeu2 . ... |
initoeu2 16489 | Initial objects are essent... |
2termoinv 16490 | Morphisms between two term... |
termoeu1 16491 | Terminal objects are essen... |
termoeu1w 16492 | Terminal objects are essen... |
homarcl 16501 | Reverse closure for an arr... |
homafval 16502 | Value of the disjointified... |
homaf 16503 | Functionality of the disjo... |
homaval 16504 | Value of the disjointified... |
elhoma 16505 | Value of the disjointified... |
elhomai 16506 | Produce an arrow from a mo... |
elhomai2 16507 | Produce an arrow from a mo... |
homarcl2 16508 | Reverse closure for the do... |
homarel 16509 | An arrow is an ordered pai... |
homa1 16510 | The first component of an ... |
homahom2 16511 | The second component of an... |
homahom 16512 | The second component of an... |
homadm 16513 | The domain of an arrow wit... |
homacd 16514 | The codomain of an arrow w... |
homadmcd 16515 | Decompose an arrow into do... |
arwval 16516 | The set of arrows is the u... |
arwrcl 16517 | The first component of an ... |
arwhoma 16518 | An arrow is contained in t... |
homarw 16519 | A hom-set is a subset of t... |
arwdm 16520 | The domain of an arrow is ... |
arwcd 16521 | The codomain of an arrow i... |
dmaf 16522 | The domain function is a f... |
cdaf 16523 | The codomain function is a... |
arwhom 16524 | The second component of an... |
arwdmcd 16525 | Decompose an arrow into do... |
idafval 16530 | Value of the identity arro... |
idaval 16531 | Value of the identity arro... |
ida2 16532 | Morphism part of the ident... |
idahom 16533 | Domain and codomain of the... |
idadm 16534 | Domain of the identity arr... |
idacd 16535 | Codomain of the identity a... |
idaf 16536 | The identity arrow functio... |
coafval 16537 | The value of the compositi... |
eldmcoa 16538 | A pair ` <. G , F >. ` is ... |
dmcoass 16539 | The domain of composition ... |
homdmcoa 16540 | If ` F : X --> Y ` and ` G... |
coaval 16541 | Value of composition for c... |
coa2 16542 | The morphism part of arrow... |
coahom 16543 | The composition of two com... |
coapm 16544 | Composition of arrows is a... |
arwlid 16545 | Left identity of a categor... |
arwrid 16546 | Right identity of a catego... |
arwass 16547 | Associativity of compositi... |
setcval 16550 | Value of the category of s... |
setcbas 16551 | Set of objects of the cate... |
setchomfval 16552 | Set of arrows of the categ... |
setchom 16553 | Set of arrows of the categ... |
elsetchom 16554 | A morphism of sets is a fu... |
setccofval 16555 | Composition in the categor... |
setcco 16556 | Composition in the categor... |
setccatid 16557 | Lemma for ~ setccat . (Co... |
setccat 16558 | The category of sets is a ... |
setcid 16559 | The identity arrow in the ... |
setcmon 16560 | A monomorphism of sets is ... |
setcepi 16561 | An epimorphism of sets is ... |
setcsect 16562 | A section in the category ... |
setcinv 16563 | An inverse in the category... |
setciso 16564 | An isomorphism in the cate... |
resssetc 16565 | The restriction of the cat... |
funcsetcres2 16566 | A functor into a smaller c... |
catcval 16569 | Value of the category of c... |
catcbas 16570 | Set of objects of the cate... |
catchomfval 16571 | Set of arrows of the categ... |
catchom 16572 | Set of arrows of the categ... |
catccofval 16573 | Composition in the categor... |
catcco 16574 | Composition in the categor... |
catccatid 16575 | Lemma for ~ catccat . (Co... |
catcid 16576 | The identity arrow in the ... |
catccat 16577 | The category of categories... |
resscatc 16578 | The restriction of the cat... |
catcisolem 16579 | Lemma for ~ catciso . (Co... |
catciso 16580 | A functor is an isomorphis... |
catcoppccl 16581 | The category of categories... |
catcfuccl 16582 | The category of categories... |
fncnvimaeqv 16583 | The inverse images of the ... |
bascnvimaeqv 16584 | The inverse image of the u... |
estrcval 16587 | Value of the category of e... |
estrcbas 16588 | Set of objects of the cate... |
estrchomfval 16589 | Set of morphisms ("arrows"... |
estrchom 16590 | The morphisms between exte... |
elestrchom 16591 | A morphism between extensi... |
estrccofval 16592 | Composition in the categor... |
estrcco 16593 | Composition in the categor... |
estrcbasbas 16594 | An element of the base set... |
estrccatid 16595 | Lemma for ~ estrccat . (C... |
estrccat 16596 | The category of extensible... |
estrcid 16597 | The identity arrow in the ... |
estrchomfn 16598 | The Hom-set operation in t... |
estrchomfeqhom 16599 | The functionalized Hom-set... |
estrreslem1 16600 | Lemma 1 for ~ estrres . (... |
estrreslem2 16601 | Lemma 2 for ~ estrres . (... |
estrres 16602 | Any restriction of a categ... |
funcestrcsetclem1 16603 | Lemma 1 for ~ funcestrcset... |
funcestrcsetclem2 16604 | Lemma 2 for ~ funcestrcset... |
funcestrcsetclem3 16605 | Lemma 3 for ~ funcestrcset... |
funcestrcsetclem4 16606 | Lemma 4 for ~ funcestrcset... |
funcestrcsetclem5 16607 | Lemma 5 for ~ funcestrcset... |
funcestrcsetclem6 16608 | Lemma 6 for ~ funcestrcset... |
funcestrcsetclem7 16609 | Lemma 7 for ~ funcestrcset... |
funcestrcsetclem8 16610 | Lemma 8 for ~ funcestrcset... |
funcestrcsetclem9 16611 | Lemma 9 for ~ funcestrcset... |
funcestrcsetc 16612 | The "natural forgetful fun... |
fthestrcsetc 16613 | The "natural forgetful fun... |
fullestrcsetc 16614 | The "natural forgetful fun... |
equivestrcsetc 16615 | The "natural forgetful fun... |
setc1strwun 16616 | A constructed one-slot str... |
funcsetcestrclem1 16617 | Lemma 1 for ~ funcsetcestr... |
funcsetcestrclem2 16618 | Lemma 2 for ~ funcsetcestr... |
funcsetcestrclem3 16619 | Lemma 3 for ~ funcsetcestr... |
embedsetcestrclem 16620 | Lemma for ~ embedsetcestrc... |
funcsetcestrclem4 16621 | Lemma 4 for ~ funcsetcestr... |
funcsetcestrclem5 16622 | Lemma 5 for ~ funcsetcestr... |
funcsetcestrclem6 16623 | Lemma 6 for ~ funcsetcestr... |
funcsetcestrclem7 16624 | Lemma 7 for ~ funcsetcestr... |
funcsetcestrclem8 16625 | Lemma 8 for ~ funcsetcestr... |
funcsetcestrclem9 16626 | Lemma 9 for ~ funcsetcestr... |
funcsetcestrc 16627 | The "embedding functor" fr... |
fthsetcestrc 16628 | The "embedding functor" fr... |
fullsetcestrc 16629 | The "embedding functor" fr... |
embedsetcestrc 16630 | The "embedding functor" fr... |
fnxpc 16639 | The binary product of cate... |
xpcval 16640 | Value of the binary produc... |
xpcbas 16641 | Set of objects of the bina... |
xpchomfval 16642 | Set of morphisms of the bi... |
xpchom 16643 | Set of morphisms of the bi... |
relxpchom 16644 | A hom-set in the binary pr... |
xpccofval 16645 | Value of composition in th... |
xpcco 16646 | Value of composition in th... |
xpcco1st 16647 | Value of composition in th... |
xpcco2nd 16648 | Value of composition in th... |
xpchom2 16649 | Value of the set of morphi... |
xpcco2 16650 | Value of composition in th... |
xpccatid 16651 | The product of two categor... |
xpcid 16652 | The identity morphism in t... |
xpccat 16653 | The product of two categor... |
1stfval 16654 | Value of the first project... |
1stf1 16655 | Value of the first project... |
1stf2 16656 | Value of the first project... |
2ndfval 16657 | Value of the first project... |
2ndf1 16658 | Value of the first project... |
2ndf2 16659 | Value of the first project... |
1stfcl 16660 | The first projection funct... |
2ndfcl 16661 | The second projection func... |
prfval 16662 | Value of the pairing funct... |
prf1 16663 | Value of the pairing funct... |
prf2fval 16664 | Value of the pairing funct... |
prf2 16665 | Value of the pairing funct... |
prfcl 16666 | The pairing of functors ` ... |
prf1st 16667 | Cancellation of pairing wi... |
prf2nd 16668 | Cancellation of pairing wi... |
1st2ndprf 16669 | Break a functor into a pro... |
catcxpccl 16670 | The category of categories... |
xpcpropd 16671 | If two categories have the... |
evlfval 16680 | Value of the evaluation fu... |
evlf2 16681 | Value of the evaluation fu... |
evlf2val 16682 | Value of the evaluation na... |
evlf1 16683 | Value of the evaluation fu... |
evlfcllem 16684 | Lemma for ~ evlfcl . (Con... |
evlfcl 16685 | The evaluation functor is ... |
curfval 16686 | Value of the curry functor... |
curf1fval 16687 | Value of the object part o... |
curf1 16688 | Value of the object part o... |
curf11 16689 | Value of the double evalua... |
curf12 16690 | The partially evaluated cu... |
curf1cl 16691 | The partially evaluated cu... |
curf2 16692 | Value of the curry functor... |
curf2val 16693 | Value of a component of th... |
curf2cl 16694 | The curry functor at a mor... |
curfcl 16695 | The curry functor of a fun... |
curfpropd 16696 | If two categories have the... |
uncfval 16697 | Value of the uncurry funct... |
uncfcl 16698 | The uncurry operation take... |
uncf1 16699 | Value of the uncurry funct... |
uncf2 16700 | Value of the uncurry funct... |
curfuncf 16701 | Cancellation of curry with... |
uncfcurf 16702 | Cancellation of uncurry wi... |
diagval 16703 | Define the diagonal functo... |
diagcl 16704 | The diagonal functor is a ... |
diag1cl 16705 | The constant functor of ` ... |
diag11 16706 | Value of the constant func... |
diag12 16707 | Value of the constant func... |
diag2 16708 | Value of the diagonal func... |
diag2cl 16709 | The diagonal functor at a ... |
curf2ndf 16710 | As shown in ~ diagval , th... |
hofval 16715 | Value of the Hom functor, ... |
hof1fval 16716 | The object part of the Hom... |
hof1 16717 | The object part of the Hom... |
hof2fval 16718 | The morphism part of the H... |
hof2val 16719 | The morphism part of the H... |
hof2 16720 | The morphism part of the H... |
hofcllem 16721 | Lemma for ~ hofcl . (Cont... |
hofcl 16722 | Closure of the Hom functor... |
oppchofcl 16723 | Closure of the opposite Ho... |
yonval 16724 | Value of the Yoneda embedd... |
yoncl 16725 | The Yoneda embedding is a ... |
yon1cl 16726 | The Yoneda embedding at an... |
yon11 16727 | Value of the Yoneda embedd... |
yon12 16728 | Value of the Yoneda embedd... |
yon2 16729 | Value of the Yoneda embedd... |
hofpropd 16730 | If two categories have the... |
yonpropd 16731 | If two categories have the... |
oppcyon 16732 | Value of the opposite Yone... |
oyoncl 16733 | The opposite Yoneda embedd... |
oyon1cl 16734 | The opposite Yoneda embedd... |
yonedalem1 16735 | Lemma for ~ yoneda . (Con... |
yonedalem21 16736 | Lemma for ~ yoneda . (Con... |
yonedalem3a 16737 | Lemma for ~ yoneda . (Con... |
yonedalem4a 16738 | Lemma for ~ yoneda . (Con... |
yonedalem4b 16739 | Lemma for ~ yoneda . (Con... |
yonedalem4c 16740 | Lemma for ~ yoneda . (Con... |
yonedalem22 16741 | Lemma for ~ yoneda . (Con... |
yonedalem3b 16742 | Lemma for ~ yoneda . (Con... |
yonedalem3 16743 | Lemma for ~ yoneda . (Con... |
yonedainv 16744 | The Yoneda Lemma with expl... |
yonffthlem 16745 | Lemma for ~ yonffth . (Co... |
yoneda 16746 | The Yoneda Lemma. There i... |
yonffth 16747 | The Yoneda Lemma. The Yon... |
yoniso 16748 | If the codomain is recover... |
isprs 16753 | Property of being a preord... |
prslem 16754 | Lemma for ~ prsref and ~ p... |
prsref 16755 | Less-or-equal is reflexive... |
prstr 16756 | Less-or-equal is transitiv... |
isdrs 16757 | Property of being a direct... |
drsdir 16758 | Direction of a directed se... |
drsprs 16759 | A directed set is a preset... |
drsbn0 16760 | The base of a directed set... |
drsdirfi 16761 | Any _finite_ number of ele... |
isdrs2 16762 | Directed sets may be defin... |
ispos 16770 | The predicate "is a poset.... |
ispos2 16771 | A poset is an antisymmetri... |
posprs 16772 | A poset is a preset. (Con... |
posi 16773 | Lemma for poset properties... |
posref 16774 | A poset ordering is reflex... |
posasymb 16775 | A poset ordering is asymme... |
postr 16776 | A poset ordering is transi... |
0pos 16777 | Technical lemma to simplif... |
isposd 16778 | Properties that determine ... |
isposi 16779 | Properties that determine ... |
isposix 16780 | Properties that determine ... |
pltfval 16782 | Value of the less-than rel... |
pltval 16783 | Less-than relation. ( ~ d... |
pltle 16784 | Less-than implies less-tha... |
pltne 16785 | Less-than relation. ( ~ d... |
pltirr 16786 | The less-than relation is ... |
pleval2i 16787 | One direction of ~ pleval2... |
pleval2 16788 | Less-than-or-equal in term... |
pltnle 16789 | Less-than implies not inve... |
pltval3 16790 | Alternate expression for l... |
pltnlt 16791 | The less-than relation imp... |
pltn2lp 16792 | The less-than relation has... |
plttr 16793 | The less-than relation is ... |
pltletr 16794 | Transitive law for chained... |
plelttr 16795 | Transitive law for chained... |
pospo 16796 | Write a poset structure in... |
lubfval 16801 | Value of the least upper b... |
lubdm 16802 | Domain of the least upper ... |
lubfun 16803 | The LUB is a function. (C... |
lubeldm 16804 | Member of the domain of th... |
lubelss 16805 | A member of the domain of ... |
lubeu 16806 | Unique existence proper of... |
lubval 16807 | Value of the least upper b... |
lubcl 16808 | The least upper bound func... |
lubprop 16809 | Properties of greatest low... |
luble 16810 | The greatest lower bound i... |
lublecllem 16811 | Lemma for ~ lublecl and ~ ... |
lublecl 16812 | The set of all elements le... |
lubid 16813 | The LUB of elements less t... |
glbfval 16814 | Value of the greatest lowe... |
glbdm 16815 | Domain of the greatest low... |
glbfun 16816 | The GLB is a function. (C... |
glbeldm 16817 | Member of the domain of th... |
glbelss 16818 | A member of the domain of ... |
glbeu 16819 | Unique existence proper of... |
glbval 16820 | Value of the greatest lowe... |
glbcl 16821 | The least upper bound func... |
glbprop 16822 | Properties of greatest low... |
glble 16823 | The greatest lower bound i... |
joinfval 16824 | Value of join function for... |
joinfval2 16825 | Value of join function for... |
joindm 16826 | Domain of join function fo... |
joindef 16827 | Two ways to say that a joi... |
joinval 16828 | Join value. Since both si... |
joincl 16829 | Closure of join of element... |
joindmss 16830 | Subset property of domain ... |
joinval2lem 16831 | Lemma for ~ joinval2 and ~... |
joinval2 16832 | Value of join for a poset ... |
joineu 16833 | Uniqueness of join of elem... |
joinlem 16834 | Lemma for join properties.... |
lejoin1 16835 | A join's first argument is... |
lejoin2 16836 | A join's second argument i... |
joinle 16837 | A join is less than or equ... |
meetfval 16838 | Value of meet function for... |
meetfval2 16839 | Value of meet function for... |
meetdm 16840 | Domain of meet function fo... |
meetdef 16841 | Two ways to say that a mee... |
meetval 16842 | Meet value. Since both si... |
meetcl 16843 | Closure of meet of element... |
meetdmss 16844 | Subset property of domain ... |
meetval2lem 16845 | Lemma for ~ meetval2 and ~... |
meetval2 16846 | Value of meet for a poset ... |
meeteu 16847 | Uniqueness of meet of elem... |
meetlem 16848 | Lemma for meet properties.... |
lemeet1 16849 | A meet's first argument is... |
lemeet2 16850 | A meet's second argument i... |
meetle 16851 | A meet is less than or equ... |
joincomALT 16852 | The join of a poset commut... |
joincom 16853 | The join of a poset commut... |
meetcomALT 16854 | The meet of a poset commut... |
meetcom 16855 | The meet of a poset commut... |
istos 16858 | The predicate "is a toset.... |
tosso 16859 | Write the totally ordered ... |
p0val 16864 | Value of poset zero. (Con... |
p1val 16865 | Value of poset zero. (Con... |
p0le 16866 | Any element is less than o... |
ple1 16867 | Any element is less than o... |
islat 16870 | The predicate "is a lattic... |
latcl2 16871 | The join and meet of any t... |
latlem 16872 | Lemma for lattice properti... |
latpos 16873 | A lattice is a poset. (Co... |
latjcl 16874 | Closure of join operation ... |
latmcl 16875 | Closure of meet operation ... |
latref 16876 | A lattice ordering is refl... |
latasymb 16877 | A lattice ordering is asym... |
latasym 16878 | A lattice ordering is asym... |
lattr 16879 | A lattice ordering is tran... |
latasymd 16880 | Deduce equality from latti... |
lattrd 16881 | A lattice ordering is tran... |
latjcom 16882 | The join of a lattice comm... |
latlej1 16883 | A join's first argument is... |
latlej2 16884 | A join's second argument i... |
latjle12 16885 | A join is less than or equ... |
latleeqj1 16886 | Less-than-or-equal-to in t... |
latleeqj2 16887 | Less-than-or-equal-to in t... |
latjlej1 16888 | Add join to both sides of ... |
latjlej2 16889 | Add join to both sides of ... |
latjlej12 16890 | Add join to both sides of ... |
latnlej 16891 | An idiom to express that a... |
latnlej1l 16892 | An idiom to express that a... |
latnlej1r 16893 | An idiom to express that a... |
latnlej2 16894 | An idiom to express that a... |
latnlej2l 16895 | An idiom to express that a... |
latnlej2r 16896 | An idiom to express that a... |
latjidm 16897 | Lattice join is idempotent... |
latmcom 16898 | The join of a lattice comm... |
latmle1 16899 | A meet is less than or equ... |
latmle2 16900 | A meet is less than or equ... |
latlem12 16901 | An element is less than or... |
latleeqm1 16902 | Less-than-or-equal-to in t... |
latleeqm2 16903 | Less-than-or-equal-to in t... |
latmlem1 16904 | Add meet to both sides of ... |
latmlem2 16905 | Add meet to both sides of ... |
latmlem12 16906 | Add join to both sides of ... |
latnlemlt 16907 | Negation of less-than-or-e... |
latnle 16908 | Equivalent expressions for... |
latmidm 16909 | Lattice join is idempotent... |
latabs1 16910 | Lattice absorption law. F... |
latabs2 16911 | Lattice absorption law. F... |
latledi 16912 | An ortholattice is distrib... |
latmlej11 16913 | Ordering of a meet and joi... |
latmlej12 16914 | Ordering of a meet and joi... |
latmlej21 16915 | Ordering of a meet and joi... |
latmlej22 16916 | Ordering of a meet and joi... |
lubsn 16917 | The least upper bound of a... |
latjass 16918 | Lattice join is associativ... |
latj12 16919 | Swap 1st and 2nd members o... |
latj32 16920 | Swap 2nd and 3rd members o... |
latj13 16921 | Swap 1st and 3rd members o... |
latj31 16922 | Swap 2nd and 3rd members o... |
latjrot 16923 | Rotate lattice join of 3 c... |
latj4 16924 | Rearrangement of lattice j... |
latj4rot 16925 | Rotate lattice join of 4 c... |
latjjdi 16926 | Lattice join distributes o... |
latjjdir 16927 | Lattice join distributes o... |
mod1ile 16928 | The weak direction of the ... |
mod2ile 16929 | The weak direction of the ... |
isclat 16932 | The predicate "is a comple... |
clatpos 16933 | A complete lattice is a po... |
clatlem 16934 | Lemma for properties of a ... |
clatlubcl 16935 | Any subset of the base set... |
clatlubcl2 16936 | Any subset of the base set... |
clatglbcl 16937 | Any subset of the base set... |
clatglbcl2 16938 | Any subset of the base set... |
clatl 16939 | A complete lattice is a la... |
isglbd 16940 | Properties that determine ... |
lublem 16941 | Lemma for the least upper ... |
lubub 16942 | The LUB of a complete latt... |
lubl 16943 | The LUB of a complete latt... |
lubss 16944 | Subset law for least upper... |
lubel 16945 | An element of a set is les... |
lubun 16946 | The LUB of a union. (Cont... |
clatglb 16947 | Properties of greatest low... |
clatglble 16948 | The greatest lower bound i... |
clatleglb 16949 | Two ways of expressing "le... |
clatglbss 16950 | Subset law for greatest lo... |
oduval 16953 | Value of an order dual str... |
oduleval 16954 | Value of the less-equal re... |
oduleg 16955 | Truth of the less-equal re... |
odubas 16956 | Base set of an order dual ... |
pospropd 16957 | Posethood is determined on... |
odupos 16958 | Being a poset is a self-du... |
oduposb 16959 | Being a poset is a self-du... |
meet0 16960 | Lemma for ~ odujoin . (Co... |
join0 16961 | Lemma for ~ odumeet . (Co... |
oduglb 16962 | Greatest lower bounds in a... |
odumeet 16963 | Meets in a dual order are ... |
odulub 16964 | Least upper bounds in a du... |
odujoin 16965 | Joins in a dual order are ... |
odulatb 16966 | Being a lattice is self-du... |
oduclatb 16967 | Being a complete lattice i... |
odulat 16968 | Being a lattice is self-du... |
poslubmo 16969 | Least upper bounds in a po... |
posglbmo 16970 | Greatest lower bounds in a... |
poslubd 16971 | Properties which determine... |
poslubdg 16972 | Properties which determine... |
posglbd 16973 | Properties which determine... |
ipostr 16976 | The structure of ~ df-ipo ... |
ipoval 16977 | Value of the inclusion pos... |
ipobas 16978 | Base set of the inclusion ... |
ipolerval 16979 | Relation of the inclusion ... |
ipotset 16980 | Topology of the inclusion ... |
ipole 16981 | Weak order condition of th... |
ipolt 16982 | Strict order condition of ... |
ipopos 16983 | The inclusion poset on a f... |
isipodrs 16984 | Condition for a family of ... |
ipodrscl 16985 | Direction by inclusion as ... |
ipodrsfi 16986 | Finite upper bound propert... |
fpwipodrs 16987 | The finite subsets of any ... |
ipodrsima 16988 | The monotone image of a di... |
isacs3lem 16989 | An algebraic closure syste... |
acsdrsel 16990 | An algebraic closure syste... |
isacs4lem 16991 | In a closure system in whi... |
isacs5lem 16992 | If closure commutes with d... |
acsdrscl 16993 | In an algebraic closure sy... |
acsficl 16994 | A closure in an algebraic ... |
isacs5 16995 | A closure system is algebr... |
isacs4 16996 | A closure system is algebr... |
isacs3 16997 | A closure system is algebr... |
acsficld 16998 | In an algebraic closure sy... |
acsficl2d 16999 | In an algebraic closure sy... |
acsfiindd 17000 | In an algebraic closure sy... |
acsmapd 17001 | In an algebraic closure sy... |
acsmap2d 17002 | In an algebraic closure sy... |
acsinfd 17003 | In an algebraic closure sy... |
acsdomd 17004 | In an algebraic closure sy... |
acsinfdimd 17005 | In an algebraic closure sy... |
acsexdimd 17006 | In an algebraic closure sy... |
mrelatglb 17007 | Greatest lower bounds in a... |
mrelatglb0 17008 | The empty intersection in ... |
mrelatlub 17009 | Least upper bounds in a Mo... |
mreclatBAD 17010 | A Moore space is a complet... |
latmass 17011 | Lattice meet is associativ... |
latdisdlem 17012 | Lemma for ~ latdisd . (Co... |
latdisd 17013 | In a lattice, joins distri... |
isdlat 17016 | Property of being a distri... |
dlatmjdi 17017 | In a distributive lattice,... |
dlatl 17018 | A distributive lattice is ... |
odudlatb 17019 | The dual of a distributive... |
dlatjmdi 17020 | In a distributive lattice,... |
isps 17025 | The predicate "is a poset"... |
psrel 17026 | A poset is a relation. (C... |
psref2 17027 | A poset is antisymmetric a... |
pstr2 17028 | A poset is transitive. (C... |
pslem 17029 | Lemma for ~ psref and othe... |
psdmrn 17030 | The domain and range of a ... |
psref 17031 | A poset is reflexive. (Co... |
psrn 17032 | The range of a poset equal... |
psasym 17033 | A poset is antisymmetric. ... |
pstr 17034 | A poset is transitive. (C... |
cnvps 17035 | The converse of a poset is... |
cnvpsb 17036 | The converse of a poset is... |
psss 17037 | Any subset of a partially ... |
psssdm2 17038 | Field of a subposet. (Con... |
psssdm 17039 | Field of a subposet. (Con... |
istsr 17040 | The predicate is a toset. ... |
istsr2 17041 | The predicate is a toset. ... |
tsrlin 17042 | A toset is a linear order.... |
tsrlemax 17043 | Two ways of saying a numbe... |
tsrps 17044 | A toset is a poset. (Cont... |
cnvtsr 17045 | The converse of a toset is... |
tsrss 17046 | Any subset of a totally or... |
ledm 17047 | domain of ` <_ ` is ` RR* ... |
lern 17048 | The range of ` <_ ` is ` R... |
lefld 17049 | The field of the 'less or ... |
letsr 17050 | The "less than or equal to... |
isdir 17055 | A condition for a relation... |
reldir 17056 | A direction is a relation.... |
dirdm 17057 | A direction's domain is eq... |
dirref 17058 | A direction is reflexive. ... |
dirtr 17059 | A direction is transitive.... |
dirge 17060 | For any two elements of a ... |
tsrdir 17061 | A totally ordered set is a... |
ismgm 17066 | The predicate "is a magma"... |
ismgmn0 17067 | The predicate "is a magma"... |
mgmcl 17068 | Closure of the operation o... |
isnmgm 17069 | A condition for a structur... |
plusffval 17070 | The group addition operati... |
plusfval 17071 | The group addition operati... |
plusfeq 17072 | If the addition operation ... |
plusffn 17073 | The group addition operati... |
mgmplusf 17074 | The group addition functio... |
issstrmgm 17075 | Characterize a substructur... |
intopsn 17076 | The internal operation for... |
mgmb1mgm1 17077 | The only magma with a base... |
mgm0 17078 | Any set with an empty base... |
mgm0b 17079 | The structure with an empt... |
mgm1 17080 | The structure with one ele... |
opifismgm 17081 | A structure with a group a... |
mgmidmo 17082 | A two-sided identity eleme... |
grpidval 17083 | The value of the identity ... |
grpidpropd 17084 | If two structures have the... |
fn0g 17085 | The group zero extractor i... |
0g0 17086 | The identity element funct... |
ismgmid 17087 | The identity element of a ... |
mgmidcl 17088 | The identity element of a ... |
mgmlrid 17089 | The identity element of a ... |
ismgmid2 17090 | Show that a given element ... |
grpidd 17091 | Deduce the identity elemen... |
mgmidsssn0 17092 | Property of the set of ide... |
gsumvalx 17093 | Expand out the substitutio... |
gsumval 17094 | Expand out the substitutio... |
gsumpropd 17095 | The group sum depends only... |
gsumpropd2lem 17096 | Lemma for ~ gsumpropd2 . ... |
gsumpropd2 17097 | A stronger version of ~ gs... |
gsummgmpropd 17098 | A stronger version of ~ gs... |
gsumress 17099 | The group sum in a substru... |
gsumval1 17100 | Value of the group sum ope... |
gsum0 17101 | Value of the empty group s... |
gsumval2a 17102 | Value of the group sum ope... |
gsumval2 17103 | Value of the group sum ope... |
gsumprval 17104 | Value of the group sum ope... |
gsumpr12val 17105 | Value of the group sum ope... |
issgrp 17108 | The predicate "is a semigr... |
issgrpv 17109 | The predicate "is a semigr... |
issgrpn0 17110 | The predicate "is a semigr... |
isnsgrp 17111 | A condition for a structur... |
sgrpmgm 17112 | A semigroup is a magma. (... |
sgrpass 17113 | A semigroup operation is a... |
sgrp0 17114 | Any set with an empty base... |
sgrp0b 17115 | The structure with an empt... |
sgrp1 17116 | The structure with one ele... |
ismnddef 17119 | The predicate "is a monoid... |
ismnd 17120 | The predicate "is a monoid... |
isnmnd 17121 | A condition for a structur... |
mndsgrp 17122 | A monoid is a semigroup. ... |
mndmgm 17123 | A monoid is a magma. (Con... |
mndcl 17124 | Closure of the operation o... |
mndass 17125 | A monoid operation is asso... |
mndid 17126 | A monoid has a two-sided i... |
mndideu 17127 | The two-sided identity ele... |
mnd32g 17128 | Commutative/associative la... |
mnd12g 17129 | Commutative/associative la... |
mnd4g 17130 | Commutative/associative la... |
mndidcl 17131 | The identity element of a ... |
mndplusf 17132 | The group addition operati... |
mndlrid 17133 | A monoid's identity elemen... |
mndlid 17134 | The identity element of a ... |
mndrid 17135 | The identity element of a ... |
ismndd 17136 | Deduce a monoid from its p... |
mndpfo 17137 | The addition operation of ... |
mndfo 17138 | The addition operation of ... |
mndpropd 17139 | If two structures have the... |
mndprop 17140 | If two structures have the... |
issubmnd 17141 | Characterize a submonoid b... |
ress0g 17142 | ` 0g ` is unaffected by re... |
submnd0 17143 | The zero of a submonoid is... |
prdsplusgcl 17144 | Structure product pointwis... |
prdsidlem 17145 | Characterization of identi... |
prdsmndd 17146 | The product of a family of... |
prds0g 17147 | Zero in a product of monoi... |
pwsmnd 17148 | The structure power of a m... |
pws0g 17149 | Zero in a product of monoi... |
imasmnd2 17150 | The image structure of a m... |
imasmnd 17151 | The image structure of a m... |
imasmndf1 17152 | The image of a monoid unde... |
xpsmnd 17153 | The binary product of mono... |
mnd1 17154 | The (smallest) structure r... |
mnd1id 17155 | The singleton element of a... |
ismhm 17160 | Property of a monoid homom... |
mhmrcl1 17161 | Reverse closure of a monoi... |
mhmrcl2 17162 | Reverse closure of a monoi... |
mhmf 17163 | A monoid homomorphism is a... |
mhmpropd 17164 | Monoid homomorphism depend... |
mhmlin 17165 | A monoid homomorphism comm... |
mhm0 17166 | A monoid homomorphism pres... |
idmhm 17167 | The identity homomorphism ... |
mhmf1o 17168 | A monoid homomorphism is b... |
submrcl 17169 | Reverse closure for submon... |
issubm 17170 | Expand definition of a sub... |
issubm2 17171 | Submonoids are subsets tha... |
issubmd 17172 | Deduction for proving a su... |
submss 17173 | Submonoids are subsets of ... |
submid 17174 | Every monoid is trivially ... |
subm0cl 17175 | Submonoids contain zero. ... |
submcl 17176 | Submonoids are closed unde... |
submmnd 17177 | Submonoids are themselves ... |
submbas 17178 | The base set of a submonoi... |
subm0 17179 | Submonoids have the same i... |
subsubm 17180 | A submonoid of a submonoid... |
0mhm 17181 | The constant zero linear f... |
resmhm 17182 | Restriction of a monoid ho... |
resmhm2 17183 | One direction of ~ resmhm2... |
resmhm2b 17184 | Restriction of the codomai... |
mhmco 17185 | The composition of monoid ... |
mhmima 17186 | The homomorphic image of a... |
mhmeql 17187 | The equalizer of two monoi... |
submacs 17188 | Submonoids are an algebrai... |
mrcmndind 17189 | (( From SO's determinants ... |
prdspjmhm 17190 | A projection from a produc... |
pwspjmhm 17191 | A projection from a produc... |
pwsdiagmhm 17192 | Diagonal monoid homomorphi... |
pwsco1mhm 17193 | Right composition with a f... |
pwsco2mhm 17194 | Left composition with a mo... |
gsumvallem2 17195 | Lemma for properties of th... |
gsumsubm 17196 | Evaluate a group sum in a ... |
gsumz 17197 | Value of a group sum over ... |
gsumwsubmcl 17198 | Closure of the composite i... |
gsumws1 17199 | A singleton composite reco... |
gsumwcl 17200 | Closure of the composite o... |
gsumccat 17201 | Homomorphic property of co... |
gsumws2 17202 | Valuation of a pair in a m... |
gsumccatsn 17203 | Homomorphic property of co... |
gsumspl 17204 | The primary purpose of the... |
gsumwmhm 17205 | Behavior of homomorphisms ... |
gsumwspan 17206 | The submonoid generated by... |
frmdval 17211 | Value of the free monoid c... |
frmdbas 17212 | The base set of a free mon... |
frmdelbas 17213 | An element of the base set... |
frmdplusg 17214 | The monoid operation of a ... |
frmdadd 17215 | Value of the monoid operat... |
vrmdfval 17216 | The canonical injection fr... |
vrmdval 17217 | The value of the generatin... |
vrmdf 17218 | The mapping from the index... |
frmdmnd 17219 | A free monoid is a monoid.... |
frmd0 17220 | The identity of the free m... |
frmdsssubm 17221 | The set of words taking va... |
frmdgsum 17222 | Any word in a free monoid ... |
frmdss2 17223 | A subset of generators is ... |
frmdup1 17224 | Any assignment of the gene... |
frmdup2 17225 | The evaluation map has the... |
frmdup3lem 17226 | Lemma for ~ frmdup3 . (Co... |
frmdup3 17227 | Universal property of the ... |
mgm2nsgrplem1 17228 | Lemma 1 for ~ mgm2nsgrp : ... |
mgm2nsgrplem2 17229 | Lemma 2 for ~ mgm2nsgrp . ... |
mgm2nsgrplem3 17230 | Lemma 3 for ~ mgm2nsgrp . ... |
mgm2nsgrplem4 17231 | Lemma 4 for ~ mgm2nsgrp : ... |
mgm2nsgrp 17232 | A small magma (with two el... |
sgrp2nmndlem1 17233 | Lemma 1 for ~ sgrp2nmnd : ... |
sgrp2nmndlem2 17234 | Lemma 2 for ~ sgrp2nmnd . ... |
sgrp2nmndlem3 17235 | Lemma 3 for ~ sgrp2nmnd . ... |
sgrp2rid2 17236 | A small semigroup (with tw... |
sgrp2rid2ex 17237 | A small semigroup (with tw... |
sgrp2nmndlem4 17238 | Lemma 4 for ~ sgrp2nmnd : ... |
sgrp2nmndlem5 17239 | Lemma 5 for ~ sgrp2nmnd : ... |
sgrp2nmnd 17240 | A small semigroup (with tw... |
mgmnsgrpex 17241 | There is a magma which is ... |
sgrpnmndex 17242 | There is a semigroup which... |
sgrpssmgm 17243 | The class of all semigroup... |
mndsssgrp 17244 | The class of all monoids i... |
isgrp 17251 | The predicate "is a group.... |
grpmnd 17252 | A group is a monoid. (Con... |
grpcl 17253 | Closure of the operation o... |
grpass 17254 | A group operation is assoc... |
grpinvex 17255 | Every member of a group ha... |
grpideu 17256 | The two-sided identity ele... |
grpplusf 17257 | The group addition operati... |
grpplusfo 17258 | The group addition operati... |
resgrpplusfrn 17259 | The underlying set of a gr... |
grppropd 17260 | If two structures have the... |
grpprop 17261 | If two structures have the... |
grppropstr 17262 | Generalize a specific 2-el... |
grpss 17263 | Show that a structure exte... |
isgrpd2e 17264 | Deduce a group from its pr... |
isgrpd2 17265 | Deduce a group from its pr... |
isgrpde 17266 | Deduce a group from its pr... |
isgrpd 17267 | Deduce a group from its pr... |
isgrpi 17268 | Properties that determine ... |
grpsgrp 17269 | A group is a semigroup. (... |
dfgrp2 17270 | Alternate definition of a ... |
dfgrp2e 17271 | Alternate definition of a ... |
isgrpix 17272 | Properties that determine ... |
grpidcl 17273 | The identity element of a ... |
grpbn0 17274 | The base set of a group is... |
grplid 17275 | The identity element of a ... |
grprid 17276 | The identity element of a ... |
grpn0 17277 | A group is not empty. (Co... |
grprcan 17278 | Right cancellation law for... |
grpinveu 17279 | The left inverse element o... |
grpid 17280 | Two ways of saying that an... |
isgrpid2 17281 | Properties showing that an... |
grpidd2 17282 | Deduce the identity elemen... |
grpinvfval 17283 | The inverse function of a ... |
grpinvval 17284 | The inverse of a group ele... |
grpinvfn 17285 | Functionality of the group... |
grpinvfvi 17286 | The group inverse function... |
grpsubfval 17287 | Group subtraction (divisio... |
grpsubval 17288 | Group subtraction (divisio... |
grpinvf 17289 | The group inversion operat... |
grpinvcl 17290 | A group element's inverse ... |
grplinv 17291 | The left inverse of a grou... |
grprinv 17292 | The right inverse of a gro... |
grpinvid1 17293 | The inverse of a group ele... |
grpinvid2 17294 | The inverse of a group ele... |
isgrpinv 17295 | Properties showing that a ... |
grplrinv 17296 | In a group, every member h... |
grpidinv2 17297 | A group's properties using... |
grpidinv 17298 | A group has a left and rig... |
grpinvid 17299 | The inverse of the identit... |
grplcan 17300 | Left cancellation law for ... |
grpasscan1 17301 | An associative cancellatio... |
grpasscan2 17302 | An associative cancellatio... |
grpidrcan 17303 | If right adding an element... |
grpidlcan 17304 | If left adding an element ... |
grpinvinv 17305 | Double inverse law for gro... |
grpinvcnv 17306 | The group inverse is its o... |
grpinv11 17307 | The group inverse is one-t... |
grpinvf1o 17308 | The group inverse is a one... |
grpinvnz 17309 | The inverse of a nonzero g... |
grpinvnzcl 17310 | The inverse of a nonzero g... |
grpsubinv 17311 | Subtraction of an inverse.... |
grplmulf1o 17312 | Left multiplication by a g... |
grpinvpropd 17313 | If two structures have the... |
grpidssd 17314 | If the base set of a group... |
grpinvssd 17315 | If the base set of a group... |
grpinvadd 17316 | The inverse of the group o... |
grpsubf 17317 | Functionality of group sub... |
grpsubcl 17318 | Closure of group subtracti... |
grpsubrcan 17319 | Right cancellation law for... |
grpinvsub 17320 | Inverse of a group subtrac... |
grpinvval2 17321 | A ~ df-neg -like equation ... |
grpsubid 17322 | Subtraction of a group ele... |
grpsubid1 17323 | Subtraction of the identit... |
grpsubeq0 17324 | If the difference between ... |
grpsubadd0sub 17325 | Subtraction expressed as a... |
grpsubadd 17326 | Relationship between group... |
grpsubsub 17327 | Double group subtraction. ... |
grpaddsubass 17328 | Associative-type law for g... |
grppncan 17329 | Cancellation law for subtr... |
grpnpcan 17330 | Cancellation law for subtr... |
grpsubsub4 17331 | Double group subtraction (... |
grppnpcan2 17332 | Cancellation law for mixed... |
grpnpncan 17333 | Cancellation law for group... |
grpnpncan0 17334 | Cancellation law for group... |
grpnnncan2 17335 | Cancellation law for group... |
dfgrp3lem 17336 | Lemma for ~ dfgrp3 . (Con... |
dfgrp3 17337 | Alternate definition of a ... |
dfgrp3e 17338 | Alternate definition of a ... |
grplactfval 17339 | The left group action of e... |
grplactval 17340 | The value of the left grou... |
grplactcnv 17341 | The left group action of e... |
grplactf1o 17342 | The left group action of e... |
grpsubpropd 17343 | Weak property deduction fo... |
grpsubpropd2 17344 | Strong property deduction ... |
grp1 17345 | The (smallest) structure r... |
grp1inv 17346 | The inverse function of th... |
prdsinvlem 17347 | Characterization of invers... |
prdsgrpd 17348 | The product of a family of... |
prdsinvgd 17349 | Negation in a product of g... |
pwsgrp 17350 | The product of a family of... |
pwsinvg 17351 | Negation in a group power.... |
pwssub 17352 | Subtraction in a group pow... |
imasgrp2 17353 | The image structure of a g... |
imasgrp 17354 | The image structure of a g... |
imasgrpf1 17355 | The image of a group under... |
qusgrp2 17356 | Prove that a quotient stru... |
xpsgrp 17357 | The binary product of grou... |
mhmlem 17358 | Lemma for ~ mhmmnd and ~ g... |
mhmid 17359 | A surjective monoid morphi... |
mhmmnd 17360 | The image of a monoid ` G ... |
mhmfmhm 17361 | The function fulfilling th... |
ghmgrp 17362 | The image of a group ` G `... |
mulgfval 17365 | Group multiple (exponentia... |
mulgval 17366 | Value of the group multipl... |
mulgfn 17367 | Functionality of the group... |
mulgfvi 17368 | The group multiple operati... |
mulg0 17369 | Group multiple (exponentia... |
mulgnn 17370 | Group multiple (exponentia... |
mulg1 17371 | Group multiple (exponentia... |
mulgnnp1 17372 | Group multiple (exponentia... |
mulg2 17373 | Group multiple (exponentia... |
mulgnegnn 17374 | Group multiple (exponentia... |
mulgnn0p1 17375 | Group multiple (exponentia... |
mulgnnsubcl 17376 | Closure of the group multi... |
mulgnn0subcl 17377 | Closure of the group multi... |
mulgsubcl 17378 | Closure of the group multi... |
mulgnncl 17379 | Closure of the group multi... |
mulgnnclOLD 17380 | Obsolete proof of ~ mulgnn... |
mulgnn0cl 17381 | Closure of the group multi... |
mulgcl 17382 | Closure of the group multi... |
mulgneg 17383 | Group multiple (exponentia... |
mulgnegneg 17384 | The inverse of a negative ... |
mulgm1 17385 | Group multiple (exponentia... |
mulgaddcomlem 17386 | Lemma for ~ mulgaddcom . ... |
mulgaddcom 17387 | The group multiple operato... |
mulginvcom 17388 | The group multiple operato... |
mulginvinv 17389 | The group multiple operato... |
mulgnn0z 17390 | A group multiple of the id... |
mulgz 17391 | A group multiple of the id... |
mulgnndir 17392 | Sum of group multiples, fo... |
mulgnndirOLD 17393 | Obsolete proof of ~ mulgnn... |
mulgnn0dir 17394 | Sum of group multiples, ge... |
mulgdirlem 17395 | Lemma for ~ mulgdir . (Co... |
mulgdir 17396 | Sum of group multiples, ge... |
mulgp1 17397 | Group multiple (exponentia... |
mulgneg2 17398 | Group multiple (exponentia... |
mulgnnass 17399 | Product of group multiples... |
mulgnnassOLD 17400 | Obsolete proof of ~ mulgnn... |
mulgnn0ass 17401 | Product of group multiples... |
mulgass 17402 | Product of group multiples... |
mulgassr 17403 | Reversed product of group ... |
mulgmodid 17404 | Casting out multiples of t... |
mulgsubdir 17405 | Subtraction of a group ele... |
mhmmulg 17406 | A homomorphism of monoids ... |
mulgpropd 17407 | Two structures with the sa... |
submmulgcl 17408 | Closure of the group multi... |
submmulg 17409 | A group multiple is the sa... |
pwsmulg 17410 | Value of a group multiple ... |
issubg 17417 | The subgroup predicate. (... |
subgss 17418 | A subgroup is a subset. (... |
subgid 17419 | A group is a subgroup of i... |
subggrp 17420 | A subgroup is a group. (C... |
subgbas 17421 | The base of the restricted... |
subgrcl 17422 | Reverse closure for the su... |
subg0 17423 | A subgroup of a group must... |
subginv 17424 | The inverse of an element ... |
subg0cl 17425 | The group identity is an e... |
subginvcl 17426 | The inverse of an element ... |
subgcl 17427 | A subgroup is closed under... |
subgsubcl 17428 | A subgroup is closed under... |
subgsub 17429 | The subtraction of element... |
subgmulgcl 17430 | Closure of the group multi... |
subgmulg 17431 | A group multiple is the sa... |
issubg2 17432 | Characterize the subgroups... |
issubgrpd2 17433 | Prove a subgroup by closur... |
issubgrpd 17434 | Prove a subgroup by closur... |
issubg3 17435 | A subgroup is a symmetric ... |
issubg4 17436 | A subgroup is a nonempty s... |
grpissubg 17437 | If the base set of a group... |
resgrpisgrp 17438 | If the base set of a group... |
subgsubm 17439 | A subgroup is a submonoid.... |
subsubg 17440 | A subgroup of a subgroup i... |
subgint 17441 | The intersection of a none... |
0subg 17442 | The zero subgroup of an ar... |
cycsubgcl 17443 | The set of integer powers ... |
cycsubgss 17444 | The cyclic subgroup genera... |
cycsubg 17445 | The cyclic group generated... |
isnsg 17446 | Property of being a normal... |
isnsg2 17447 | Weaken the condition of ~ ... |
nsgbi 17448 | Defining property of a nor... |
nsgsubg 17449 | A normal subgroup is a sub... |
nsgconj 17450 | The conjugation of an elem... |
isnsg3 17451 | A subgroup is normal iff t... |
subgacs 17452 | Subgroups are an algebraic... |
nsgacs 17453 | Normal subgroups form an a... |
cycsubg2 17454 | The subgroup generated by ... |
cycsubg2cl 17455 | Any multiple of an element... |
elnmz 17456 | Elementhood in the normali... |
nmzbi 17457 | Defining property of the n... |
nmzsubg 17458 | The normalizer N_G(S) of a... |
ssnmz 17459 | A subgroup is a subset of ... |
isnsg4 17460 | A subgroup is normal iff i... |
nmznsg 17461 | Any subgroup is a normal s... |
0nsg 17462 | The zero subgroup is norma... |
nsgid 17463 | The whole group is a norma... |
releqg 17464 | The left coset equivalence... |
eqgfval 17465 | Value of the subgroup left... |
eqgval 17466 | Value of the subgroup left... |
eqger 17467 | The subgroup coset equival... |
eqglact 17468 | A left coset can be expres... |
eqgid 17469 | The left coset containing ... |
eqgen 17470 | Each coset is equipotent t... |
eqgcpbl 17471 | The subgroup coset equival... |
qusgrp 17472 | If ` Y ` is a normal subgr... |
quseccl 17473 | Closure of the quotient ma... |
qusadd 17474 | Value of the group operati... |
qus0 17475 | Value of the group identit... |
qusinv 17476 | Value of the group inverse... |
qussub 17477 | Value of the group subtrac... |
lagsubg2 17478 | Lagrange's theorem for fin... |
lagsubg 17479 | Lagrange theorem for Group... |
reldmghm 17482 | Lemma for group homomorphi... |
isghm 17483 | Property of being a homomo... |
isghm3 17484 | Property of a group homomo... |
ghmgrp1 17485 | A group homomorphism is on... |
ghmgrp2 17486 | A group homomorphism is on... |
ghmf 17487 | A group homomorphism is a ... |
ghmlin 17488 | A homomorphism of groups i... |
ghmid 17489 | A homomorphism of groups p... |
ghminv 17490 | A homomorphism of groups p... |
ghmsub 17491 | Linearity of subtraction t... |
isghmd 17492 | Deduction for a group homo... |
ghmmhm 17493 | A group homomorphism is a ... |
ghmmhmb 17494 | Group homomorphisms and mo... |
ghmmulg 17495 | A homomorphism of monoids ... |
ghmrn 17496 | The range of a homomorphis... |
0ghm 17497 | The constant zero linear f... |
idghm 17498 | The identity homomorphism ... |
resghm 17499 | Restriction of a homomorph... |
resghm2 17500 | One direction of ~ resghm2... |
resghm2b 17501 | Restriction of the codomai... |
ghmghmrn 17502 | A group homomorphism from ... |
ghmco 17503 | The composition of group h... |
ghmima 17504 | The image of a subgroup un... |
ghmpreima 17505 | The inverse image of a sub... |
ghmeql 17506 | The equalizer of two group... |
ghmnsgima 17507 | The image of a normal subg... |
ghmnsgpreima 17508 | The inverse image of a nor... |
ghmker 17509 | The kernel of a homomorphi... |
ghmeqker 17510 | Two source points map to t... |
pwsdiagghm 17511 | Diagonal homomorphism into... |
ghmf1 17512 | Two ways of saying a group... |
ghmf1o 17513 | A bijective group homomorp... |
conjghm 17514 | Conjugation is an automorp... |
conjsubg 17515 | A conjugated subgroup is a... |
conjsubgen 17516 | A conjugated subgroup is e... |
conjnmz 17517 | A subgroup is unchanged un... |
conjnmzb 17518 | Alternative condition for ... |
conjnsg 17519 | A normal subgroup is uncha... |
qusghm 17520 | If ` Y ` is a normal subgr... |
ghmpropd 17521 | Group homomorphism depends... |
gimfn 17526 | The group isomorphism func... |
isgim 17527 | An isomorphism of groups i... |
gimf1o 17528 | An isomorphism of groups i... |
gimghm 17529 | An isomorphism of groups i... |
isgim2 17530 | A group isomorphism is a h... |
subggim 17531 | Behavior of subgroups unde... |
gimcnv 17532 | The converse of a bijectiv... |
gimco 17533 | The composition of group i... |
brgic 17534 | The relation "is isomorphi... |
brgici 17535 | Prove isomorphic by an exp... |
gicref 17536 | Isomorphism is reflexive. ... |
giclcl 17537 | Isomorphism implies the le... |
gicrcl 17538 | Isomorphism implies the ri... |
gicsym 17539 | Isomorphism is symmetric. ... |
gictr 17540 | Isomorphism is transitive.... |
gicer 17541 | Isomorphism is an equivale... |
gicerOLD 17542 | Obsolete proof of ~ gicer ... |
gicen 17543 | Isomorphic groups have equ... |
gicsubgen 17544 | A less trivial example of ... |
isga 17547 | The predicate "is a (left)... |
gagrp 17548 | The left argument of a gro... |
gaset 17549 | The right argument of a gr... |
gagrpid 17550 | The identity of the group ... |
gaf 17551 | The mapping of the group a... |
gafo 17552 | A group action is onto its... |
gaass 17553 | An "associative" property ... |
ga0 17554 | The action of a group on t... |
gaid 17555 | The trivial action of a gr... |
subgga 17556 | A subgroup acts on its par... |
gass 17557 | A subset of a group action... |
gasubg 17558 | The restriction of a group... |
gaid2 17559 | A group operation is a lef... |
galcan 17560 | The action of a particular... |
gacan 17561 | Group inverses cancel in a... |
gapm 17562 | The action of a particular... |
gaorb 17563 | The orbit equivalence rela... |
gaorber 17564 | The orbit equivalence rela... |
gastacl 17565 | The stabilizer subgroup in... |
gastacos 17566 | Write the coset relation f... |
orbstafun 17567 | Existence and uniqueness f... |
orbstaval 17568 | Value of the function at a... |
orbsta 17569 | The Orbit-Stabilizer theor... |
orbsta2 17570 | Relation between the size ... |
cntrval 17575 | Substitute definition of t... |
cntzfval 17576 | First level substitution f... |
cntzval 17577 | Definition substitution fo... |
elcntz 17578 | Elementhood in the central... |
cntzel 17579 | Membership in a centralize... |
cntzsnval 17580 | Special substitution for t... |
elcntzsn 17581 | Value of the centralizer o... |
sscntz 17582 | A centralizer expression f... |
cntzrcl 17583 | Reverse closure for elemen... |
cntzssv 17584 | The centralizer is uncondi... |
cntzi 17585 | Membership in a centralize... |
cntri 17586 | Defining property of the c... |
resscntz 17587 | Centralizer in a substruct... |
cntz2ss 17588 | Centralizers reverse the s... |
cntzrec 17589 | Reciprocity relationship f... |
cntziinsn 17590 | Express any centralizer as... |
cntzsubm 17591 | Centralizers in a monoid a... |
cntzsubg 17592 | Centralizers in a group ar... |
cntzidss 17593 | If the elements of ` S ` c... |
cntzmhm 17594 | Centralizers in a monoid a... |
cntzmhm2 17595 | Centralizers in a monoid a... |
cntrsubgnsg 17596 | A central subgroup is norm... |
cntrnsg 17597 | The center of a group is a... |
oppgval 17600 | Value of the opposite grou... |
oppgplusfval 17601 | Value of the addition oper... |
oppgplus 17602 | Value of the addition oper... |
oppglem 17603 | Lemma for ~ oppgbas . (Co... |
oppgbas 17604 | Base set of an opposite gr... |
oppgtset 17605 | Topology of an opposite gr... |
oppgtopn 17606 | Topology of an opposite gr... |
oppgmnd 17607 | The opposite of a monoid i... |
oppgmndb 17608 | Bidirectional form of ~ op... |
oppgid 17609 | Zero in a monoid is a symm... |
oppggrp 17610 | The opposite of a group is... |
oppggrpb 17611 | Bidirectional form of ~ op... |
oppginv 17612 | Inverses in a group are a ... |
invoppggim 17613 | The inverse is an antiauto... |
oppggic 17614 | Every group is (naturally)... |
oppgsubm 17615 | Being a submonoid is a sym... |
oppgsubg 17616 | Being a subgroup is a symm... |
oppgcntz 17617 | A centralizer in a group i... |
oppgcntr 17618 | The center of a group is t... |
gsumwrev 17619 | A sum in an opposite monoi... |
symgval 17622 | The value of the symmetric... |
symgbas 17623 | The base set of the symmet... |
elsymgbas2 17624 | Two ways of saying a funct... |
elsymgbas 17625 | Two ways of saying a funct... |
symgbasf1o 17626 | Elements in the symmetric ... |
symgbasf 17627 | A permutation (element of ... |
symghash 17628 | The symmetric group on ` n... |
symgbasfi 17629 | The symmetric group on a f... |
symgfv 17630 | The function value of a pe... |
symgfvne 17631 | The function values of a p... |
symgplusg 17632 | The group operation of a s... |
symgov 17633 | The value of the group ope... |
symgcl 17634 | The group operation of the... |
symgmov1 17635 | For a permutation of a set... |
symgmov2 17636 | For a permutation of a set... |
symgbas0 17637 | The base set of the symmet... |
symg1hash 17638 | The symmetric group on a s... |
symg1bas 17639 | The symmetric group on a s... |
symg2hash 17640 | The symmetric group on a (... |
symg2bas 17641 | The symmetric group on a p... |
symgtset 17642 | The topology of the symmet... |
symggrp 17643 | The symmetric group on a s... |
symgid 17644 | The group identity element... |
symginv 17645 | The group inverse in the s... |
galactghm 17646 | The currying of a group ac... |
lactghmga 17647 | The converse of ~ galactgh... |
symgtopn 17648 | The topology of the symmet... |
symgga 17649 | The symmetric group induce... |
pgrpsubgsymgbi 17650 | Every permutation group is... |
pgrpsubgsymg 17651 | Every permutation group is... |
idresperm 17652 | The identity function rest... |
idressubgsymg 17653 | The singleton containing o... |
idrespermg 17654 | The structure with the sin... |
cayleylem1 17655 | Lemma for ~ cayley . (Con... |
cayleylem2 17656 | Lemma for ~ cayley . (Con... |
cayley 17657 | Cayley's Theorem (construc... |
cayleyth 17658 | Cayley's Theorem (existenc... |
symgfix2 17659 | If a permutation does not ... |
symgextf 17660 | The extension of a permuta... |
symgextfv 17661 | The function value of the ... |
symgextfve 17662 | The function value of the ... |
symgextf1lem 17663 | Lemma for ~ symgextf1 . (... |
symgextf1 17664 | The extension of a permuta... |
symgextfo 17665 | The extension of a permuta... |
symgextf1o 17666 | The extension of a permuta... |
symgextsymg 17667 | The extension of a permuta... |
symgextres 17668 | The restriction of the ext... |
gsumccatsymgsn 17669 | Homomorphic property of co... |
gsmsymgrfixlem1 17670 | Lemma 1 for ~ gsmsymgrfix ... |
gsmsymgrfix 17671 | The composition of permuta... |
fvcosymgeq 17672 | The values of two composit... |
gsmsymgreqlem1 17673 | Lemma 1 for ~ gsmsymgreq .... |
gsmsymgreqlem2 17674 | Lemma 2 for ~ gsmsymgreq .... |
gsmsymgreq 17675 | Two combination of permuta... |
symgfixelq 17676 | A permutation of a set fix... |
symgfixels 17677 | The restriction of a permu... |
symgfixelsi 17678 | The restriction of a permu... |
symgfixf 17679 | The mapping of a permutati... |
symgfixf1 17680 | The mapping of a permutati... |
symgfixfolem1 17681 | Lemma 1 for ~ symgfixfo . ... |
symgfixfo 17682 | The mapping of a permutati... |
symgfixf1o 17683 | The mapping of a permutati... |
f1omvdmvd 17686 | A permutation of any class... |
f1omvdcnv 17687 | A permutation and its inve... |
mvdco 17688 | Composing two permutations... |
f1omvdconj 17689 | Conjugation of a permutati... |
f1otrspeq 17690 | A transposition is charact... |
f1omvdco2 17691 | If exactly one of two perm... |
f1omvdco3 17692 | If a point is moved by exa... |
pmtrfval 17693 | The function generating tr... |
pmtrval 17694 | A generated transposition,... |
pmtrfv 17695 | General value of mapping a... |
pmtrprfv 17696 | In a transposition of two ... |
pmtrprfv3 17697 | In a transposition of two ... |
pmtrf 17698 | Functionality of a transpo... |
pmtrmvd 17699 | A transposition moves prec... |
pmtrrn 17700 | Transposing two points giv... |
pmtrfrn 17701 | A transposition (as a kind... |
pmtrffv 17702 | Mapping of a point under a... |
pmtrrn2 17703 | For any transposition ther... |
pmtrfinv 17704 | A transposition function i... |
pmtrfmvdn0 17705 | A transposition moves at l... |
pmtrff1o 17706 | A transposition function i... |
pmtrfcnv 17707 | A transposition function i... |
pmtrfb 17708 | An intrinsic characterizat... |
pmtrfconj 17709 | Any conjugate of a transpo... |
symgsssg 17710 | The symmetric group has su... |
symgfisg 17711 | The symmetric group has a ... |
symgtrf 17712 | Transpositions are element... |
symggen 17713 | The span of the transposit... |
symggen2 17714 | A finite permutation group... |
symgtrinv 17715 | To invert a permutation re... |
pmtr3ncomlem1 17716 | Lemma 1 for ~ pmtr3ncom . ... |
pmtr3ncomlem2 17717 | Lemma 2 for ~ pmtr3ncom . ... |
pmtr3ncom 17718 | Transpositions over sets w... |
pmtrdifellem1 17719 | Lemma 1 for ~ pmtrdifel . ... |
pmtrdifellem2 17720 | Lemma 2 for ~ pmtrdifel . ... |
pmtrdifellem3 17721 | Lemma 3 for ~ pmtrdifel . ... |
pmtrdifellem4 17722 | Lemma 4 for ~ pmtrdifel . ... |
pmtrdifel 17723 | A transposition of element... |
pmtrdifwrdellem1 17724 | Lemma 1 for ~ pmtrdifwrdel... |
pmtrdifwrdellem2 17725 | Lemma 2 for ~ pmtrdifwrdel... |
pmtrdifwrdellem3 17726 | Lemma 3 for ~ pmtrdifwrdel... |
pmtrdifwrdel2lem1 17727 | Lemma 1 for ~ pmtrdifwrdel... |
pmtrdifwrdel 17728 | A sequence of transpositio... |
pmtrdifwrdel2 17729 | A sequence of transpositio... |
pmtrprfval 17730 | The transpositions on a pa... |
pmtrprfvalrn 17731 | The range of the transposi... |
psgnunilem1 17736 | Lemma for ~ psgnuni . Giv... |
psgnunilem5 17737 | Lemma for ~ psgnuni . It ... |
psgnunilem2 17738 | Lemma for ~ psgnuni . Ind... |
psgnunilem3 17739 | Lemma for ~ psgnuni . Any... |
psgnunilem4 17740 | Lemma for ~ psgnuni . An ... |
m1expaddsub 17741 | Addition and subtraction o... |
psgnuni 17742 | If the same permutation ca... |
psgnfval 17743 | Function definition of the... |
psgnfn 17744 | Functionality and domain o... |
psgndmsubg 17745 | The finitary permutations ... |
psgneldm 17746 | Property of being a finita... |
psgneldm2 17747 | The finitary permutations ... |
psgneldm2i 17748 | A sequence of transpositio... |
psgneu 17749 | A finitary permutation has... |
psgnval 17750 | Value of the permutation s... |
psgnvali 17751 | A finitary permutation has... |
psgnvalii 17752 | Any representation of a pe... |
psgnpmtr 17753 | All transpositions are odd... |
psgn0fv0 17754 | The permutation sign funct... |
sygbasnfpfi 17755 | The class of non-fixed poi... |
psgnfvalfi 17756 | Function definition of the... |
psgnvalfi 17757 | Value of the permutation s... |
psgnran 17758 | The range of the permutati... |
gsmtrcl 17759 | The group sum of transposi... |
psgnfitr 17760 | A permutation of a finite ... |
psgnfieu 17761 | A permutation of a finite ... |
pmtrsn 17762 | The value of the transposi... |
psgnsn 17763 | The permutation sign funct... |
psgnprfval 17764 | The permutation sign funct... |
psgnprfval1 17765 | The permutation sign of th... |
psgnprfval2 17766 | The permutation sign of th... |
odfval 17775 | Value of the order functio... |
odval 17776 | Second substitution for th... |
odlem1 17777 | The group element order is... |
odcl 17778 | The order of a group eleme... |
odf 17779 | Functionality of the group... |
odid 17780 | Any element to the power o... |
odlem2 17781 | Any positive annihilator o... |
odmodnn0 17782 | Reduce the argument of a g... |
mndodconglem 17783 | Lemma for ~ mndodcong . (... |
mndodcong 17784 | If two multipliers are con... |
mndodcongi 17785 | If two multipliers are con... |
oddvdsnn0 17786 | The only multiples of ` A ... |
odnncl 17787 | If a nonzero multiple of a... |
odmod 17788 | Reduce the argument of a g... |
oddvds 17789 | The only multiples of ` A ... |
oddvdsi 17790 | Any group element is annih... |
odcong 17791 | If two multipliers are con... |
odeq 17792 | The ~ oddvds property uniq... |
odval2 17793 | A non-conditional definiti... |
odmulgid 17794 | A relationship between the... |
odmulg2 17795 | The order of a multiple di... |
odmulg 17796 | Relationship between the o... |
odmulgeq 17797 | A multiple of a point of f... |
odbezout 17798 | If ` N ` is coprime to the... |
od1 17799 | The order of the group ide... |
odeq1 17800 | The group identity is the ... |
odinv 17801 | The order of the inverse o... |
odf1 17802 | The multiples of an elemen... |
odinf 17803 | The multiples of an elemen... |
dfod2 17804 | An alternative definition ... |
odcl2 17805 | The order of an element of... |
oddvds2 17806 | The order of an element of... |
submod 17807 | The order of an element is... |
subgod 17808 | The order of an element is... |
odsubdvds 17809 | The order of an element of... |
odf1o1 17810 | An element with zero order... |
odf1o2 17811 | An element with nonzero or... |
odhash 17812 | An element of zero order g... |
odhash2 17813 | If an element has nonzero ... |
odhash3 17814 | An element which generates... |
odngen 17815 | A cyclic subgroup of size ... |
gexval 17816 | Value of the exponent of a... |
gexlem1 17817 | The group element order is... |
gexcl 17818 | The exponent of a group is... |
gexid 17819 | Any element to the power o... |
gexlem2 17820 | Any positive annihilator o... |
gexdvdsi 17821 | Any group element is annih... |
gexdvds 17822 | The only ` N ` that annihi... |
gexdvds2 17823 | An integer divides the gro... |
gexod 17824 | Any group element is annih... |
gexcl3 17825 | If the order of every grou... |
gexnnod 17826 | Every group element has fi... |
gexcl2 17827 | The exponent of a finite g... |
gexdvds3 17828 | The exponent of a finite g... |
gex1 17829 | A group or monoid has expo... |
ispgp 17830 | A group is a ` P ` -group ... |
pgpprm 17831 | Reverse closure for the fi... |
pgpgrp 17832 | Reverse closure for the se... |
pgpfi1 17833 | A finite group with order ... |
pgp0 17834 | The identity subgroup is a... |
subgpgp 17835 | A subgroup of a p-group is... |
sylow1lem1 17836 | Lemma for ~ sylow1 . The ... |
sylow1lem2 17837 | Lemma for ~ sylow1 . The ... |
sylow1lem3 17838 | Lemma for ~ sylow1 . One ... |
sylow1lem4 17839 | Lemma for ~ sylow1 . The ... |
sylow1lem5 17840 | Lemma for ~ sylow1 . Usin... |
sylow1 17841 | Sylow's first theorem. If... |
odcau 17842 | Cauchy's theorem for the o... |
pgpfi 17843 | The converse to ~ pgpfi1 .... |
pgpfi2 17844 | Alternate version of ~ pgp... |
pgphash 17845 | The order of a p-group. (... |
isslw 17846 | The property of being a Sy... |
slwprm 17847 | Reverse closure for the fi... |
slwsubg 17848 | A Sylow ` P ` -subgroup is... |
slwispgp 17849 | Defining property of a Syl... |
slwpss 17850 | A proper superset of a Syl... |
slwpgp 17851 | A Sylow ` P ` -subgroup is... |
pgpssslw 17852 | Every ` P ` -subgroup is c... |
slwn0 17853 | Every finite group contain... |
subgslw 17854 | A Sylow subgroup that is c... |
sylow2alem1 17855 | Lemma for ~ sylow2a . An ... |
sylow2alem2 17856 | Lemma for ~ sylow2a . All... |
sylow2a 17857 | A named lemma of Sylow's s... |
sylow2blem1 17858 | Lemma for ~ sylow2b . Eva... |
sylow2blem2 17859 | Lemma for ~ sylow2b . Lef... |
sylow2blem3 17860 | Sylow's second theorem. P... |
sylow2b 17861 | Sylow's second theorem. A... |
slwhash 17862 | A sylow subgroup has cardi... |
fislw 17863 | The sylow subgroups of a f... |
sylow2 17864 | Sylow's second theorem. S... |
sylow3lem1 17865 | Lemma for ~ sylow3 , first... |
sylow3lem2 17866 | Lemma for ~ sylow3 , first... |
sylow3lem3 17867 | Lemma for ~ sylow3 , first... |
sylow3lem4 17868 | Lemma for ~ sylow3 , first... |
sylow3lem5 17869 | Lemma for ~ sylow3 , secon... |
sylow3lem6 17870 | Lemma for ~ sylow3 , secon... |
sylow3 17871 | Sylow's third theorem. Th... |
lsmfval 17876 | The subgroup sum function ... |
lsmvalx 17877 | Subspace sum value (for a ... |
lsmelvalx 17878 | Subspace sum membership (f... |
lsmelvalix 17879 | Subspace sum membership (f... |
oppglsm 17880 | The subspace sum operation... |
lsmssv 17881 | Subgroup sum is a subset o... |
lsmless1x 17882 | Subset implies subgroup su... |
lsmless2x 17883 | Subset implies subgroup su... |
lsmub1x 17884 | Subgroup sum is an upper b... |
lsmub2x 17885 | Subgroup sum is an upper b... |
lsmval 17886 | Subgroup sum value (for a ... |
lsmelval 17887 | Subgroup sum membership (f... |
lsmelvali 17888 | Subgroup sum membership (f... |
lsmelvalm 17889 | Subgroup sum membership an... |
lsmelvalmi 17890 | Membership of vector subtr... |
lsmsubm 17891 | The sum of two commuting s... |
lsmsubg 17892 | The sum of two commuting s... |
lsmcom2 17893 | Subgroup sum commutes. (C... |
lsmub1 17894 | Subgroup sum is an upper b... |
lsmub2 17895 | Subgroup sum is an upper b... |
lsmunss 17896 | Union of subgroups is a su... |
lsmless1 17897 | Subset implies subgroup su... |
lsmless2 17898 | Subset implies subgroup su... |
lsmless12 17899 | Subset implies subgroup su... |
lsmidm 17900 | Subgroup sum is idempotent... |
lsmlub 17901 | The least upper bound prop... |
lsmss1 17902 | Subgroup sum with a subset... |
lsmss1b 17903 | Subgroup sum with a subset... |
lsmss2 17904 | Subgroup sum with a subset... |
lsmss2b 17905 | Subgroup sum with a subset... |
lsmass 17906 | Subgroup sum is associativ... |
lsm01 17907 | Subgroup sum with the zero... |
lsm02 17908 | Subgroup sum with the zero... |
subglsm 17909 | The subgroup sum evaluated... |
lssnle 17910 | Equivalent expressions for... |
lsmmod 17911 | The modular law holds for ... |
lsmmod2 17912 | Modular law dual for subgr... |
lsmpropd 17913 | If two structures have the... |
cntzrecd 17914 | Commute the "subgroups com... |
lsmcntz 17915 | The "subgroups commute" pr... |
lsmcntzr 17916 | The "subgroups commute" pr... |
lsmdisj 17917 | Disjointness from a subgro... |
lsmdisj2 17918 | Association of the disjoin... |
lsmdisj3 17919 | Association of the disjoin... |
lsmdisjr 17920 | Disjointness from a subgro... |
lsmdisj2r 17921 | Association of the disjoin... |
lsmdisj3r 17922 | Association of the disjoin... |
lsmdisj2a 17923 | Association of the disjoin... |
lsmdisj2b 17924 | Association of the disjoin... |
lsmdisj3a 17925 | Association of the disjoin... |
lsmdisj3b 17926 | Association of the disjoin... |
subgdisj1 17927 | Vectors belonging to disjo... |
subgdisj2 17928 | Vectors belonging to disjo... |
subgdisjb 17929 | Vectors belonging to disjo... |
pj1fval 17930 | The left projection functi... |
pj1val 17931 | The left projection functi... |
pj1eu 17932 | Uniqueness of a left proje... |
pj1f 17933 | The left projection functi... |
pj2f 17934 | The right projection funct... |
pj1id 17935 | Any element of a direct su... |
pj1eq 17936 | Any element of a direct su... |
pj1lid 17937 | The left projection functi... |
pj1rid 17938 | The left projection functi... |
pj1ghm 17939 | The left projection functi... |
pj1ghm2 17940 | The left projection functi... |
lsmhash 17941 | The order of the direct pr... |
efgmval 17948 | Value of the formal invers... |
efgmf 17949 | The formal inverse operati... |
efgmnvl 17950 | The inversion function on ... |
efgrcl 17951 | Lemma for ~ efgval . (Con... |
efglem 17952 | Lemma for ~ efgval . (Con... |
efgval 17953 | Value of the free group co... |
efger 17954 | Value of the free group co... |
efgi 17955 | Value of the free group co... |
efgi0 17956 | Value of the free group co... |
efgi1 17957 | Value of the free group co... |
efgtf 17958 | Value of the free group co... |
efgtval 17959 | Value of the extension fun... |
efgval2 17960 | Value of the free group co... |
efgi2 17961 | Value of the free group co... |
efgtlen 17962 | Value of the free group co... |
efginvrel2 17963 | The inverse of the reverse... |
efginvrel1 17964 | The inverse of the reverse... |
efgsf 17965 | Value of the auxiliary fun... |
efgsdm 17966 | Elementhood in the domain ... |
efgsval 17967 | Value of the auxiliary fun... |
efgsdmi 17968 | Property of the last link ... |
efgsval2 17969 | Value of the auxiliary fun... |
efgsrel 17970 | The start and end of any e... |
efgs1 17971 | A singleton of an irreduci... |
efgs1b 17972 | Every extension sequence e... |
efgsp1 17973 | If ` F ` is an extension s... |
efgsres 17974 | An initial segment of an e... |
efgsfo 17975 | For any word, there is a s... |
efgredlema 17976 | The reduced word that form... |
efgredlemf 17977 | Lemma for ~ efgredleme . ... |
efgredlemg 17978 | Lemma for ~ efgred . (Con... |
efgredleme 17979 | Lemma for ~ efgred . (Con... |
efgredlemd 17980 | The reduced word that form... |
efgredlemc 17981 | The reduced word that form... |
efgredlemb 17982 | The reduced word that form... |
efgredlem 17983 | The reduced word that form... |
efgred 17984 | The reduced word that form... |
efgrelexlema 17985 | If two words ` A , B ` are... |
efgrelexlemb 17986 | If two words ` A , B ` are... |
efgrelex 17987 | If two words ` A , B ` are... |
efgredeu 17988 | There is a unique reduced ... |
efgred2 17989 | Two extension sequences ha... |
efgcpbllema 17990 | Lemma for ~ efgrelex . De... |
efgcpbllemb 17991 | Lemma for ~ efgrelex . Sh... |
efgcpbl 17992 | Two extension sequences ha... |
efgcpbl2 17993 | Two extension sequences ha... |
frgpval 17994 | Value of the free group co... |
frgpcpbl 17995 | Compatibility of the group... |
frgp0 17996 | The free group is a group.... |
frgpeccl 17997 | Closure of the quotient ma... |
frgpgrp 17998 | The free group is a group.... |
frgpadd 17999 | Addition in the free group... |
frgpinv 18000 | The inverse of an element ... |
frgpmhm 18001 | The "natural map" from wor... |
vrgpfval 18002 | The canonical injection fr... |
vrgpval 18003 | The value of the generatin... |
vrgpf 18004 | The mapping from the index... |
vrgpinv 18005 | The inverse of a generatin... |
frgpuptf 18006 | Any assignment of the gene... |
frgpuptinv 18007 | Any assignment of the gene... |
frgpuplem 18008 | Any assignment of the gene... |
frgpupf 18009 | Any assignment of the gene... |
frgpupval 18010 | Any assignment of the gene... |
frgpup1 18011 | Any assignment of the gene... |
frgpup2 18012 | The evaluation map has the... |
frgpup3lem 18013 | The evaluation map has the... |
frgpup3 18014 | Universal property of the ... |
0frgp 18015 | The free group on zero gen... |
isabl 18020 | The predicate "is an Abeli... |
ablgrp 18021 | An Abelian group is a grou... |
ablcmn 18022 | An Abelian group is a comm... |
iscmn 18023 | The predicate "is a commut... |
isabl2 18024 | The predicate "is an Abeli... |
cmnpropd 18025 | If two structures have the... |
ablpropd 18026 | If two structures have the... |
ablprop 18027 | If two structures have the... |
iscmnd 18028 | Properties that determine ... |
isabld 18029 | Properties that determine ... |
isabli 18030 | Properties that determine ... |
cmnmnd 18031 | A commutative monoid is a ... |
cmncom 18032 | A commutative monoid is co... |
ablcom 18033 | An Abelian group operation... |
cmn32 18034 | Commutative/associative la... |
cmn4 18035 | Commutative/associative la... |
cmn12 18036 | Commutative/associative la... |
abl32 18037 | Commutative/associative la... |
ablinvadd 18038 | The inverse of an Abelian ... |
ablsub2inv 18039 | Abelian group subtraction ... |
ablsubadd 18040 | Relationship between Abeli... |
ablsub4 18041 | Commutative/associative su... |
abladdsub4 18042 | Abelian group addition/sub... |
abladdsub 18043 | Associative-type law for g... |
ablpncan2 18044 | Cancellation law for subtr... |
ablpncan3 18045 | A cancellation law for com... |
ablsubsub 18046 | Law for double subtraction... |
ablsubsub4 18047 | Law for double subtraction... |
ablpnpcan 18048 | Cancellation law for mixed... |
ablnncan 18049 | Cancellation law for group... |
ablsub32 18050 | Swap the second and third ... |
ablnnncan 18051 | Cancellation law for group... |
ablnnncan1 18052 | Cancellation law for group... |
ablsubsub23 18053 | Swap subtrahend and result... |
mulgnn0di 18054 | Group multiple of a sum, f... |
mulgdi 18055 | Group multiple of a sum. ... |
mulgmhm 18056 | The map from ` x ` to ` n ... |
mulgghm 18057 | The map from ` x ` to ` n ... |
mulgsubdi 18058 | Group multiple of a differ... |
ghmfghm 18059 | The function fulfilling th... |
ghmcmn 18060 | The image of a commutative... |
ghmabl 18061 | The image of an abelian gr... |
invghm 18062 | The inversion map is a gro... |
eqgabl 18063 | Value of the subgroup cose... |
subgabl 18064 | A subgroup of an abelian g... |
subcmn 18065 | A submonoid of a commutati... |
submcmn 18066 | A submonoid of a commutati... |
submcmn2 18067 | A submonoid is commutative... |
cntzcmn 18068 | The centralizer of any sub... |
cntzcmnss 18069 | Any subset in a commutativ... |
cntzspan 18070 | If the generators commute,... |
cntzcmnf 18071 | Discharge the centralizer ... |
ghmplusg 18072 | The pointwise sum of two l... |
ablnsg 18073 | Every subgroup of an abeli... |
odadd1 18074 | The order of a product in ... |
odadd2 18075 | The order of a product in ... |
odadd 18076 | The order of a product is ... |
gex2abl 18077 | A group with exponent 2 (o... |
gexexlem 18078 | Lemma for ~ gexex . (Cont... |
gexex 18079 | In an abelian group with f... |
torsubg 18080 | The set of all elements of... |
oddvdssubg 18081 | The set of all elements wh... |
lsmcomx 18082 | Subgroup sum commutes (ext... |
ablcntzd 18083 | All subgroups in an abelia... |
lsmcom 18084 | Subgroup sum commutes. (C... |
lsmsubg2 18085 | The sum of two subgroups i... |
lsm4 18086 | Commutative/associative la... |
prdscmnd 18087 | The product of a family of... |
prdsabld 18088 | The product of a family of... |
pwscmn 18089 | The structure power on a c... |
pwsabl 18090 | The structure power on an ... |
qusabl 18091 | If ` Y ` is a subgroup of ... |
abl1 18092 | The (smallest) structure r... |
abln0 18093 | Abelian groups (and theref... |
cnaddablx 18094 | The complex numbers are an... |
cnaddabl 18095 | The complex numbers are an... |
cnaddid 18096 | The group identity element... |
cnaddinv 18097 | Value of the group inverse... |
zaddablx 18098 | The integers are an Abelia... |
frgpnabllem1 18099 | Lemma for ~ frgpnabl . (C... |
frgpnabllem2 18100 | Lemma for ~ frgpnabl . (C... |
frgpnabl 18101 | The free group on two or m... |
iscyg 18104 | Definition of a cyclic gro... |
iscyggen 18105 | The property of being a cy... |
iscyggen2 18106 | The property of being a cy... |
iscyg2 18107 | A cyclic group is a group ... |
cyggeninv 18108 | The inverse of a cyclic ge... |
cyggenod 18109 | An element is the generato... |
cyggenod2 18110 | In an infinite cyclic grou... |
iscyg3 18111 | Definition of a cyclic gro... |
iscygd 18112 | Definition of a cyclic gro... |
iscygodd 18113 | Show that a group with an ... |
cyggrp 18114 | A cyclic group is a group.... |
cygabl 18115 | A cyclic group is abelian.... |
cygctb 18116 | A cyclic group is countabl... |
0cyg 18117 | The trivial group is cycli... |
prmcyg 18118 | A group with prime order i... |
lt6abl 18119 | A group with fewer than ` ... |
ghmcyg 18120 | The image of a cyclic grou... |
cyggex2 18121 | The exponent of a cyclic g... |
cyggex 18122 | The exponent of a finite c... |
cyggexb 18123 | A finite abelian group is ... |
giccyg 18124 | Cyclicity is a group prope... |
cycsubgcyg 18125 | The cyclic subgroup genera... |
cycsubgcyg2 18126 | The cyclic subgroup genera... |
gsumval3a 18127 | Value of the group sum ope... |
gsumval3eu 18128 | The group sum as defined i... |
gsumval3lem1 18129 | Lemma 1 for ~ gsumval3 . ... |
gsumval3lem2 18130 | Lemma 2 for ~ gsumval3 . ... |
gsumval3 18131 | Value of the group sum ope... |
gsumcllem 18132 | Lemma for ~ gsumcl and rel... |
gsumzres 18133 | Extend a finite group sum ... |
gsumzcl2 18134 | Closure of a finite group ... |
gsumzcl 18135 | Closure of a finite group ... |
gsumzf1o 18136 | Re-index a finite group su... |
gsumres 18137 | Extend a finite group sum ... |
gsumcl2 18138 | Closure of a finite group ... |
gsumcl 18139 | Closure of a finite group ... |
gsumf1o 18140 | Re-index a finite group su... |
gsumzsubmcl 18141 | Closure of a group sum in ... |
gsumsubmcl 18142 | Closure of a group sum in ... |
gsumsubgcl 18143 | Closure of a group sum in ... |
gsumzaddlem 18144 | The sum of two group sums.... |
gsumzadd 18145 | The sum of two group sums.... |
gsumadd 18146 | The sum of two group sums.... |
gsummptfsadd 18147 | The sum of two group sums ... |
gsummptfidmadd 18148 | The sum of two group sums ... |
gsummptfidmadd2 18149 | The sum of two group sums ... |
gsumzsplit 18150 | Split a group sum into two... |
gsumsplit 18151 | Split a group sum into two... |
gsumsplit2 18152 | Split a group sum into two... |
gsummptfidmsplit 18153 | Split a group sum expresse... |
gsummptfidmsplitres 18154 | Split a group sum expresse... |
gsummptfzsplit 18155 | Split a group sum expresse... |
gsummptfzsplitl 18156 | Split a group sum expresse... |
gsumconst 18157 | Sum of a constant series. ... |
gsumconstf 18158 | Sum of a constant series. ... |
gsummptshft 18159 | Index shift of a finite gr... |
gsumzmhm 18160 | Apply a group homomorphism... |
gsummhm 18161 | Apply a group homomorphism... |
gsummhm2 18162 | Apply a group homomorphism... |
gsummptmhm 18163 | Apply a group homomorphism... |
gsummulglem 18164 | Lemma for ~ gsummulg and ~... |
gsummulg 18165 | Nonnegative multiple of a ... |
gsummulgz 18166 | Integer multiple of a grou... |
gsumzoppg 18167 | The opposite of a group su... |
gsumzinv 18168 | Inverse of a group sum. (... |
gsuminv 18169 | Inverse of a group sum. (... |
gsummptfidminv 18170 | Inverse of a group sum exp... |
gsumsub 18171 | The difference of two grou... |
gsummptfssub 18172 | The difference of two grou... |
gsummptfidmsub 18173 | The difference of two grou... |
gsumsnfd 18174 | Group sum of a singleton, ... |
gsumsnd 18175 | Group sum of a singleton, ... |
gsumsnf 18176 | Group sum of a singleton, ... |
gsumsn 18177 | Group sum of a singleton. ... |
gsumzunsnd 18178 | Append an element to a fin... |
gsumunsnfd 18179 | Append an element to a fin... |
gsumunsnd 18180 | Append an element to a fin... |
gsumunsnf 18181 | Append an element to a fin... |
gsumunsn 18182 | Append an element to a fin... |
gsumdifsnd 18183 | Extract a summand from a f... |
gsumpt 18184 | Sum of a family that is no... |
gsummptf1o 18185 | Re-index a finite group su... |
gsummptun 18186 | Group sum of a disjoint un... |
gsummpt1n0 18187 | If only one summand in a f... |
gsummptif1n0 18188 | If only one summand in a f... |
gsummptcl 18189 | Closure of a finite group ... |
gsummptfif1o 18190 | Re-index a finite group su... |
gsummptfzcl 18191 | Closure of a finite group ... |
gsum2dlem1 18192 | Lemma 1 for ~ gsum2d . (C... |
gsum2dlem2 18193 | Lemma for ~ gsum2d . (Con... |
gsum2d 18194 | Write a sum over a two-dim... |
gsum2d2lem 18195 | Lemma for ~ gsum2d2 : show... |
gsum2d2 18196 | Write a group sum over a t... |
gsumcom2 18197 | Two-dimensional commutatio... |
gsumxp 18198 | Write a group sum over a c... |
gsumcom 18199 | Commute the arguments of a... |
prdsgsum 18200 | Finite commutative sums in... |
pwsgsum 18201 | Finite commutative sums in... |
fsfnn0gsumfsffz 18202 | Replacing a finitely suppo... |
nn0gsumfz 18203 | Replacing a finitely suppo... |
nn0gsumfz0 18204 | Replacing a finitely suppo... |
gsummptnn0fz 18205 | A final group sum over a f... |
gsummptnn0fzv 18206 | A final group sum over a f... |
gsummptnn0fzfv 18207 | A final group sum over a f... |
telgsumfzslem 18208 | Lemma for ~ telgsumfzs (in... |
telgsumfzs 18209 | Telescoping group sum rang... |
telgsumfz 18210 | Telescoping group sum rang... |
telgsumfz0s 18211 | Telescoping finite group s... |
telgsumfz0 18212 | Telescoping finite group s... |
telgsums 18213 | Telescoping finitely suppo... |
telgsum 18214 | Telescoping finitely suppo... |
reldmdprd 18219 | The domain of the internal... |
dmdprd 18220 | The domain of definition o... |
dmdprdd 18221 | Show that a given family i... |
dprddomprc 18222 | A family of subgroups inde... |
dprddomcld 18223 | If a family of subgroups i... |
dprdval0prc 18224 | The internal direct produc... |
dprdval 18225 | The value of the internal ... |
eldprd 18226 | A class ` A ` is an intern... |
dprdgrp 18227 | Reverse closure for the in... |
dprdf 18228 | The function ` S ` is a fa... |
dprdf2 18229 | The function ` S ` is a fa... |
dprdcntz 18230 | The function ` S ` is a fa... |
dprddisj 18231 | The function ` S ` is a fa... |
dprdw 18232 | The property of being a fi... |
dprdwd 18233 | A mapping being a finitely... |
dprdff 18234 | A finitely supported funct... |
dprdfcl 18235 | A finitely supported funct... |
dprdffsupp 18236 | A finitely supported funct... |
dprdfcntz 18237 | A function on the elements... |
dprdssv 18238 | The internal direct produc... |
dprdfid 18239 | A function mapping all but... |
eldprdi 18240 | The domain of definition o... |
dprdfinv 18241 | Take the inverse of a grou... |
dprdfadd 18242 | Take the sum of group sums... |
dprdfsub 18243 | Take the difference of gro... |
dprdfeq0 18244 | The zero function is the o... |
dprdf11 18245 | Two group sums over a dire... |
dprdsubg 18246 | The internal direct produc... |
dprdub 18247 | Each factor is a subset of... |
dprdlub 18248 | The direct product is smal... |
dprdspan 18249 | The direct product is the ... |
dprdres 18250 | Restriction of a direct pr... |
dprdss 18251 | Create a direct product by... |
dprdz 18252 | A family consisting entire... |
dprd0 18253 | The empty family is an int... |
dprdf1o 18254 | Rearrange the index set of... |
dprdf1 18255 | Rearrange the index set of... |
subgdmdprd 18256 | A direct product in a subg... |
subgdprd 18257 | A direct product in a subg... |
dprdsn 18258 | A singleton family is an i... |
dmdprdsplitlem 18259 | Lemma for ~ dmdprdsplit . ... |
dprdcntz2 18260 | The function ` S ` is a fa... |
dprddisj2 18261 | The function ` S ` is a fa... |
dprd2dlem2 18262 | The direct product of a co... |
dprd2dlem1 18263 | The direct product of a co... |
dprd2da 18264 | The direct product of a co... |
dprd2db 18265 | The direct product of a co... |
dprd2d2 18266 | The direct product of a co... |
dmdprdsplit2lem 18267 | Lemma for ~ dmdprdsplit . ... |
dmdprdsplit2 18268 | The direct product splits ... |
dmdprdsplit 18269 | The direct product splits ... |
dprdsplit 18270 | The direct product is the ... |
dmdprdpr 18271 | A singleton family is an i... |
dprdpr 18272 | A singleton family is an i... |
dpjlem 18273 | Lemma for theorems about d... |
dpjcntz 18274 | The two subgroups that app... |
dpjdisj 18275 | The two subgroups that app... |
dpjlsm 18276 | The two subgroups that app... |
dpjfval 18277 | Value of the direct produc... |
dpjval 18278 | Value of the direct produc... |
dpjf 18279 | The ` X ` -th index projec... |
dpjidcl 18280 | The key property of projec... |
dpjeq 18281 | Decompose a group sum into... |
dpjid 18282 | The key property of projec... |
dpjlid 18283 | The ` X ` -th index projec... |
dpjrid 18284 | The ` Y ` -th index projec... |
dpjghm 18285 | The direct product is the ... |
dpjghm2 18286 | The direct product is the ... |
ablfacrplem 18287 | Lemma for ~ ablfacrp2 . (... |
ablfacrp 18288 | A finite abelian group who... |
ablfacrp2 18289 | The factors ` K , L ` of ~... |
ablfac1lem 18290 | Lemma for ~ ablfac1b . Sa... |
ablfac1a 18291 | The factors of ~ ablfac1b ... |
ablfac1b 18292 | Any abelian group is the d... |
ablfac1c 18293 | The factors of ~ ablfac1b ... |
ablfac1eulem 18294 | Lemma for ~ ablfac1eu . (... |
ablfac1eu 18295 | The factorization of ~ abl... |
pgpfac1lem1 18296 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem2 18297 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem3a 18298 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem3 18299 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem4 18300 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1lem5 18301 | Lemma for ~ pgpfac1 . (Co... |
pgpfac1 18302 | Factorization of a finite ... |
pgpfaclem1 18303 | Lemma for ~ pgpfac . (Con... |
pgpfaclem2 18304 | Lemma for ~ pgpfac . (Con... |
pgpfaclem3 18305 | Lemma for ~ pgpfac . (Con... |
pgpfac 18306 | Full factorization of a fi... |
ablfaclem1 18307 | Lemma for ~ ablfac . (Con... |
ablfaclem2 18308 | Lemma for ~ ablfac . (Con... |
ablfaclem3 18309 | Lemma for ~ ablfac . (Con... |
ablfac 18310 | The Fundamental Theorem of... |
ablfac2 18311 | Choose generators for each... |
fnmgp 18314 | The multiplicative group o... |
mgpval 18315 | Value of the multiplicatio... |
mgpplusg 18316 | Value of the group operati... |
mgplem 18317 | Lemma for ~ mgpbas . (Con... |
mgpbas 18318 | Base set of the multiplica... |
mgpsca 18319 | The multiplication monoid ... |
mgptset 18320 | Topology component of the ... |
mgptopn 18321 | Topology of the multiplica... |
mgpds 18322 | Distance function of the m... |
mgpress 18323 | Subgroup commutes with the... |
ringidval 18326 | The value of the unity ele... |
dfur2 18327 | The multiplicative identit... |
issrg 18330 | The predicate "is a semiri... |
srgcmn 18331 | A semiring is a commutativ... |
srgmnd 18332 | A semiring is a monoid. (... |
srgmgp 18333 | A semiring is a monoid und... |
srgi 18334 | Properties of a semiring. ... |
srgcl 18335 | Closure of the multiplicat... |
srgass 18336 | Associative law for the mu... |
srgideu 18337 | The unit element of a semi... |
srgfcl 18338 | Functionality of the multi... |
srgdi 18339 | Distributive law for the m... |
srgdir 18340 | Distributive law for the m... |
srgidcl 18341 | The unit element of a semi... |
srg0cl 18342 | The zero element of a semi... |
srgidmlem 18343 | Lemma for ~ srglidm and ~ ... |
srglidm 18344 | The unit element of a semi... |
srgridm 18345 | The unit element of a semi... |
issrgid 18346 | Properties showing that an... |
srgacl 18347 | Closure of the addition op... |
srgcom 18348 | Commutativity of the addit... |
srgrz 18349 | The zero of a semiring is ... |
srglz 18350 | The zero of a semiring is ... |
srgisid 18351 | In a semiring, the only le... |
srg1zr 18352 | The only semiring with a b... |
srgen1zr 18353 | The only semiring with one... |
srgmulgass 18354 | An associative property be... |
srgpcomp 18355 | If two elements of a semir... |
srgpcompp 18356 | If two elements of a semir... |
srgpcomppsc 18357 | If two elements of a semir... |
srglmhm 18358 | Left-multiplication in a s... |
srgrmhm 18359 | Right-multiplication in a ... |
srgsummulcr 18360 | A finite semiring sum mult... |
sgsummulcl 18361 | A finite semiring sum mult... |
srg1expzeq1 18362 | The exponentiation (by a n... |
srgbinomlem1 18363 | Lemma 1 for ~ srgbinomlem ... |
srgbinomlem2 18364 | Lemma 2 for ~ srgbinomlem ... |
srgbinomlem3 18365 | Lemma 3 for ~ srgbinomlem ... |
srgbinomlem4 18366 | Lemma 4 for ~ srgbinomlem ... |
srgbinomlem 18367 | Lemma for ~ srgbinom . In... |
srgbinom 18368 | The binomial theorem for c... |
csrgbinom 18369 | The binomial theorem for c... |
isring 18374 | The predicate "is a (unita... |
ringgrp 18375 | A ring is a group. (Contr... |
ringmgp 18376 | A ring is a monoid under m... |
iscrng 18377 | A commutative ring is a ri... |
crngmgp 18378 | A commutative ring's multi... |
ringmnd 18379 | A ring is a monoid under a... |
ringmgm 18380 | A ring is a magma. (Contr... |
crngring 18381 | A commutative ring is a ri... |
mgpf 18382 | Restricted functionality o... |
ringi 18383 | Properties of a unital rin... |
ringcl 18384 | Closure of the multiplicat... |
crngcom 18385 | A commutative ring's multi... |
iscrng2 18386 | A commutative ring is a ri... |
ringass 18387 | Associative law for the mu... |
ringideu 18388 | The unit element of a ring... |
ringdi 18389 | Distributive law for the m... |
ringdir 18390 | Distributive law for the m... |
ringidcl 18391 | The unit element of a ring... |
ring0cl 18392 | The zero element of a ring... |
ringidmlem 18393 | Lemma for ~ ringlidm and ~... |
ringlidm 18394 | The unit element of a ring... |
ringridm 18395 | The unit element of a ring... |
isringid 18396 | Properties showing that an... |
ringid 18397 | The multiplication operati... |
ringadd2 18398 | A ring element plus itself... |
rngo2times 18399 | A ring element plus itself... |
ringidss 18400 | A subset of the multiplica... |
ringacl 18401 | Closure of the addition op... |
ringcom 18402 | Commutativity of the addit... |
ringabl 18403 | A ring is an Abelian group... |
ringcmn 18404 | A ring is a commutative mo... |
ringpropd 18405 | If two structures have the... |
crngpropd 18406 | If two structures have the... |
ringprop 18407 | If two structures have the... |
isringd 18408 | Properties that determine ... |
iscrngd 18409 | Properties that determine ... |
ringlz 18410 | The zero of a unital ring ... |
ringrz 18411 | The zero of a unital ring ... |
ringsrg 18412 | Any ring is also a semirin... |
ring1eq0 18413 | If one and zero are equal,... |
ring1ne0 18414 | If a ring has at least two... |
ringinvnz1ne0 18415 | In a unitary ring, a left ... |
ringinvnzdiv 18416 | In a unitary ring, a left ... |
ringnegl 18417 | Negation in a ring is the ... |
rngnegr 18418 | Negation in a ring is the ... |
ringmneg1 18419 | Negation of a product in a... |
ringmneg2 18420 | Negation of a product in a... |
ringm2neg 18421 | Double negation of a produ... |
ringsubdi 18422 | Ring multiplication distri... |
rngsubdir 18423 | Ring multiplication distri... |
mulgass2 18424 | An associative property be... |
ring1 18425 | The (smallest) structure r... |
ringn0 18426 | Rings exist. (Contributed... |
ringlghm 18427 | Left-multiplication in a r... |
ringrghm 18428 | Right-multiplication in a ... |
gsummulc1 18429 | A finite ring sum multipli... |
gsummulc2 18430 | A finite ring sum multipli... |
gsummgp0 18431 | If one factor in a finite ... |
gsumdixp 18432 | Distribute a binary produc... |
prdsmgp 18433 | The multiplicative monoid ... |
prdsmulrcl 18434 | A structure product of rin... |
prdsringd 18435 | A product of rings is a ri... |
prdscrngd 18436 | A product of commutative r... |
prds1 18437 | Value of the ring unit in ... |
pwsring 18438 | A structure power of a rin... |
pws1 18439 | Value of the ring unit in ... |
pwscrng 18440 | A structure power of a com... |
pwsmgp 18441 | The multiplicative group o... |
imasring 18442 | The image structure of a r... |
qusring2 18443 | The quotient structure of ... |
crngbinom 18444 | The binomial theorem for c... |
opprval 18447 | Value of the opposite ring... |
opprmulfval 18448 | Value of the multiplicatio... |
opprmul 18449 | Value of the multiplicatio... |
crngoppr 18450 | In a commutative ring, the... |
opprlem 18451 | Lemma for ~ opprbas and ~ ... |
opprbas 18452 | Base set of an opposite ri... |
oppradd 18453 | Addition operation of an o... |
opprring 18454 | An opposite ring is a ring... |
opprringb 18455 | Bidirectional form of ~ op... |
oppr0 18456 | Additive identity of an op... |
oppr1 18457 | Multiplicative identity of... |
opprneg 18458 | The negative function in a... |
opprsubg 18459 | Being a subgroup is a symm... |
mulgass3 18460 | An associative property be... |
reldvdsr 18467 | The divides relation is a ... |
dvdsrval 18468 | Value of the divides relat... |
dvdsr 18469 | Value of the divides relat... |
dvdsr2 18470 | Value of the divides relat... |
dvdsrmul 18471 | A left-multiple of ` X ` i... |
dvdsrcl 18472 | Closure of a dividing elem... |
dvdsrcl2 18473 | Closure of a dividing elem... |
dvdsrid 18474 | An element in a (unital) r... |
dvdsrtr 18475 | Divisibility is transitive... |
dvdsrmul1 18476 | The divisibility relation ... |
dvdsrneg 18477 | An element divides its neg... |
dvdsr01 18478 | In a ring, zero is divisib... |
dvdsr02 18479 | Only zero is divisible by ... |
isunit 18480 | Property of being a unit o... |
1unit 18481 | The multiplicative identit... |
unitcl 18482 | A unit is an element of th... |
unitss 18483 | The set of units is contai... |
opprunit 18484 | Being a unit is a symmetri... |
crngunit 18485 | Property of being a unit i... |
dvdsunit 18486 | A divisor of a unit is a u... |
unitmulcl 18487 | The product of units is a ... |
unitmulclb 18488 | Reversal of ~ unitmulcl in... |
unitgrpbas 18489 | The base set of the group ... |
unitgrp 18490 | The group of units is a gr... |
unitabl 18491 | The group of units of a co... |
unitgrpid 18492 | The identity of the multip... |
unitsubm 18493 | The group of units is a su... |
invrfval 18496 | Multiplicative inverse fun... |
unitinvcl 18497 | The inverse of a unit exis... |
unitinvinv 18498 | The inverse of the inverse... |
ringinvcl 18499 | The inverse of a unit is a... |
unitlinv 18500 | A unit times its inverse i... |
unitrinv 18501 | A unit times its inverse i... |
1rinv 18502 | The inverse of the identit... |
0unit 18503 | The additive identity is a... |
unitnegcl 18504 | The negative of a unit is ... |
dvrfval 18507 | Division operation in a ri... |
dvrval 18508 | Division operation in a ri... |
dvrcl 18509 | Closure of division operat... |
unitdvcl 18510 | The units are closed under... |
dvrid 18511 | A cancellation law for div... |
dvr1 18512 | A cancellation law for div... |
dvrass 18513 | An associative law for div... |
dvrcan1 18514 | A cancellation law for div... |
dvrcan3 18515 | A cancellation law for div... |
dvreq1 18516 | A cancellation law for div... |
ringinvdv 18517 | Write the inverse function... |
rngidpropd 18518 | The ring identity depends ... |
dvdsrpropd 18519 | The divisibility relation ... |
unitpropd 18520 | The set of units depends o... |
invrpropd 18521 | The ring inverse function ... |
isirred 18522 | An irreducible element of ... |
isnirred 18523 | The property of being a no... |
isirred2 18524 | Expand out the class diffe... |
opprirred 18525 | Irreducibility is symmetri... |
irredn0 18526 | The additive identity is n... |
irredcl 18527 | An irreducible element is ... |
irrednu 18528 | An irreducible element is ... |
irredn1 18529 | The multiplicative identit... |
irredrmul 18530 | The product of an irreduci... |
irredlmul 18531 | The product of a unit and ... |
irredmul 18532 | If product of two elements... |
irredneg 18533 | The negative of an irreduc... |
irrednegb 18534 | An element is irreducible ... |
dfrhm2 18540 | The property of a ring hom... |
rhmrcl1 18542 | Reverse closure of a ring ... |
rhmrcl2 18543 | Reverse closure of a ring ... |
isrhm 18544 | A function is a ring homom... |
rhmmhm 18545 | A ring homomorphism is a h... |
isrim0 18546 | An isomorphism of rings is... |
rimrcl 18547 | Reverse closure for an iso... |
rhmghm 18548 | A ring homomorphism is an ... |
rhmf 18549 | A ring homomorphism is a f... |
rhmmul 18550 | A homomorphism of rings pr... |
isrhm2d 18551 | Demonstration of ring homo... |
isrhmd 18552 | Demonstration of ring homo... |
rhm1 18553 | Ring homomorphisms are req... |
idrhm 18554 | The identity homomorphism ... |
rhmf1o 18555 | A ring homomorphism is bij... |
isrim 18556 | An isomorphism of rings is... |
rimf1o 18557 | An isomorphism of rings is... |
rimrhm 18558 | An isomorphism of rings is... |
rimgim 18559 | An isomorphism of rings is... |
rhmco 18560 | The composition of ring ho... |
pwsco1rhm 18561 | Right composition with a f... |
pwsco2rhm 18562 | Left composition with a ri... |
f1rhm0to0 18563 | If a ring homomorphism ` F... |
f1rhm0to0ALT 18564 | Alternate proof for ~ f1rh... |
rim0to0 18565 | A ring isomorphism maps th... |
kerf1hrm 18566 | A ring homomorphism ` F ` ... |
brric 18567 | The relation "is isomorphi... |
brric2 18568 | The relation "is isomorphi... |
ricgic 18569 | If two rings are (ring) is... |
isdrng 18574 | The predicate "is a divisi... |
drngunit 18575 | Elementhood in the set of ... |
drngui 18576 | The set of units of a divi... |
drngring 18577 | A division ring is a ring.... |
drnggrp 18578 | A division ring is a group... |
isfld 18579 | A field is a commutative d... |
isdrng2 18580 | A division ring can equiva... |
drngprop 18581 | If two structures have the... |
drngmgp 18582 | A division ring contains a... |
drngmcl 18583 | The product of two nonzero... |
drngid 18584 | A division ring's unit is ... |
drngunz 18585 | A division ring's unit is ... |
drngid2 18586 | Properties showing that an... |
drnginvrcl 18587 | Closure of the multiplicat... |
drnginvrn0 18588 | The multiplicative inverse... |
drnginvrl 18589 | Property of the multiplica... |
drnginvrr 18590 | Property of the multiplica... |
drngmul0or 18591 | A product is zero iff one ... |
drngmulne0 18592 | A product is nonzero iff b... |
drngmuleq0 18593 | An element is zero iff its... |
opprdrng 18594 | The opposite of a division... |
isdrngd 18595 | Properties that determine ... |
isdrngrd 18596 | Properties that determine ... |
drngpropd 18597 | If two structures have the... |
fldpropd 18598 | If two structures have the... |
issubrg 18603 | The subring predicate. (C... |
subrgss 18604 | A subring is a subset. (C... |
subrgid 18605 | Every ring is a subring of... |
subrgring 18606 | A subring is a ring. (Con... |
subrgcrng 18607 | A subring of a commutative... |
subrgrcl 18608 | Reverse closure for a subr... |
subrgsubg 18609 | A subring is a subgroup. ... |
subrg0 18610 | A subring always has the s... |
subrg1cl 18611 | A subring contains the mul... |
subrgbas 18612 | Base set of a subring stru... |
subrg1 18613 | A subring always has the s... |
subrgacl 18614 | A subring is closed under ... |
subrgmcl 18615 | A subgroup is closed under... |
subrgsubm 18616 | A subring is a submonoid o... |
subrgdvds 18617 | If an element divides anot... |
subrguss 18618 | A unit of a subring is a u... |
subrginv 18619 | A subring always has the s... |
subrgdv 18620 | A subring always has the s... |
subrgunit 18621 | An element of a ring is a ... |
subrgugrp 18622 | The units of a subring for... |
issubrg2 18623 | Characterize the subrings ... |
opprsubrg 18624 | Being a subring is a symme... |
subrgint 18625 | The intersection of a none... |
subrgin 18626 | The intersection of two su... |
subrgmre 18627 | The subrings of a ring are... |
issubdrg 18628 | Characterize the subfields... |
subsubrg 18629 | A subring of a subring is ... |
subsubrg2 18630 | The set of subrings of a s... |
issubrg3 18631 | A subring is an additive s... |
resrhm 18632 | Restriction of a ring homo... |
rhmeql 18633 | The equalizer of two ring ... |
rhmima 18634 | The homomorphic image of a... |
cntzsubr 18635 | Centralizers in a ring are... |
pwsdiagrhm 18636 | Diagonal homomorphism into... |
subrgpropd 18637 | If two structures have the... |
rhmpropd 18638 | Ring homomorphism depends ... |
abvfval 18641 | Value of the set of absolu... |
isabv 18642 | Elementhood in the set of ... |
isabvd 18643 | Properties that determine ... |
abvrcl 18644 | Reverse closure for the ab... |
abvfge0 18645 | An absolute value is a fun... |
abvf 18646 | An absolute value is a fun... |
abvcl 18647 | An absolute value is a fun... |
abvge0 18648 | The absolute value of a nu... |
abveq0 18649 | The value of an absolute v... |
abvne0 18650 | The absolute value of a no... |
abvgt0 18651 | The absolute value of a no... |
abvmul 18652 | An absolute value distribu... |
abvtri 18653 | An absolute value satisfie... |
abv0 18654 | The absolute value of zero... |
abv1z 18655 | The absolute value of one ... |
abv1 18656 | The absolute value of one ... |
abvneg 18657 | The absolute value of a ne... |
abvsubtri 18658 | An absolute value satisfie... |
abvrec 18659 | The absolute value distrib... |
abvdiv 18660 | The absolute value distrib... |
abvdom 18661 | Any ring with an absolute ... |
abvres 18662 | The restriction of an abso... |
abvtrivd 18663 | The trivial absolute value... |
abvtriv 18664 | The trivial absolute value... |
abvpropd 18665 | If two structures have the... |
staffval 18670 | The functionalization of t... |
stafval 18671 | The functionalization of t... |
staffn 18672 | The functionalization is e... |
issrng 18673 | The predicate "is a star r... |
srngrhm 18674 | The involution function in... |
srngring 18675 | A star ring is a ring. (C... |
srngcnv 18676 | The involution function in... |
srngf1o 18677 | The involution function in... |
srngcl 18678 | The involution function in... |
srngnvl 18679 | The involution function in... |
srngadd 18680 | The involution function in... |
srngmul 18681 | The involution function in... |
srng1 18682 | The conjugate of the ring ... |
srng0 18683 | The conjugate of the ring ... |
issrngd 18684 | Properties that determine ... |
idsrngd 18685 | A commutative ring is a st... |
islmod 18690 | The predicate "is a left m... |
lmodlema 18691 | Lemma for properties of a ... |
islmodd 18692 | Properties that determine ... |
lmodgrp 18693 | A left module is a group. ... |
lmodring 18694 | The scalar component of a ... |
lmodfgrp 18695 | The scalar component of a ... |
lmodbn0 18696 | The base set of a left mod... |
lmodacl 18697 | Closure of ring addition f... |
lmodmcl 18698 | Closure of ring multiplica... |
lmodsn0 18699 | The set of scalars in a le... |
lmodvacl 18700 | Closure of vector addition... |
lmodass 18701 | Left module vector sum is ... |
lmodlcan 18702 | Left cancellation law for ... |
lmodvscl 18703 | Closure of scalar product ... |
scaffval 18704 | The scalar multiplication ... |
scafval 18705 | The scalar multiplication ... |
scafeq 18706 | If the scalar multiplicati... |
scaffn 18707 | The scalar multiplication ... |
lmodscaf 18708 | The scalar multiplication ... |
lmodvsdi 18709 | Distributive law for scala... |
lmodvsdir 18710 | Distributive law for scala... |
lmodvsass 18711 | Associative law for scalar... |
lmod0cl 18712 | The ring zero in a left mo... |
lmod1cl 18713 | The ring unit in a left mo... |
lmodvs1 18714 | Scalar product with ring u... |
lmod0vcl 18715 | The zero vector is a vecto... |
lmod0vlid 18716 | Left identity law for the ... |
lmod0vrid 18717 | Right identity law for the... |
lmod0vid 18718 | Identity equivalent to the... |
lmod0vs 18719 | Zero times a vector is the... |
lmodvs0 18720 | Anything times the zero ve... |
lmodvsmmulgdi 18721 | Distributive law for a gro... |
lmodfopnelem1 18722 | Lemma 1 for ~ lmodfopne . ... |
lmodfopnelem2 18723 | Lemma 2 for ~ lmodfopne . ... |
lmodfopne 18724 | The (functionalized) opera... |
lcomf 18725 | A linear-combination sum i... |
lcomfsupp 18726 | A linear-combination sum i... |
lmodvnegcl 18727 | Closure of vector negative... |
lmodvnegid 18728 | Addition of a vector with ... |
lmodvneg1 18729 | Minus 1 times a vector is ... |
lmodvsneg 18730 | Multiplication of a vector... |
lmodvsubcl 18731 | Closure of vector subtract... |
lmodcom 18732 | Left module vector sum is ... |
lmodabl 18733 | A left module is an abelia... |
lmodcmn 18734 | A left module is a commuta... |
lmodnegadd 18735 | Distribute negation throug... |
lmod4 18736 | Commutative/associative la... |
lmodvsubadd 18737 | Relationship between vecto... |
lmodvaddsub4 18738 | Vector addition/subtractio... |
lmodvpncan 18739 | Addition/subtraction cance... |
lmodvnpcan 18740 | Cancellation law for vecto... |
lmodvsubval2 18741 | Value of vector subtractio... |
lmodsubvs 18742 | Subtraction of a scalar pr... |
lmodsubdi 18743 | Scalar multiplication dist... |
lmodsubdir 18744 | Scalar multiplication dist... |
lmodsubeq0 18745 | If the difference between ... |
lmodsubid 18746 | Subtraction of a vector fr... |
lmodvsghm 18747 | Scalar multiplication of t... |
lmodprop2d 18748 | If two structures have the... |
lmodpropd 18749 | If two structures have the... |
gsumvsmul 18750 | Pull a scalar multiplicati... |
mptscmfsupp0 18751 | A mapping to a scalar prod... |
mptscmfsuppd 18752 | A function mapping to a sc... |
lssset 18755 | The set of all (not necess... |
islss 18756 | The predicate "is a subspa... |
islssd 18757 | Properties that determine ... |
lssss 18758 | A subspace is a set of vec... |
lssel 18759 | A subspace member is a vec... |
lss1 18760 | The set of vectors in a le... |
lssuni 18761 | The union of all subspaces... |
lssn0 18762 | A subspace is not empty. ... |
00lss 18763 | The empty structure has no... |
lsscl 18764 | Closure property of a subs... |
lssvsubcl 18765 | Closure of vector subtract... |
lssvancl1 18766 | Non-closure: if one vector... |
lssvancl2 18767 | Non-closure: if one vector... |
lss0cl 18768 | The zero vector belongs to... |
lsssn0 18769 | The singleton of the zero ... |
lss0ss 18770 | The zero subspace is inclu... |
lssle0 18771 | No subspace is smaller tha... |
lssne0 18772 | A nonzero subspace has a n... |
lssneln0 18773 | A vector which doesn't bel... |
lssssr 18774 | Conclude subspace ordering... |
lssvacl 18775 | Closure of vector addition... |
lssvscl 18776 | Closure of scalar product ... |
lssvnegcl 18777 | Closure of negative vector... |
lsssubg 18778 | All subspaces are subgroup... |
lsssssubg 18779 | All subspaces are subgroup... |
islss3 18780 | A linear subspace of a mod... |
lsslmod 18781 | A submodule is a module. ... |
lsslss 18782 | The subspaces of a subspac... |
islss4 18783 | A linear subspace is a sub... |
lss1d 18784 | One-dimensional subspace (... |
lssintcl 18785 | The intersection of a none... |
lssincl 18786 | The intersection of two su... |
lssmre 18787 | The subspaces of a module ... |
lssacs 18788 | Submodules are an algebrai... |
prdsvscacl 18789 | Pointwise scalar multiplic... |
prdslmodd 18790 | The product of a family of... |
pwslmod 18791 | The product of a family of... |
lspfval 18794 | The span function for a le... |
lspf 18795 | The span operator on a lef... |
lspval 18796 | The span of a set of vecto... |
lspcl 18797 | The span of a set of vecto... |
lspsncl 18798 | The span of a singleton is... |
lspprcl 18799 | The span of a pair is a su... |
lsptpcl 18800 | The span of an unordered t... |
lspsnsubg 18801 | The span of a singleton is... |
00lsp 18802 | ~ fvco4i lemma for linear ... |
lspid 18803 | The span of a subspace is ... |
lspssv 18804 | A span is a set of vectors... |
lspss 18805 | Span preserves subset orde... |
lspssid 18806 | A set of vectors is a subs... |
lspidm 18807 | The span of a set of vecto... |
lspun 18808 | The span of union is the s... |
lspssp 18809 | If a set of vectors is a s... |
mrclsp 18810 | Moore closure generalizes ... |
lspsnss 18811 | The span of the singleton ... |
lspsnel3 18812 | A member of the span of th... |
lspprss 18813 | The span of a pair of vect... |
lspsnid 18814 | A vector belongs to the sp... |
lspsnel6 18815 | Relationship between a vec... |
lspsnel5 18816 | Relationship between a vec... |
lspsnel5a 18817 | Relationship between a vec... |
lspprid1 18818 | A member of a pair of vect... |
lspprid2 18819 | A member of a pair of vect... |
lspprvacl 18820 | The sum of two vectors bel... |
lssats2 18821 | A way to express atomistic... |
lspsneli 18822 | A scalar product with a ve... |
lspsn 18823 | Span of the singleton of a... |
lspsnel 18824 | Member of span of the sing... |
lspsnvsi 18825 | Span of a scalar product o... |
lspsnss2 18826 | Comparable spans of single... |
lspsnneg 18827 | Negation does not change t... |
lspsnsub 18828 | Swapping subtraction order... |
lspsn0 18829 | Span of the singleton of t... |
lsp0 18830 | Span of the empty set. (C... |
lspuni0 18831 | Union of the span of the e... |
lspun0 18832 | The span of a union with t... |
lspsneq0 18833 | Span of the singleton is t... |
lspsneq0b 18834 | Equal singleton spans impl... |
lmodindp1 18835 | Two independent (non-colin... |
lsslsp 18836 | Spans in submodules corres... |
lss0v 18837 | The zero vector in a submo... |
lsspropd 18838 | If two structures have the... |
lsppropd 18839 | If two structures have the... |
reldmlmhm 18846 | Lemma for module homomorph... |
lmimfn 18847 | Lemma for module isomorphi... |
islmhm 18848 | Property of being a homomo... |
islmhm3 18849 | Property of a module homom... |
lmhmlem 18850 | Non-quantified consequence... |
lmhmsca 18851 | A homomorphism of left mod... |
lmghm 18852 | A homomorphism of left mod... |
lmhmlmod2 18853 | A homomorphism of left mod... |
lmhmlmod1 18854 | A homomorphism of left mod... |
lmhmf 18855 | A homomorphism of left mod... |
lmhmlin 18856 | A homomorphism of left mod... |
lmodvsinv 18857 | Multiplication of a vector... |
lmodvsinv2 18858 | Multiplying a negated vect... |
islmhm2 18859 | A one-equation proof of li... |
islmhmd 18860 | Deduction for a module hom... |
0lmhm 18861 | The constant zero linear f... |
idlmhm 18862 | The identity function on a... |
invlmhm 18863 | The negative function on a... |
lmhmco 18864 | The composition of two mod... |
lmhmplusg 18865 | The pointwise sum of two l... |
lmhmvsca 18866 | The pointwise scalar produ... |
lmhmf1o 18867 | A bijective module homomor... |
lmhmima 18868 | The image of a subspace un... |
lmhmpreima 18869 | The inverse image of a sub... |
lmhmlsp 18870 | Homomorphisms preserve spa... |
lmhmrnlss 18871 | The range of a homomorphis... |
lmhmkerlss 18872 | The kernel of a homomorphi... |
reslmhm 18873 | Restriction of a homomorph... |
reslmhm2 18874 | Expansion of the codomain ... |
reslmhm2b 18875 | Expansion of the codomain ... |
lmhmeql 18876 | The equalizer of two modul... |
lspextmo 18877 | A linear function is compl... |
pwsdiaglmhm 18878 | Diagonal homomorphism into... |
pwssplit0 18879 | Splitting for structure po... |
pwssplit1 18880 | Splitting for structure po... |
pwssplit2 18881 | Splitting for structure po... |
pwssplit3 18882 | Splitting for structure po... |
islmim 18883 | An isomorphism of left mod... |
lmimf1o 18884 | An isomorphism of left mod... |
lmimlmhm 18885 | An isomorphism of modules ... |
lmimgim 18886 | An isomorphism of modules ... |
islmim2 18887 | An isomorphism of left mod... |
lmimcnv 18888 | The converse of a bijectiv... |
brlmic 18889 | The relation "is isomorphi... |
brlmici 18890 | Prove isomorphic by an exp... |
lmiclcl 18891 | Isomorphism implies the le... |
lmicrcl 18892 | Isomorphism implies the ri... |
lmicsym 18893 | Module isomorphism is symm... |
lmhmpropd 18894 | Module homomorphism depend... |
islbs 18897 | The predicate " ` B ` is a... |
lbsss 18898 | A basis is a set of vector... |
lbsel 18899 | An element of a basis is a... |
lbssp 18900 | The span of a basis is the... |
lbsind 18901 | A basis is linearly indepe... |
lbsind2 18902 | A basis is linearly indepe... |
lbspss 18903 | No proper subset of a basi... |
lsmcl 18904 | The sum of two subspaces i... |
lsmspsn 18905 | Member of subspace sum of ... |
lsmelval2 18906 | Subspace sum membership in... |
lsmsp 18907 | Subspace sum in terms of s... |
lsmsp2 18908 | Subspace sum of spans of s... |
lsmssspx 18909 | Subspace sum (in its exten... |
lsmpr 18910 | The span of a pair of vect... |
lsppreli 18911 | A vector expressed as a su... |
lsmelpr 18912 | Two ways to say that a vec... |
lsppr0 18913 | The span of a vector paire... |
lsppr 18914 | Span of a pair of vectors.... |
lspprel 18915 | Member of the span of a pa... |
lspprabs 18916 | Absorption of vector sum i... |
lspvadd 18917 | The span of a vector sum i... |
lspsntri 18918 | Triangle-type inequality f... |
lspsntrim 18919 | Triangle-type inequality f... |
lbspropd 18920 | If two structures have the... |
pj1lmhm 18921 | The left projection functi... |
pj1lmhm2 18922 | The left projection functi... |
islvec 18925 | The predicate "is a left v... |
lvecdrng 18926 | The set of scalars of a le... |
lveclmod 18927 | A left vector space is a l... |
lsslvec 18928 | A vector subspace is a vec... |
lvecvs0or 18929 | If a scalar product is zer... |
lvecvsn0 18930 | A scalar product is nonzer... |
lssvs0or 18931 | If a scalar product belong... |
lvecvscan 18932 | Cancellation law for scala... |
lvecvscan2 18933 | Cancellation law for scala... |
lvecinv 18934 | Invert coefficient of scal... |
lspsnvs 18935 | A nonzero scalar product d... |
lspsneleq 18936 | Membership relation that i... |
lspsncmp 18937 | Comparable spans of nonzer... |
lspsnne1 18938 | Two ways to express that v... |
lspsnne2 18939 | Two ways to express that v... |
lspsnnecom 18940 | Swap two vectors with diff... |
lspabs2 18941 | Absorption law for span of... |
lspabs3 18942 | Absorption law for span of... |
lspsneq 18943 | Equal spans of singletons ... |
lspsneu 18944 | Nonzero vectors with equal... |
lspsnel4 18945 | A member of the span of th... |
lspdisj 18946 | The span of a vector not i... |
lspdisjb 18947 | A nonzero vector is not in... |
lspdisj2 18948 | Unequal spans are disjoint... |
lspfixed 18949 | Show membership in the spa... |
lspexch 18950 | Exchange property for span... |
lspexchn1 18951 | Exchange property for span... |
lspexchn2 18952 | Exchange property for span... |
lspindpi 18953 | Partial independence prope... |
lspindp1 18954 | Alternate way to say 3 vec... |
lspindp2l 18955 | Alternate way to say 3 vec... |
lspindp2 18956 | Alternate way to say 3 vec... |
lspindp3 18957 | Independence of 2 vectors ... |
lspindp4 18958 | (Partial) independence of ... |
lvecindp 18959 | Compute the ` X ` coeffici... |
lvecindp2 18960 | Sums of independent vector... |
lspsnsubn0 18961 | Unequal singleton spans im... |
lsmcv 18962 | Subspace sum has the cover... |
lspsolvlem 18963 | Lemma for ~ lspsolv . (Co... |
lspsolv 18964 | If ` X ` is in the span of... |
lssacsex 18965 | In a vector space, subspac... |
lspsnat 18966 | There is no subspace stric... |
lspsncv0 18967 | The span of a singleton co... |
lsppratlem1 18968 | Lemma for ~ lspprat . Let... |
lsppratlem2 18969 | Lemma for ~ lspprat . Sho... |
lsppratlem3 18970 | Lemma for ~ lspprat . In ... |
lsppratlem4 18971 | Lemma for ~ lspprat . In ... |
lsppratlem5 18972 | Lemma for ~ lspprat . Com... |
lsppratlem6 18973 | Lemma for ~ lspprat . Neg... |
lspprat 18974 | A proper subspace of the s... |
islbs2 18975 | An equivalent formulation ... |
islbs3 18976 | An equivalent formulation ... |
lbsacsbs 18977 | Being a basis in a vector ... |
lvecdim 18978 | The dimension theorem for ... |
lbsextlem1 18979 | Lemma for ~ lbsext . The ... |
lbsextlem2 18980 | Lemma for ~ lbsext . Sinc... |
lbsextlem3 18981 | Lemma for ~ lbsext . A ch... |
lbsextlem4 18982 | Lemma for ~ lbsext . ~ lbs... |
lbsextg 18983 | For any linearly independe... |
lbsext 18984 | For any linearly independe... |
lbsexg 18985 | Every vector space has a b... |
lbsex 18986 | Every vector space has a b... |
lvecprop2d 18987 | If two structures have the... |
lvecpropd 18988 | If two structures have the... |
sraval 18997 | Lemma for ~ srabase throug... |
sralem 18998 | Lemma for ~ srabase and si... |
srabase 18999 | Base set of a subring alge... |
sraaddg 19000 | Additive operation of a su... |
sramulr 19001 | Multiplicative operation o... |
srasca 19002 | The set of scalars of a su... |
sravsca 19003 | The scalar product operati... |
sraip 19004 | The inner product operatio... |
sratset 19005 | Topology component of a su... |
sratopn 19006 | Topology component of a su... |
srads 19007 | Distance function of a sub... |
sralmod 19008 | The subring algebra is a l... |
sralmod0 19009 | The subring module inherit... |
issubrngd2 19010 | Prove a subring by closure... |
rlmfn 19011 | ` ringLMod ` is a function... |
rlmval 19012 | Value of the ring module. ... |
lidlval 19013 | Value of the set of ring i... |
rspval 19014 | Value of the ring span fun... |
rlmval2 19015 | Value of the ring module e... |
rlmbas 19016 | Base set of the ring modul... |
rlmplusg 19017 | Vector addition in the rin... |
rlm0 19018 | Zero vector in the ring mo... |
rlmsub 19019 | Subtraction in the ring mo... |
rlmmulr 19020 | Ring multiplication in the... |
rlmsca 19021 | Scalars in the ring module... |
rlmsca2 19022 | Scalars in the ring module... |
rlmvsca 19023 | Scalar multiplication in t... |
rlmtopn 19024 | Topology component of the ... |
rlmds 19025 | Metric component of the ri... |
rlmlmod 19026 | The ring module is a modul... |
rlmlvec 19027 | The ring module over a div... |
rlmvneg 19028 | Vector negation in the rin... |
rlmscaf 19029 | Functionalized scalar mult... |
ixpsnbasval 19030 | The value of an infinite C... |
lidlss 19031 | An ideal is a subset of th... |
islidl 19032 | Predicate of being a (left... |
lidl0cl 19033 | An ideal contains 0. (Con... |
lidlacl 19034 | An ideal is closed under a... |
lidlnegcl 19035 | An ideal contains negative... |
lidlsubg 19036 | An ideal is a subgroup of ... |
lidlsubcl 19037 | An ideal is closed under s... |
lidlmcl 19038 | An ideal is closed under l... |
lidl1el 19039 | An ideal contains 1 iff it... |
lidl0 19040 | Every ring contains a zero... |
lidl1 19041 | Every ring contains a unit... |
lidlacs 19042 | The ideal system is an alg... |
rspcl 19043 | The span of a set of ring ... |
rspssid 19044 | The span of a set of ring ... |
rsp1 19045 | The span of the identity e... |
rsp0 19046 | The span of the zero eleme... |
rspssp 19047 | The ideal span of a set of... |
mrcrsp 19048 | Moore closure generalizes ... |
lidlnz 19049 | A nonzero ideal contains a... |
drngnidl 19050 | A division ring has only t... |
lidlrsppropd 19051 | The left ideals and ring s... |
2idlval 19054 | Definition of a two-sided ... |
2idlcpbl 19055 | The coset equivalence rela... |
qus1 19056 | The multiplicative identit... |
qusring 19057 | If ` S ` is a two-sided id... |
qusrhm 19058 | If ` S ` is a two-sided id... |
crngridl 19059 | In a commutative ring, the... |
crng2idl 19060 | In a commutative ring, a t... |
quscrng 19061 | The quotient of a commutat... |
lpival 19066 | Value of the set of princi... |
islpidl 19067 | Property of being a princi... |
lpi0 19068 | The zero ideal is always p... |
lpi1 19069 | The unit ideal is always p... |
islpir 19070 | Principal ideal rings are ... |
lpiss 19071 | Principal ideals are a sub... |
islpir2 19072 | Principal ideal rings are ... |
lpirring 19073 | Principal ideal rings are ... |
drnglpir 19074 | Division rings are princip... |
rspsn 19075 | Membership in principal id... |
lidldvgen 19076 | An element generates an id... |
lpigen 19077 | An ideal is principal iff ... |
isnzr 19080 | Property of a nonzero ring... |
nzrnz 19081 | One and zero are different... |
nzrring 19082 | A nonzero ring is a ring. ... |
drngnzr 19083 | All division rings are non... |
isnzr2 19084 | Equivalent characterizatio... |
isnzr2hash 19085 | Equivalent characterizatio... |
opprnzr 19086 | The opposite of a nonzero ... |
ringelnzr 19087 | A ring is nonzero if it ha... |
nzrunit 19088 | A unit is nonzero in any n... |
subrgnzr 19089 | A subring of a nonzero rin... |
0ringnnzr 19090 | A ring is a zero ring iff ... |
0ring 19091 | If a ring has only one ele... |
0ring01eq 19092 | In a ring with only one el... |
01eq0ring 19093 | If the zero and the identi... |
0ring01eqbi 19094 | In a unital ring the zero ... |
rng1nnzr 19095 | The (smallest) structure r... |
ring1zr 19096 | The only (unital) ring wit... |
rngen1zr 19097 | The only (unital) ring wit... |
ringen1zr 19098 | The only unital ring with ... |
rng1nfld 19099 | The zero ring is not a fie... |
rrgval 19108 | Value of the set or left-r... |
isrrg 19109 | Membership in the set of l... |
rrgeq0i 19110 | Property of a left-regular... |
rrgeq0 19111 | Left-multiplication by a l... |
rrgsupp 19112 | Left multiplication by a l... |
rrgss 19113 | Left-regular elements are ... |
unitrrg 19114 | Units are regular elements... |
isdomn 19115 | Expand definition of a dom... |
domnnzr 19116 | A domain is a nonzero ring... |
domnring 19117 | A domain is a ring. (Cont... |
domneq0 19118 | In a domain, a product is ... |
domnmuln0 19119 | In a domain, a product of ... |
isdomn2 19120 | A ring is a domain iff all... |
domnrrg 19121 | In a domain, any nonzero e... |
opprdomn 19122 | The opposite of a domain i... |
abvn0b 19123 | Another characterization o... |
drngdomn 19124 | A division ring is a domai... |
isidom 19125 | An integral domain is a co... |
fldidom 19126 | A field is an integral dom... |
fidomndrnglem 19127 | Lemma for ~ fidomndrng . ... |
fidomndrng 19128 | A finite domain is a divis... |
fiidomfld 19129 | A finite integral domain i... |
isassa 19136 | The properties of an assoc... |
assalem 19137 | The properties of an assoc... |
assaass 19138 | Left-associative property ... |
assaassr 19139 | Right-associative property... |
assalmod 19140 | An associative algebra is ... |
assaring 19141 | An associative algebra is ... |
assasca 19142 | An associative algebra's s... |
assa2ass 19143 | Left- and right-associativ... |
isassad 19144 | Sufficient condition for b... |
issubassa 19145 | The subalgebras of an asso... |
sraassa 19146 | The subring algebra over a... |
rlmassa 19147 | The ring module over a com... |
assapropd 19148 | If two structures have the... |
aspval 19149 | Value of the algebraic clo... |
asplss 19150 | The algebraic span of a se... |
aspid 19151 | The algebraic span of a su... |
aspsubrg 19152 | The algebraic span of a se... |
aspss 19153 | Span preserves subset orde... |
aspssid 19154 | A set of vectors is a subs... |
asclfval 19155 | Function value of the alge... |
asclval 19156 | Value of a mapped algebra ... |
asclfn 19157 | Unconditional functionalit... |
asclf 19158 | The algebra scalars functi... |
asclghm 19159 | The algebra scalars functi... |
asclmul1 19160 | Left multiplication by a l... |
asclmul2 19161 | Right multiplication by a ... |
asclinvg 19162 | The group inverse (negatio... |
asclrhm 19163 | The scalar injection is a ... |
rnascl 19164 | The set of injected scalar... |
ressascl 19165 | The injection of scalars i... |
issubassa2 19166 | A subring of a unital alge... |
asclpropd 19167 | If two structures have the... |
aspval2 19168 | The algebraic closure is t... |
assamulgscmlem1 19169 | Lemma 1 for ~ assamulgscm ... |
assamulgscmlem2 19170 | Lemma for ~ assamulgscm (i... |
assamulgscm 19171 | Exponentiation of a scalar... |
reldmpsr 19182 | The multivariate power ser... |
psrval 19183 | Value of the multivariate ... |
psrvalstr 19184 | The multivariate power ser... |
psrbag 19185 | Elementhood in the set of ... |
psrbagf 19186 | A finite bag is a function... |
snifpsrbag 19187 | A bag containing one eleme... |
fczpsrbag 19188 | The constant function equa... |
psrbaglesupp 19189 | The support of a dominated... |
psrbaglecl 19190 | The set of finite bags is ... |
psrbagaddcl 19191 | The sum of two finite bags... |
psrbagcon 19192 | The analogue of the statem... |
psrbaglefi 19193 | There are finitely many ba... |
psrbagconcl 19194 | The complement of a bag is... |
psrbagconf1o 19195 | Bag complementation is a b... |
gsumbagdiaglem 19196 | Lemma for ~ gsumbagdiag . ... |
gsumbagdiag 19197 | Two-dimensional commutatio... |
psrass1lem 19198 | A group sum commutation us... |
psrbas 19199 | The base set of the multiv... |
psrelbas 19200 | An element of the set of p... |
psrelbasfun 19201 | An element of the set of p... |
psrplusg 19202 | The addition operation of ... |
psradd 19203 | The addition operation of ... |
psraddcl 19204 | Closure of the power serie... |
psrmulr 19205 | The multiplication operati... |
psrmulfval 19206 | The multiplication operati... |
psrmulval 19207 | The multiplication operati... |
psrmulcllem 19208 | Closure of the power serie... |
psrmulcl 19209 | Closure of the power serie... |
psrsca 19210 | The scalar field of the mu... |
psrvscafval 19211 | The scalar multiplication ... |
psrvsca 19212 | The scalar multiplication ... |
psrvscaval 19213 | The scalar multiplication ... |
psrvscacl 19214 | Closure of the power serie... |
psr0cl 19215 | The zero element of the ri... |
psr0lid 19216 | The zero element of the ri... |
psrnegcl 19217 | The negative function in t... |
psrlinv 19218 | The negative function in t... |
psrgrp 19219 | The ring of power series i... |
psr0 19220 | The zero element of the ri... |
psrneg 19221 | The negative function of t... |
psrlmod 19222 | The ring of power series i... |
psr1cl 19223 | The identity element of th... |
psrlidm 19224 | The identity element of th... |
psrridm 19225 | The identity element of th... |
psrass1 19226 | Associative identity for t... |
psrdi 19227 | Distributive law for the r... |
psrdir 19228 | Distributive law for the r... |
psrass23l 19229 | Associative identity for t... |
psrcom 19230 | Commutative law for the ri... |
psrass23 19231 | Associative identities for... |
psrring 19232 | The ring of power series i... |
psr1 19233 | The identity element of th... |
psrcrng 19234 | The ring of power series i... |
psrassa 19235 | The ring of power series i... |
resspsrbas 19236 | A restricted power series ... |
resspsradd 19237 | A restricted power series ... |
resspsrmul 19238 | A restricted power series ... |
resspsrvsca 19239 | A restricted power series ... |
subrgpsr 19240 | A subring of the base ring... |
mvrfval 19241 | Value of the generating el... |
mvrval 19242 | Value of the generating el... |
mvrval2 19243 | Value of the generating el... |
mvrid 19244 | The ` X i ` -th coefficien... |
mvrf 19245 | The power series variable ... |
mvrf1 19246 | The power series variable ... |
mvrcl2 19247 | A power series variable is... |
reldmmpl 19248 | The multivariate polynomia... |
mplval 19249 | Value of the set of multiv... |
mplbas 19250 | Base set of the set of mul... |
mplelbas 19251 | Property of being a polyno... |
mplval2 19252 | Self-referential expressio... |
mplbasss 19253 | The set of polynomials is ... |
mplelf 19254 | A polynomial is defined as... |
mplsubglem 19255 | If ` A ` is an ideal of se... |
mpllsslem 19256 | If ` A ` is an ideal of su... |
mplsubglem2 19257 | Lemma for ~ mplsubg and ~ ... |
mplsubg 19258 | The set of polynomials is ... |
mpllss 19259 | The set of polynomials is ... |
mplsubrglem 19260 | Lemma for ~ mplsubrg . (C... |
mplsubrg 19261 | The set of polynomials is ... |
mpl0 19262 | The zero polynomial. (Con... |
mpladd 19263 | The addition operation on ... |
mplmul 19264 | The multiplication operati... |
mpl1 19265 | The identity element of th... |
mplsca 19266 | The scalar field of a mult... |
mplvsca2 19267 | The scalar multiplication ... |
mplvsca 19268 | The scalar multiplication ... |
mplvscaval 19269 | The scalar multiplication ... |
mvrcl 19270 | A power series variable is... |
mplgrp 19271 | The polynomial ring is a g... |
mpllmod 19272 | The polynomial ring is a l... |
mplring 19273 | The polynomial ring is a r... |
mplcrng 19274 | The polynomial ring is a c... |
mplassa 19275 | The polynomial ring is an ... |
ressmplbas2 19276 | The base set of a restrict... |
ressmplbas 19277 | A restricted polynomial al... |
ressmpladd 19278 | A restricted polynomial al... |
ressmplmul 19279 | A restricted polynomial al... |
ressmplvsca 19280 | A restricted power series ... |
subrgmpl 19281 | A subring of the base ring... |
subrgmvr 19282 | The variables in a subring... |
subrgmvrf 19283 | The variables in a polynom... |
mplmon 19284 | A monomial is a polynomial... |
mplmonmul 19285 | The product of two monomia... |
mplcoe1 19286 | Decompose a polynomial int... |
mplcoe3 19287 | Decompose a monomial in on... |
mplcoe5lem 19288 | Lemma for ~ mplcoe4 . (Co... |
mplcoe5 19289 | Decompose a monomial into ... |
mplcoe2 19290 | Decompose a monomial into ... |
mplbas2 19291 | An alternative expression ... |
ltbval 19292 | Value of the well-order on... |
ltbwe 19293 | The finite bag order is a ... |
reldmopsr 19294 | Lemma for ordered power se... |
opsrval 19295 | The value of the "ordered ... |
opsrle 19296 | An alternative expression ... |
opsrval2 19297 | Self-referential expressio... |
opsrbaslem 19298 | Get a component of the ord... |
opsrbaslemOLD 19299 | Obsolete version of ~ opsr... |
opsrbas 19300 | The base set of the ordere... |
opsrplusg 19301 | The addition operation of ... |
opsrmulr 19302 | The multiplication operati... |
opsrvsca 19303 | The scalar product operati... |
opsrsca 19304 | The scalar ring of the ord... |
opsrtoslem1 19305 | Lemma for ~ opsrtos . (Co... |
opsrtoslem2 19306 | Lemma for ~ opsrtos . (Co... |
opsrtos 19307 | The ordered power series s... |
opsrso 19308 | The ordered power series s... |
opsrcrng 19309 | The ring of ordered power ... |
opsrassa 19310 | The ring of ordered power ... |
mplrcl 19311 | Reverse closure for the po... |
mplelsfi 19312 | A polynomial treated as a ... |
mvrf2 19313 | The power series/polynomia... |
mplmon2 19314 | Express a scaled monomial.... |
psrbag0 19315 | The empty bag is a bag. (... |
psrbagsn 19316 | A singleton bag is a bag. ... |
mplascl 19317 | Value of the scalar inject... |
mplasclf 19318 | The scalar injection is a ... |
subrgascl 19319 | The scalar injection funct... |
subrgasclcl 19320 | The scalars in a polynomia... |
mplmon2cl 19321 | A scaled monomial is a pol... |
mplmon2mul 19322 | Product of scaled monomial... |
mplind 19323 | Prove a property of polyno... |
mplcoe4 19324 | Decompose a polynomial int... |
evlslem4 19329 | The support of a tensor pr... |
psrbagfsupp 19330 | Finite bags have finite no... |
psrbagev1 19331 | A bag of multipliers provi... |
psrbagev2 19332 | Closure of a sum using a b... |
evlslem2 19333 | A linear function on the p... |
evlslem6 19334 | Lemma for ~ evlseu . Fini... |
evlslem3 19335 | Lemma for ~ evlseu . Poly... |
evlslem1 19336 | Lemma for ~ evlseu , give ... |
evlseu 19337 | For a given interpretation... |
reldmevls 19338 | Well-behaved binary operat... |
mpfrcl 19339 | Reverse closure for the se... |
evlsval 19340 | Value of the polynomial ev... |
evlsval2 19341 | Characterizing properties ... |
evlsrhm 19342 | Polynomial evaluation is a... |
evlssca 19343 | Polynomial evaluation maps... |
evlsvar 19344 | Polynomial evaluation maps... |
evlval 19345 | Value of the simple/same r... |
evlrhm 19346 | The simple evaluation map ... |
evlsscasrng 19347 | The evaluation of a scalar... |
evlsca 19348 | Simple polynomial evaluati... |
evlsvarsrng 19349 | The evaluation of the vari... |
evlvar 19350 | Simple polynomial evaluati... |
mpfconst 19351 | Constants are multivariate... |
mpfproj 19352 | Projections are multivaria... |
mpfsubrg 19353 | Polynomial functions are a... |
mpff 19354 | Polynomial functions are f... |
mpfaddcl 19355 | The sum of multivariate po... |
mpfmulcl 19356 | The product of multivariat... |
mpfind 19357 | Prove a property of polyno... |
psr1baslem 19376 | The set of finite bags on ... |
psr1val 19377 | Value of the ring of univa... |
psr1crng 19378 | The ring of univariate pow... |
psr1assa 19379 | The ring of univariate pow... |
psr1tos 19380 | The ordered power series s... |
psr1bas2 19381 | The base set of the ring o... |
psr1bas 19382 | The base set of the ring o... |
vr1val 19383 | The value of the generator... |
vr1cl2 19384 | The variable ` X ` is a me... |
ply1val 19385 | The value of the set of un... |
ply1bas 19386 | The value of the base set ... |
ply1lss 19387 | Univariate polynomials for... |
ply1subrg 19388 | Univariate polynomials for... |
ply1crng 19389 | The ring of univariate pol... |
ply1assa 19390 | The ring of univariate pol... |
psr1bascl 19391 | A univariate power series ... |
psr1basf 19392 | Univariate power series ba... |
ply1basf 19393 | Univariate polynomial base... |
ply1bascl 19394 | A univariate polynomial is... |
ply1bascl2 19395 | A univariate polynomial is... |
coe1fval 19396 | Value of the univariate po... |
coe1fv 19397 | Value of an evaluated coef... |
fvcoe1 19398 | Value of a multivariate co... |
coe1fval3 19399 | Univariate power series co... |
coe1f2 19400 | Functionality of univariat... |
coe1fval2 19401 | Univariate polynomial coef... |
coe1f 19402 | Functionality of univariat... |
coe1fvalcl 19403 | A coefficient of a univari... |
coe1sfi 19404 | Finite support of univaria... |
coe1fsupp 19405 | The coefficient vector of ... |
mptcoe1fsupp 19406 | A mapping involving coeffi... |
coe1ae0 19407 | The coefficient vector of ... |
vr1cl 19408 | The generator of a univari... |
opsr0 19409 | Zero in the ordered power ... |
opsr1 19410 | One in the ordered power s... |
mplplusg 19411 | Value of addition in a pol... |
mplmulr 19412 | Value of multiplication in... |
psr1plusg 19413 | Value of addition in a uni... |
psr1vsca 19414 | Value of scalar multiplica... |
psr1mulr 19415 | Value of multiplication in... |
ply1plusg 19416 | Value of addition in a uni... |
ply1vsca 19417 | Value of scalar multiplica... |
ply1mulr 19418 | Value of multiplication in... |
ressply1bas2 19419 | The base set of a restrict... |
ressply1bas 19420 | A restricted polynomial al... |
ressply1add 19421 | A restricted polynomial al... |
ressply1mul 19422 | A restricted polynomial al... |
ressply1vsca 19423 | A restricted power series ... |
subrgply1 19424 | A subring of the base ring... |
gsumply1subr 19425 | Evaluate a group sum in a ... |
psrbaspropd 19426 | Property deduction for pow... |
psrplusgpropd 19427 | Property deduction for pow... |
mplbaspropd 19428 | Property deduction for pol... |
psropprmul 19429 | Reversing multiplication i... |
ply1opprmul 19430 | Reversing multiplication i... |
00ply1bas 19431 | Lemma for ~ ply1basfvi and... |
ply1basfvi 19432 | Protection compatibility o... |
ply1plusgfvi 19433 | Protection compatibility o... |
ply1baspropd 19434 | Property deduction for uni... |
ply1plusgpropd 19435 | Property deduction for uni... |
opsrring 19436 | Ordered power series form ... |
opsrlmod 19437 | Ordered power series form ... |
psr1ring 19438 | Univariate power series fo... |
ply1ring 19439 | Univariate polynomials for... |
psr1lmod 19440 | Univariate power series fo... |
psr1sca 19441 | Scalars of a univariate po... |
psr1sca2 19442 | Scalars of a univariate po... |
ply1lmod 19443 | Univariate polynomials for... |
ply1sca 19444 | Scalars of a univariate po... |
ply1sca2 19445 | Scalars of a univariate po... |
ply1mpl0 19446 | The univariate polynomial ... |
ply10s0 19447 | Zero times a univariate po... |
ply1mpl1 19448 | The univariate polynomial ... |
ply1ascl 19449 | The univariate polynomial ... |
subrg1ascl 19450 | The scalar injection funct... |
subrg1asclcl 19451 | The scalars in a polynomia... |
subrgvr1 19452 | The variables in a subring... |
subrgvr1cl 19453 | The variables in a polynom... |
coe1z 19454 | The coefficient vector of ... |
coe1add 19455 | The coefficient vector of ... |
coe1addfv 19456 | A particular coefficient o... |
coe1subfv 19457 | A particular coefficient o... |
coe1mul2lem1 19458 | An equivalence for ~ coe1m... |
coe1mul2lem2 19459 | An equivalence for ~ coe1m... |
coe1mul2 19460 | The coefficient vector of ... |
coe1mul 19461 | The coefficient vector of ... |
ply1moncl 19462 | Closure of the expression ... |
ply1tmcl 19463 | Closure of the expression ... |
coe1tm 19464 | Coefficient vector of a po... |
coe1tmfv1 19465 | Nonzero coefficient of a p... |
coe1tmfv2 19466 | Zero coefficient of a poly... |
coe1tmmul2 19467 | Coefficient vector of a po... |
coe1tmmul 19468 | Coefficient vector of a po... |
coe1tmmul2fv 19469 | Function value of a right-... |
coe1pwmul 19470 | Coefficient vector of a po... |
coe1pwmulfv 19471 | Function value of a right-... |
ply1scltm 19472 | A scalar is a term with ze... |
coe1sclmul 19473 | Coefficient vector of a po... |
coe1sclmulfv 19474 | A single coefficient of a ... |
coe1sclmul2 19475 | Coefficient vector of a po... |
ply1sclf 19476 | A scalar polynomial is a p... |
ply1sclcl 19477 | The value of the algebra s... |
coe1scl 19478 | Coefficient vector of a sc... |
ply1sclid 19479 | Recover the base scalar fr... |
ply1sclf1 19480 | The polynomial scalar func... |
ply1scl0 19481 | The zero scalar is zero. ... |
ply1scln0 19482 | Nonzero scalars create non... |
ply1scl1 19483 | The one scalar is the unit... |
ply1idvr1 19484 | The identity of a polynomi... |
cply1mul 19485 | The product of two constan... |
ply1coefsupp 19486 | The decomposition of a uni... |
ply1coe 19487 | Decompose a univariate pol... |
eqcoe1ply1eq 19488 | Two polynomials over the s... |
ply1coe1eq 19489 | Two polynomials over the s... |
cply1coe0 19490 | All but the first coeffici... |
cply1coe0bi 19491 | A polynomial is constant (... |
coe1fzgsumdlem 19492 | Lemma for ~ coe1fzgsumd (i... |
coe1fzgsumd 19493 | Value of an evaluated coef... |
gsumsmonply1 19494 | A finite group sum of scal... |
gsummoncoe1 19495 | A coefficient of the polyn... |
gsumply1eq 19496 | Two univariate polynomials... |
lply1binom 19497 | The binomial theorem for l... |
lply1binomsc 19498 | The binomial theorem for l... |
reldmevls1 19503 | Well-behaved binary operat... |
ply1frcl 19504 | Reverse closure for the se... |
evls1fval 19505 | Value of the univariate po... |
evls1val 19506 | Value of the univariate po... |
evls1rhmlem 19507 | Lemma for ~ evl1rhm and ~ ... |
evls1rhm 19508 | Polynomial evaluation is a... |
evls1sca 19509 | Univariate polynomial eval... |
evls1gsumadd 19510 | Univariate polynomial eval... |
evls1gsummul 19511 | Univariate polynomial eval... |
evls1varpw 19512 | Univariate polynomial eval... |
evl1fval 19513 | Value of the simple/same r... |
evl1val 19514 | Value of the simple/same r... |
evl1fval1lem 19515 | Lemma for ~ evl1fval1 . (... |
evl1fval1 19516 | Value of the simple/same r... |
evl1rhm 19517 | Polynomial evaluation is a... |
fveval1fvcl 19518 | The function value of the ... |
evl1sca 19519 | Polynomial evaluation maps... |
evl1scad 19520 | Polynomial evaluation buil... |
evl1var 19521 | Polynomial evaluation maps... |
evl1vard 19522 | Polynomial evaluation buil... |
evls1var 19523 | Univariate polynomial eval... |
evls1scasrng 19524 | The evaluation of a scalar... |
evls1varsrng 19525 | The evaluation of the vari... |
evl1addd 19526 | Polynomial evaluation buil... |
evl1subd 19527 | Polynomial evaluation buil... |
evl1muld 19528 | Polynomial evaluation buil... |
evl1vsd 19529 | Polynomial evaluation buil... |
evl1expd 19530 | Polynomial evaluation buil... |
pf1const 19531 | Constants are polynomial f... |
pf1id 19532 | The identity is a polynomi... |
pf1subrg 19533 | Polynomial functions are a... |
pf1rcl 19534 | Reverse closure for the se... |
pf1f 19535 | Polynomial functions are f... |
mpfpf1 19536 | Convert a multivariate pol... |
pf1mpf 19537 | Convert a univariate polyn... |
pf1addcl 19538 | The sum of multivariate po... |
pf1mulcl 19539 | The product of multivariat... |
pf1ind 19540 | Prove a property of polyno... |
evl1gsumdlem 19541 | Lemma for ~ evl1gsumd (ind... |
evl1gsumd 19542 | Polynomial evaluation buil... |
evl1gsumadd 19543 | Univariate polynomial eval... |
evl1gsumaddval 19544 | Value of a univariate poly... |
evl1gsummul 19545 | Univariate polynomial eval... |
evl1varpw 19546 | Univariate polynomial eval... |
evl1varpwval 19547 | Value of a univariate poly... |
evl1scvarpw 19548 | Univariate polynomial eval... |
evl1scvarpwval 19549 | Value of a univariate poly... |
evl1gsummon 19550 | Value of a univariate poly... |
cnfldstr 19569 | The field of complex numbe... |
cnfldex 19570 | The field of complex numbe... |
cnfldbas 19571 | The base set of the field ... |
cnfldadd 19572 | The addition operation of ... |
cnfldmul 19573 | The multiplication operati... |
cnfldcj 19574 | The conjugation operation ... |
cnfldtset 19575 | The topology component of ... |
cnfldle 19576 | The ordering of the field ... |
cnfldds 19577 | The metric of the field of... |
cnfldunif 19578 | The uniform structure comp... |
xrsstr 19579 | The extended real structur... |
xrsex 19580 | The extended real structur... |
xrsbas 19581 | The base set of the extend... |
xrsadd 19582 | The addition operation of ... |
xrsmul 19583 | The multiplication operati... |
xrstset 19584 | The topology component of ... |
xrsle 19585 | The ordering of the extend... |
cncrng 19586 | The complex numbers form a... |
cnring 19587 | The complex numbers form a... |
xrsmcmn 19588 | The multiplicative group o... |
cnfld0 19589 | The zero element of the fi... |
cnfld1 19590 | The unit element of the fi... |
cnfldneg 19591 | The additive inverse in th... |
cnfldplusf 19592 | The functionalized additio... |
cnfldsub 19593 | The subtraction operator i... |
cndrng 19594 | The complex numbers form a... |
cnflddiv 19595 | The division operation in ... |
cnfldinv 19596 | The multiplicative inverse... |
cnfldmulg 19597 | The group multiple functio... |
cnfldexp 19598 | The exponentiation operato... |
cnsrng 19599 | The complex numbers form a... |
xrsmgm 19600 | The (additive group of the... |
xrsnsgrp 19601 | The (additive group of the... |
xrsmgmdifsgrp 19602 | The (additive group of the... |
xrs1mnd 19603 | The extended real numbers,... |
xrs10 19604 | The zero of the extended r... |
xrs1cmn 19605 | The extended real numbers ... |
xrge0subm 19606 | The nonnegative extended r... |
xrge0cmn 19607 | The nonnegative extended r... |
xrsds 19608 | The metric of the extended... |
xrsdsval 19609 | The metric of the extended... |
xrsdsreval 19610 | The metric of the extended... |
xrsdsreclblem 19611 | Lemma for ~ xrsdsreclb . ... |
xrsdsreclb 19612 | The metric of the extended... |
cnsubmlem 19613 | Lemma for ~ nn0subm and fr... |
cnsubglem 19614 | Lemma for ~ resubdrg and f... |
cnsubrglem 19615 | Lemma for ~ resubdrg and f... |
cnsubdrglem 19616 | Lemma for ~ resubdrg and f... |
qsubdrg 19617 | The rational numbers form ... |
zsubrg 19618 | The integers form a subrin... |
gzsubrg 19619 | The gaussian integers form... |
nn0subm 19620 | The nonnegative integers f... |
rege0subm 19621 | The nonnegative reals form... |
absabv 19622 | The regular absolute value... |
zsssubrg 19623 | The integers are a subset ... |
qsssubdrg 19624 | The rational numbers are a... |
cnsubrg 19625 | There are no subrings of t... |
cnmgpabl 19626 | The unit group of the comp... |
cnmgpid 19627 | The group identity element... |
cnmsubglem 19628 | Lemma for ~ rpmsubg and fr... |
rpmsubg 19629 | The positive reals form a ... |
gzrngunitlem 19630 | Lemma for ~ gzrngunit . (... |
gzrngunit 19631 | The units on ` ZZ [ _i ] `... |
gsumfsum 19632 | Relate a group sum on ` CC... |
regsumfsum 19633 | Relate a group sum on ` ( ... |
expmhm 19634 | Exponentiation is a monoid... |
nn0srg 19635 | The nonnegative integers f... |
rge0srg 19636 | The nonnegative real numbe... |
zringcrng 19639 | The ring of integers is a ... |
zringring 19640 | The ring of integers is a ... |
zringabl 19641 | The ring of integers is an... |
zringgrp 19642 | The ring of integers is an... |
zringbas 19643 | The integers are the base ... |
zringplusg 19644 | The addition operation of ... |
zringmulg 19645 | The multiplication (group ... |
zringmulr 19646 | The multiplication operati... |
zring0 19647 | The neutral element of the... |
zring1 19648 | The multiplicative neutral... |
zringnzr 19649 | The ring of integers is a ... |
dvdsrzring 19650 | Ring divisibility in the r... |
zringlpirlem1 19651 | Lemma for ~ zringlpir . A... |
zringlpirlem2 19652 | Lemma for ~ zringlpir . A... |
zringlpirlem3 19653 | Lemma for ~ zringlpir . A... |
zringinvg 19654 | The additive inverse of an... |
zringunit 19655 | The units of ` ZZ ` are th... |
zringlpir 19656 | The integers are a princip... |
zringndrg 19657 | The integers are not a div... |
zringcyg 19658 | The integers are a cyclic ... |
zringmpg 19659 | The multiplication group o... |
prmirredlem 19660 | A positive integer is irre... |
dfprm2 19661 | The positive irreducible e... |
prmirred 19662 | The irreducible elements o... |
expghm 19663 | Exponentiation is a group ... |
mulgghm2 19664 | The powers of a group elem... |
mulgrhm 19665 | The powers of the element ... |
mulgrhm2 19666 | The powers of the element ... |
zrhval 19675 | Define the unique homomorp... |
zrhval2 19676 | Alternate value of the ` Z... |
zrhmulg 19677 | Value of the ` ZRHom ` hom... |
zrhrhmb 19678 | The ` ZRHom ` homomorphism... |
zrhrhm 19679 | The ` ZRHom ` homomorphism... |
zrh1 19680 | Interpretation of 1 in a r... |
zrh0 19681 | Interpretation of 0 in a r... |
zrhpropd 19682 | The ` ZZ ` ring homomorphi... |
zlmval 19683 | Augment an abelian group w... |
zlmlem 19684 | Lemma for ~ zlmbas and ~ z... |
zlmbas 19685 | Base set of a ` ZZ ` -modu... |
zlmplusg 19686 | Group operation of a ` ZZ ... |
zlmmulr 19687 | Ring operation of a ` ZZ `... |
zlmsca 19688 | Scalar ring of a ` ZZ ` -m... |
zlmvsca 19689 | Scalar multiplication oper... |
zlmlmod 19690 | The ` ZZ ` -module operati... |
zlmassa 19691 | The ` ZZ ` -module operati... |
chrval 19692 | Definition substitution of... |
chrcl 19693 | Closure of the characteris... |
chrid 19694 | The canonical ` ZZ ` ring ... |
chrdvds 19695 | The ` ZZ ` ring homomorphi... |
chrcong 19696 | If two integers are congru... |
chrnzr 19697 | Nonzero rings are precisel... |
chrrhm 19698 | The characteristic restric... |
domnchr 19699 | The characteristic of a do... |
znlidl 19700 | The set ` n ZZ ` is an ide... |
zncrng2 19701 | The value of the ` Z/nZ ` ... |
znval 19702 | The value of the ` Z/nZ ` ... |
znle 19703 | The value of the ` Z/nZ ` ... |
znval2 19704 | Self-referential expressio... |
znbaslem 19705 | Lemma for ~ znbas . (Cont... |
znbaslemOLD 19706 | Obsolete version of ~ znba... |
znbas2 19707 | The base set of ` Z/nZ ` i... |
znadd 19708 | The additive structure of ... |
znmul 19709 | The multiplicative structu... |
znzrh 19710 | The ` ZZ ` ring homomorphi... |
znbas 19711 | The base set of ` Z/nZ ` s... |
zncrng 19712 | ` Z/nZ ` is a commutative ... |
znzrh2 19713 | The ` ZZ ` ring homomorphi... |
znzrhval 19714 | The ` ZZ ` ring homomorphi... |
znzrhfo 19715 | The ` ZZ ` ring homomorphi... |
zncyg 19716 | The group ` ZZ / n ZZ ` is... |
zndvds 19717 | Express equality of equiva... |
zndvds0 19718 | Special case of ~ zndvds w... |
znf1o 19719 | The function ` F ` enumera... |
zzngim 19720 | The ` ZZ ` ring homomorphi... |
znle2 19721 | The ordering of the ` Z/nZ... |
znleval 19722 | The ordering of the ` Z/nZ... |
znleval2 19723 | The ordering of the ` Z/nZ... |
zntoslem 19724 | Lemma for ~ zntos . (Cont... |
zntos 19725 | The ` Z/nZ ` structure is ... |
znhash 19726 | The ` Z/nZ ` structure has... |
znfi 19727 | The ` Z/nZ ` structure is ... |
znfld 19728 | The ` Z/nZ ` structure is ... |
znidomb 19729 | The ` Z/nZ ` structure is ... |
znchr 19730 | Cyclic rings are defined b... |
znunit 19731 | The units of ` Z/nZ ` are ... |
znunithash 19732 | The size of the unit group... |
znrrg 19733 | The regular elements of ` ... |
cygznlem1 19734 | Lemma for ~ cygzn . (Cont... |
cygznlem2a 19735 | Lemma for ~ cygzn . (Cont... |
cygznlem2 19736 | Lemma for ~ cygzn . (Cont... |
cygznlem3 19737 | A cyclic group with ` n ` ... |
cygzn 19738 | A cyclic group with ` n ` ... |
cygth 19739 | The "fundamental theorem o... |
cyggic 19740 | Cyclic groups are isomorph... |
frgpcyg 19741 | A free group is cyclic iff... |
cnmsgnsubg 19742 | The signs form a multiplic... |
cnmsgnbas 19743 | The base set of the sign s... |
cnmsgngrp 19744 | The group of signs under m... |
psgnghm 19745 | The sign is a homomorphism... |
psgnghm2 19746 | The sign is a homomorphism... |
psgninv 19747 | The sign of a permutation ... |
psgnco 19748 | Multiplicativity of the pe... |
zrhpsgnmhm 19749 | Embedding of permutation s... |
zrhpsgninv 19750 | The embedded sign of a per... |
evpmss 19751 | Even permutations are perm... |
psgnevpmb 19752 | A class is an even permuta... |
psgnodpm 19753 | A permutation which is odd... |
psgnevpm 19754 | A permutation which is eve... |
psgnodpmr 19755 | If a permutation has sign ... |
zrhpsgnevpm 19756 | The sign of an even permut... |
zrhpsgnodpm 19757 | The sign of an odd permuta... |
zrhcofipsgn 19758 | Composition of a ` ZRHom `... |
zrhpsgnelbas 19759 | Embedding of permutation s... |
zrhcopsgnelbas 19760 | Embedding of permutation s... |
evpmodpmf1o 19761 | The function for performin... |
pmtrodpm 19762 | A transposition is an odd ... |
psgnfix1 19763 | A permutation of a finite ... |
psgnfix2 19764 | A permutation of a finite ... |
psgndiflemB 19765 | Lemma 1 for ~ psgndif . (... |
psgndiflemA 19766 | Lemma 2 for ~ psgndif . (... |
psgndif 19767 | Embedding of permutation s... |
zrhcopsgndif 19768 | Embedding of permutation s... |
rebase 19771 | The base of the field of r... |
remulg 19772 | The multiplication (group ... |
resubdrg 19773 | The real numbers form a di... |
resubgval 19774 | Subtraction in the field o... |
replusg 19775 | The addition operation of ... |
remulr 19776 | The multiplication operati... |
re0g 19777 | The neutral element of the... |
re1r 19778 | The multiplicative neutral... |
rele2 19779 | The ordering relation of t... |
relt 19780 | The ordering relation of t... |
reds 19781 | The distance of the field ... |
redvr 19782 | The division operation of ... |
retos 19783 | The real numbers are a tot... |
refld 19784 | The real numbers form a fi... |
refldcj 19785 | The conjugation operation ... |
recrng 19786 | The real numbers form a st... |
regsumsupp 19787 | The group sum over the rea... |
isphl 19792 | The predicate "is a genera... |
phllvec 19793 | A pre-Hilbert space is a l... |
phllmod 19794 | A pre-Hilbert space is a l... |
phlsrng 19795 | The scalar ring of a pre-H... |
phllmhm 19796 | The inner product of a pre... |
ipcl 19797 | Closure of the inner produ... |
ipcj 19798 | Conjugate of an inner prod... |
iporthcom 19799 | Orthogonality (meaning inn... |
ip0l 19800 | Inner product with a zero ... |
ip0r 19801 | Inner product with a zero ... |
ipeq0 19802 | The inner product of a vec... |
ipdir 19803 | Distributive law for inner... |
ipdi 19804 | Distributive law for inner... |
ip2di 19805 | Distributive law for inner... |
ipsubdir 19806 | Distributive law for inner... |
ipsubdi 19807 | Distributive law for inner... |
ip2subdi 19808 | Distributive law for inner... |
ipass 19809 | Associative law for inner ... |
ipassr 19810 | "Associative" law for seco... |
ipassr2 19811 | "Associative" law for inne... |
ipffval 19812 | The inner product operatio... |
ipfval 19813 | The inner product operatio... |
ipfeq 19814 | If the inner product opera... |
ipffn 19815 | The inner product operatio... |
phlipf 19816 | The inner product operatio... |
ip2eq 19817 | Two vectors are equal iff ... |
isphld 19818 | Properties that determine ... |
phlpropd 19819 | If two structures have the... |
ssipeq 19820 | The inner product on a sub... |
phssipval 19821 | The inner product on a sub... |
phssip 19822 | The inner product (as a fu... |
ocvfval 19829 | The orthocomplement operat... |
ocvval 19830 | Value of the orthocompleme... |
elocv 19831 | Elementhood in the orthoco... |
ocvi 19832 | Property of a member of th... |
ocvss 19833 | The orthocomplement of a s... |
ocvocv 19834 | A set is contained in its ... |
ocvlss 19835 | The orthocomplement of a s... |
ocv2ss 19836 | Orthocomplements reverse s... |
ocvin 19837 | An orthocomplement has tri... |
ocvsscon 19838 | Two ways to say that ` S `... |
ocvlsp 19839 | The orthocomplement of a l... |
ocv0 19840 | The orthocomplement of the... |
ocvz 19841 | The orthocomplement of the... |
ocv1 19842 | The orthocomplement of the... |
unocv 19843 | The orthocomplement of a u... |
iunocv 19844 | The orthocomplement of an ... |
cssval 19845 | The set of closed subspace... |
iscss 19846 | The predicate "is a closed... |
cssi 19847 | Property of a closed subsp... |
cssss 19848 | A closed subspace is a sub... |
iscss2 19849 | It is sufficient to prove ... |
ocvcss 19850 | The orthocomplement of any... |
cssincl 19851 | The zero subspace is a clo... |
css0 19852 | The zero subspace is a clo... |
css1 19853 | The whole space is a close... |
csslss 19854 | A closed subspace of a pre... |
lsmcss 19855 | A subset of a pre-Hilbert ... |
cssmre 19856 | The closed subspaces of a ... |
mrccss 19857 | The Moore closure correspo... |
thlval 19858 | Value of the Hilbert latti... |
thlbas 19859 | Base set of the Hilbert la... |
thlle 19860 | Ordering on the Hilbert la... |
thlleval 19861 | Ordering on the Hilbert la... |
thloc 19862 | Orthocomplement on the Hil... |
pjfval 19869 | The value of the projectio... |
pjdm 19870 | A subspace is in the domai... |
pjpm 19871 | The projection map is a pa... |
pjfval2 19872 | Value of the projection ma... |
pjval 19873 | Value of the projection ma... |
pjdm2 19874 | A subspace is in the domai... |
pjff 19875 | A projection is a linear o... |
pjf 19876 | A projection is a function... |
pjf2 19877 | A projection is a function... |
pjfo 19878 | A projection is a surjecti... |
pjcss 19879 | A projection subspace is a... |
ocvpj 19880 | The orthocomplement of a p... |
ishil 19881 | The predicate "is a Hilber... |
ishil2 19882 | The predicate "is a Hilber... |
isobs 19883 | The predicate "is an ortho... |
obsip 19884 | The inner product of two e... |
obsipid 19885 | A basis element has unit l... |
obsrcl 19886 | Reverse closure for an ort... |
obsss 19887 | An orthonormal basis is a ... |
obsne0 19888 | A basis element is nonzero... |
obsocv 19889 | An orthonormal basis has t... |
obs2ocv 19890 | The double orthocomplement... |
obselocv 19891 | A basis element is in the ... |
obs2ss 19892 | A basis has no proper subs... |
obslbs 19893 | An orthogonal basis is a l... |
reldmdsmm 19896 | The direct sum is a well-b... |
dsmmval 19897 | Value of the module direct... |
dsmmbase 19898 | Base set of the module dir... |
dsmmval2 19899 | Self-referential definitio... |
dsmmbas2 19900 | Base set of the direct sum... |
dsmmfi 19901 | For finite products, the d... |
dsmmelbas 19902 | Membership in the finitely... |
dsmm0cl 19903 | The all-zero vector is con... |
dsmmacl 19904 | The finite hull is closed ... |
prdsinvgd2 19905 | Negation of a single coord... |
dsmmsubg 19906 | The finite hull of a produ... |
dsmmlss 19907 | The finite hull of a produ... |
dsmmlmod 19908 | The direct sum of a family... |
frlmval 19911 | Value of the free module. ... |
frlmlmod 19912 | The free module is a modul... |
frlmpws 19913 | The free module as a restr... |
frlmlss 19914 | The base set of the free m... |
frlmpwsfi 19915 | The finite free module is ... |
frlmsca 19916 | The ring of scalars of a f... |
frlm0 19917 | Zero in a free module (rin... |
frlmbas 19918 | Base set of the free modul... |
frlmelbas 19919 | Membership in the base set... |
frlmrcl 19920 | If a free module is inhabi... |
frlmbasfsupp 19921 | Elements of the free modul... |
frlmbasmap 19922 | Elements of the free modul... |
frlmbasf 19923 | Elements of the free modul... |
frlmfibas 19924 | The base set of the finite... |
elfrlmbasn0 19925 | If the dimension of a free... |
frlmplusgval 19926 | Addition in a free module.... |
frlmsubgval 19927 | Subtraction in a free modu... |
frlmvscafval 19928 | Scalar multiplication in a... |
frlmvscaval 19929 | Scalar multiplication in a... |
frlmgsum 19930 | Finite commutative sums in... |
frlmsplit2 19931 | Restriction is homomorphic... |
frlmsslss 19932 | A subset of a free module ... |
frlmsslss2 19933 | A subset of a free module ... |
frlmbas3 19934 | An element of the base set... |
mpt2frlmd 19935 | Elements of the free modul... |
frlmip 19936 | The inner product of a fre... |
frlmipval 19937 | The inner product of a fre... |
frlmphllem 19938 | Lemma for ~ frlmphl . (Co... |
frlmphl 19939 | Conditions for a free modu... |
uvcfval 19942 | Value of the unit-vector g... |
uvcval 19943 | Value of a single unit vec... |
uvcvval 19944 | Value of a unit vector coo... |
uvcvvcl 19945 | A coodinate of a unit vect... |
uvcvvcl2 19946 | A unit vector coordinate i... |
uvcvv1 19947 | The unit vector is one at ... |
uvcvv0 19948 | The unit vector is zero at... |
uvcff 19949 | Domain and range of the un... |
uvcf1 19950 | In a nonzero ring, each un... |
uvcresum 19951 | Any element of a free modu... |
frlmssuvc1 19952 | A scalar multiple of a uni... |
frlmssuvc2 19953 | A nonzero scalar multiple ... |
frlmsslsp 19954 | A subset of a free module ... |
frlmlbs 19955 | The unit vectors comprise ... |
frlmup1 19956 | Any assignment of unit vec... |
frlmup2 19957 | The evaluation map has the... |
frlmup3 19958 | The range of such an evalu... |
frlmup4 19959 | Universal property of the ... |
ellspd 19960 | The elements of the span o... |
elfilspd 19961 | Simplified version of ~ el... |
rellindf 19966 | The independent-family pre... |
islinds 19967 | Property of an independent... |
linds1 19968 | An independent set of vect... |
linds2 19969 | An independent set of vect... |
islindf 19970 | Property of an independent... |
islinds2 19971 | Expanded property of an in... |
islindf2 19972 | Property of an independent... |
lindff 19973 | Functional property of a l... |
lindfind 19974 | A linearly independent fam... |
lindsind 19975 | A linearly independent set... |
lindfind2 19976 | In a linearly independent ... |
lindsind2 19977 | In a linearly independent ... |
lindff1 19978 | A linearly independent fam... |
lindfrn 19979 | The range of an independen... |
f1lindf 19980 | Rearranging and deleting e... |
lindfres 19981 | Any restriction of an inde... |
lindsss 19982 | Any subset of an independe... |
f1linds 19983 | A family constructed from ... |
islindf3 19984 | In a nonzero ring, indepen... |
lindfmm 19985 | Linear independence of a f... |
lindsmm 19986 | Linear independence of a s... |
lindsmm2 19987 | The monomorphic image of a... |
lsslindf 19988 | Linear independence is unc... |
lsslinds 19989 | Linear independence is unc... |
islbs4 19990 | A basis is an independent ... |
lbslinds 19991 | A basis is independent. (... |
islinds3 19992 | A subset is linearly indep... |
islinds4 19993 | A set is independent in a ... |
lmimlbs 19994 | The isomorphic image of a ... |
lmiclbs 19995 | Having a basis is an isomo... |
islindf4 19996 | A family is independent if... |
islindf5 19997 | A family is independent if... |
indlcim 19998 | An independent, spanning f... |
lbslcic 19999 | A module with a basis is i... |
lmisfree 20000 | A module has a basis iff i... |
lvecisfrlm 20001 | Every vector space is isom... |
lmimco 20002 | The composition of two iso... |
lmictra 20003 | Module isomorphism is tran... |
uvcf1o 20004 | In a nonzero ring, the map... |
uvcendim 20005 | In a nonzero ring, the num... |
frlmisfrlm 20006 | A free module is isomorphi... |
frlmiscvec 20007 | Every free module is isomo... |
mamufval 20010 | Functional value of the ma... |
mamuval 20011 | Multiplication of two matr... |
mamufv 20012 | A cell in the multiplicati... |
mamudm 20013 | The domain of the matrix m... |
mamufacex 20014 | Every solution of the equa... |
mamures 20015 | Rows in a matrix product a... |
mndvcl 20016 | Tuple-wise additive closur... |
mndvass 20017 | Tuple-wise associativity i... |
mndvlid 20018 | Tuple-wise left identity i... |
mndvrid 20019 | Tuple-wise right identity ... |
grpvlinv 20020 | Tuple-wise left inverse in... |
grpvrinv 20021 | Tuple-wise right inverse i... |
mhmvlin 20022 | Tuple extension of monoid ... |
ringvcl 20023 | Tuple-wise multiplication ... |
gsumcom3 20024 | A commutative law for fini... |
gsumcom3fi 20025 | A commutative law for fini... |
mamucl 20026 | Operation closure of matri... |
mamuass 20027 | Matrix multiplication is a... |
mamudi 20028 | Matrix multiplication dist... |
mamudir 20029 | Matrix multiplication dist... |
mamuvs1 20030 | Matrix multiplication dist... |
mamuvs2 20031 | Matrix multiplication dist... |
matbas0pc 20034 | There is no matrix with a ... |
matbas0 20035 | There is no matrix for a n... |
matval 20036 | Value of the matrix algebr... |
matrcl 20037 | Reverse closure for the ma... |
matbas 20038 | The matrix ring has the sa... |
matplusg 20039 | The matrix ring has the sa... |
matsca 20040 | The matrix ring has the sa... |
matvsca 20041 | The matrix ring has the sa... |
mat0 20042 | The matrix ring has the sa... |
matinvg 20043 | The matrix ring has the sa... |
mat0op 20044 | Value of a zero matrix as ... |
matsca2 20045 | The scalars of the matrix ... |
matbas2 20046 | The base set of the matrix... |
matbas2i 20047 | A matrix is a function. (... |
matbas2d 20048 | The base set of the matrix... |
eqmat 20049 | Two square matrices of the... |
matecl 20050 | Each entry (according to W... |
matecld 20051 | Each entry (according to W... |
matplusg2 20052 | Addition in the matrix rin... |
matvsca2 20053 | Scalar multiplication in t... |
matlmod 20054 | The matrix ring is a linea... |
matgrp 20055 | The matrix ring is a group... |
matvscl 20056 | Closure of the scalar mult... |
matsubg 20057 | The matrix ring has the sa... |
matplusgcell 20058 | Addition in the matrix rin... |
matsubgcell 20059 | Subtraction in the matrix ... |
matinvgcell 20060 | Additive inversion in the ... |
matvscacell 20061 | Scalar multiplication in t... |
matgsum 20062 | Finite commutative sums in... |
matmulr 20063 | Multiplication in the matr... |
mamumat1cl 20064 | The identity matrix (as op... |
mat1comp 20065 | The components of the iden... |
mamulid 20066 | The identity matrix (as op... |
mamurid 20067 | The identity matrix (as op... |
matring 20068 | Existence of the matrix ri... |
matassa 20069 | Existence of the matrix al... |
matmulcell 20070 | Multiplication in the matr... |
mpt2matmul 20071 | Multiplication of two N x ... |
mat1 20072 | Value of an identity matri... |
mat1ov 20073 | Entries of an identity mat... |
mat1bas 20074 | The identity matrix is a m... |
matsc 20075 | The identity matrix multip... |
ofco2 20076 | Distribution law for the f... |
oftpos 20077 | The transposition of the v... |
mattposcl 20078 | The transpose of a square ... |
mattpostpos 20079 | The transpose of the trans... |
mattposvs 20080 | The transposition of a mat... |
mattpos1 20081 | The transposition of the i... |
tposmap 20082 | The transposition of an I ... |
mamutpos 20083 | Behavior of transposes in ... |
mattposm 20084 | Multiplying two transposed... |
matgsumcl 20085 | Closure of a group sum ove... |
madetsumid 20086 | The identity summand in th... |
matepmcl 20087 | Each entry of a matrix wit... |
matepm2cl 20088 | Each entry of a matrix wit... |
madetsmelbas 20089 | A summand of the determina... |
madetsmelbas2 20090 | A summand of the determina... |
mat0dimbas0 20091 | The empty set is the one a... |
mat0dim0 20092 | The zero of the algebra of... |
mat0dimid 20093 | The identity of the algebr... |
mat0dimscm 20094 | The scalar multiplication ... |
mat0dimcrng 20095 | The algebra of matrices wi... |
mat1dimelbas 20096 | A matrix with dimension 1 ... |
mat1dimbas 20097 | A matrix with dimension 1 ... |
mat1dim0 20098 | The zero of the algebra of... |
mat1dimid 20099 | The identity of the algebr... |
mat1dimscm 20100 | The scalar multiplication ... |
mat1dimmul 20101 | The ring multiplication in... |
mat1dimcrng 20102 | The algebra of matrices wi... |
mat1f1o 20103 | There is a 1-1 function fr... |
mat1rhmval 20104 | The value of the ring homo... |
mat1rhmelval 20105 | The value of the ring homo... |
mat1rhmcl 20106 | The value of the ring homo... |
mat1f 20107 | There is a function from a... |
mat1ghm 20108 | There is a group homomorph... |
mat1mhm 20109 | There is a monoid homomorp... |
mat1rhm 20110 | There is a ring homomorphi... |
mat1rngiso 20111 | There is a ring isomorphis... |
mat1ric 20112 | A ring is isomorphic to th... |
dmatval 20117 | The set of ` N ` x ` N ` d... |
dmatel 20118 | A ` N ` x ` N ` diagonal m... |
dmatmat 20119 | An ` N ` x ` N ` diagonal ... |
dmatid 20120 | The identity matrix is a d... |
dmatelnd 20121 | An extradiagonal entry of ... |
dmatmul 20122 | The product of two diagona... |
dmatsubcl 20123 | The difference of two diag... |
dmatsgrp 20124 | The set of diagonal matric... |
dmatmulcl 20125 | The product of two diagona... |
dmatsrng 20126 | The set of diagonal matric... |
dmatcrng 20127 | The subring of diagonal ma... |
dmatscmcl 20128 | The multiplication of a di... |
scmatval 20129 | The set of ` N ` x ` N ` s... |
scmatel 20130 | An ` N ` x ` N ` scalar ma... |
scmatscmid 20131 | A scalar matrix can be exp... |
scmatscmide 20132 | An entry of a scalar matri... |
scmatscmiddistr 20133 | Distributive law for scala... |
scmatmat 20134 | An ` N ` x ` N ` scalar ma... |
scmate 20135 | An entry of an ` N ` x ` N... |
scmatmats 20136 | The set of an ` N ` x ` N ... |
scmateALT 20137 | Alternate proof of ~ scmat... |
scmatscm 20138 | The multiplication of a ma... |
scmatid 20139 | The identity matrix is a s... |
scmatdmat 20140 | A scalar matrix is a diago... |
scmataddcl 20141 | The sum of two scalar matr... |
scmatsubcl 20142 | The difference of two scal... |
scmatmulcl 20143 | The product of two scalar ... |
scmatsgrp 20144 | The set of scalar matrices... |
scmatsrng 20145 | The set of scalar matrices... |
scmatcrng 20146 | The subring of scalar matr... |
scmatsgrp1 20147 | The set of scalar matrices... |
scmatsrng1 20148 | The set of scalar matrices... |
smatvscl 20149 | Closure of the scalar mult... |
scmatlss 20150 | The set of scalar matrices... |
scmatstrbas 20151 | The set of scalar matrices... |
scmatrhmval 20152 | The value of the ring homo... |
scmatrhmcl 20153 | The value of the ring homo... |
scmatf 20154 | There is a function from a... |
scmatfo 20155 | There is a function from a... |
scmatf1 20156 | There is a 1-1 function fr... |
scmatf1o 20157 | There is a bijection betwe... |
scmatghm 20158 | There is a group homomorph... |
scmatmhm 20159 | There is a monoid homomorp... |
scmatrhm 20160 | There is a ring homomorphi... |
scmatrngiso 20161 | There is a ring isomorphis... |
scmatric 20162 | A ring is isomorphic to ev... |
mat0scmat 20163 | The empty matrix over a ri... |
mat1scmat 20164 | A 1-dimensional matrix ove... |
mvmulfval 20167 | Functional value of the ma... |
mvmulval 20168 | Multiplication of a vector... |
mvmulfv 20169 | A cell/element in the vect... |
mavmulval 20170 | Multiplication of a vector... |
mavmulfv 20171 | A cell/element in the vect... |
mavmulcl 20172 | Multiplication of an NxN m... |
1mavmul 20173 | Multiplication of the iden... |
mavmulass 20174 | Associativity of the multi... |
mavmuldm 20175 | The domain of the matrix v... |
mavmulsolcl 20176 | Every solution of the equa... |
mavmul0 20177 | Multiplication of a 0-dime... |
mavmul0g 20178 | The result of the 0-dimens... |
mvmumamul1 20179 | The multiplication of an M... |
mavmumamul1 20180 | The multiplication of an N... |
marrepfval 20185 | First substitution for the... |
marrepval0 20186 | Second substitution for th... |
marrepval 20187 | Third substitution for the... |
marrepeval 20188 | An entry of a matrix with ... |
marrepcl 20189 | Closure of the row replace... |
marepvfval 20190 | First substitution for the... |
marepvval0 20191 | Second substitution for th... |
marepvval 20192 | Third substitution for the... |
marepveval 20193 | An entry of a matrix with ... |
marepvcl 20194 | Closure of the column repl... |
ma1repvcl 20195 | Closure of the column repl... |
ma1repveval 20196 | An entry of an identity ma... |
mulmarep1el 20197 | Element by element multipl... |
mulmarep1gsum1 20198 | The sum of element by elem... |
mulmarep1gsum2 20199 | The sum of element by elem... |
1marepvmarrepid 20200 | Replacing the ith row by 0... |
submabas 20203 | Any subset of the index se... |
submafval 20204 | First substitution for a s... |
submaval0 20205 | Second substitution for a ... |
submaval 20206 | Third substitution for a s... |
submaeval 20207 | An entry of a submatrix of... |
1marepvsma1 20208 | The submatrix of the ident... |
mdetfval 20211 | First substitution for the... |
mdetleib 20212 | Full substitution of our d... |
mdetleib2 20213 | Leibniz' formula can also ... |
nfimdetndef 20214 | The determinant is not def... |
mdetfval1 20215 | First substitution of an a... |
mdetleib1 20216 | Full substitution of an al... |
mdet0pr 20217 | The determinant for 0-dime... |
mdet0f1o 20218 | The determinant for 0-dime... |
mdet0fv0 20219 | The determinant of a 0-dim... |
mdetf 20220 | Functionality of the deter... |
mdetcl 20221 | The determinant evaluates ... |
m1detdiag 20222 | The determinant of a 1-dim... |
mdetdiaglem 20223 | Lemma for ~ mdetdiag . Pr... |
mdetdiag 20224 | The determinant of a diago... |
mdetdiagid 20225 | The determinant of a diago... |
mdet1 20226 | The determinant of the ide... |
mdetrlin 20227 | The determinant function i... |
mdetrsca 20228 | The determinant function i... |
mdetrsca2 20229 | The determinant function i... |
mdetr0 20230 | The determinant of a matri... |
mdet0 20231 | The determinant of the zer... |
mdetrlin2 20232 | The determinant function i... |
mdetralt 20233 | The determinant function i... |
mdetralt2 20234 | The determinant function i... |
mdetero 20235 | The determinant function i... |
mdettpos 20236 | Determinant is invariant u... |
mdetunilem1 20237 | Lemma for ~ mdetuni . (Co... |
mdetunilem2 20238 | Lemma for ~ mdetuni . (Co... |
mdetunilem3 20239 | Lemma for ~ mdetuni . (Co... |
mdetunilem4 20240 | Lemma for ~ mdetuni . (Co... |
mdetunilem5 20241 | Lemma for ~ mdetuni . (Co... |
mdetunilem6 20242 | Lemma for ~ mdetuni . (Co... |
mdetunilem7 20243 | Lemma for ~ mdetuni . (Co... |
mdetunilem8 20244 | Lemma for ~ mdetuni . (Co... |
mdetunilem9 20245 | Lemma for ~ mdetuni . (Co... |
mdetuni0 20246 | Lemma for ~ mdetuni . (Co... |
mdetuni 20247 | According to the definitio... |
mdetmul 20248 | Multiplicativity of the de... |
m2detleiblem1 20249 | Lemma 1 for ~ m2detleib . ... |
m2detleiblem5 20250 | Lemma 5 for ~ m2detleib . ... |
m2detleiblem6 20251 | Lemma 6 for ~ m2detleib . ... |
m2detleiblem7 20252 | Lemma 7 for ~ m2detleib . ... |
m2detleiblem2 20253 | Lemma 2 for ~ m2detleib . ... |
m2detleiblem3 20254 | Lemma 3 for ~ m2detleib . ... |
m2detleiblem4 20255 | Lemma 4 for ~ m2detleib . ... |
m2detleib 20256 | Leibniz' Formula for 2x2-m... |
mndifsplit 20261 | Lemma for ~ maducoeval2 . ... |
madufval 20262 | First substitution for the... |
maduval 20263 | Second substitution for th... |
maducoeval 20264 | An entry of the adjunct (c... |
maducoeval2 20265 | An entry of the adjunct (c... |
maduf 20266 | Creating the adjunct of ma... |
madutpos 20267 | The adjuct of a transposed... |
madugsum 20268 | The determinant of a matri... |
madurid 20269 | Multiplying a matrix with ... |
madulid 20270 | Multiplying the adjunct of... |
minmar1fval 20271 | First substitution for the... |
minmar1val0 20272 | Second substitution for th... |
minmar1val 20273 | Third substitution for the... |
minmar1eval 20274 | An entry of a matrix for a... |
minmar1marrep 20275 | The minor matrix is a spec... |
minmar1cl 20276 | Closure of the row replace... |
maducoevalmin1 20277 | The coefficients of an adj... |
symgmatr01lem 20278 | Lemma for ~ symgmatr01 . ... |
symgmatr01 20279 | Applying a permutation tha... |
gsummatr01lem1 20280 | Lemma A for ~ gsummatr01 .... |
gsummatr01lem2 20281 | Lemma B for ~ gsummatr01 .... |
gsummatr01lem3 20282 | Lemma 1 for ~ gsummatr01 .... |
gsummatr01lem4 20283 | Lemma 2 for ~ gsummatr01 .... |
gsummatr01 20284 | Lemma 1 for ~ smadiadetlem... |
marep01ma 20285 | Replacing a row of a squar... |
smadiadetlem0 20286 | Lemma 0 for ~ smadiadet : ... |
smadiadetlem1 20287 | Lemma 1 for ~ smadiadet : ... |
smadiadetlem1a 20288 | Lemma 1a for ~ smadiadet :... |
smadiadetlem2 20289 | Lemma 2 for ~ smadiadet : ... |
smadiadetlem3lem0 20290 | Lemma 0 for ~ smadiadetlem... |
smadiadetlem3lem1 20291 | Lemma 1 for ~ smadiadetlem... |
smadiadetlem3lem2 20292 | Lemma 2 for ~ smadiadetlem... |
smadiadetlem3 20293 | Lemma 3 for ~ smadiadet . ... |
smadiadetlem4 20294 | Lemma 4 for ~ smadiadet . ... |
smadiadet 20295 | The determinant of a subma... |
smadiadetglem1 20296 | Lemma 1 for ~ smadiadetg .... |
smadiadetglem2 20297 | Lemma 2 for ~ smadiadetg .... |
smadiadetg 20298 | The determinant of a squar... |
smadiadetg0 20299 | Lemma for ~ smadiadetr : v... |
smadiadetr 20300 | The determinant of a squar... |
invrvald 20301 | If a matrix multiplied wit... |
matinv 20302 | The inverse of a matrix is... |
matunit 20303 | A matrix is a unit in the ... |
slesolvec 20304 | Every solution of a system... |
slesolinv 20305 | The solution of a system o... |
slesolinvbi 20306 | The solution of a system o... |
slesolex 20307 | Every system of linear equ... |
cramerimplem1 20308 | Lemma 1 for ~ cramerimp : ... |
cramerimplem2 20309 | Lemma 2 for ~ cramerimp : ... |
cramerimplem3 20310 | Lemma 3 for ~ cramerimp : ... |
cramerimp 20311 | One direction of Cramer's ... |
cramerlem1 20312 | Lemma 1 for ~ cramer . (C... |
cramerlem2 20313 | Lemma 2 for ~ cramer . (C... |
cramerlem3 20314 | Lemma 3 for ~ cramer . (C... |
cramer0 20315 | Special case of Cramer's r... |
cramer 20316 | Cramer's rule. According ... |
pmatring 20317 | The set of polynomial matr... |
pmatlmod 20318 | The set of polynomial matr... |
pmat0op 20319 | The zero polynomial matrix... |
pmat1op 20320 | The identity polynomial ma... |
pmat1ovd 20321 | Entries of the identity po... |
pmat0opsc 20322 | The zero polynomial matrix... |
pmat1opsc 20323 | The identity polynomial ma... |
pmat1ovscd 20324 | Entries of the identity po... |
pmatcoe1fsupp 20325 | For a polynomial matrix th... |
1pmatscmul 20326 | The scalar product of the ... |
cpmat 20333 | Value of the constructor o... |
cpmatpmat 20334 | A constant polynomial matr... |
cpmatel 20335 | Property of a constant pol... |
cpmatelimp 20336 | Implication of a set being... |
cpmatel2 20337 | Another property of a cons... |
cpmatelimp2 20338 | Another implication of a s... |
1elcpmat 20339 | The identity of the ring o... |
cpmatacl 20340 | The set of all constant po... |
cpmatinvcl 20341 | The set of all constant po... |
cpmatmcllem 20342 | Lemma for ~ cpmatmcl . (C... |
cpmatmcl 20343 | The set of all constant po... |
cpmatsubgpmat 20344 | The set of all constant po... |
cpmatsrgpmat 20345 | The set of all constant po... |
0elcpmat 20346 | The zero of the ring of al... |
mat2pmatfval 20347 | Value of the matrix transf... |
mat2pmatval 20348 | The result of a matrix tra... |
mat2pmatvalel 20349 | A (matrix) element of the ... |
mat2pmatbas 20350 | The result of a matrix tra... |
mat2pmatbas0 20351 | The result of a matrix tra... |
mat2pmatf 20352 | The matrix transformation ... |
mat2pmatf1 20353 | The matrix transformation ... |
mat2pmatghm 20354 | The transformation of matr... |
mat2pmatmul 20355 | The transformation of matr... |
mat2pmat1 20356 | The transformation of the ... |
mat2pmatmhm 20357 | The transformation of matr... |
mat2pmatrhm 20358 | The transformation of matr... |
mat2pmatlin 20359 | The transformation of matr... |
0mat2pmat 20360 | The transformed zero matri... |
idmatidpmat 20361 | The transformed identity m... |
d0mat2pmat 20362 | The transformed empty set ... |
d1mat2pmat 20363 | The transformation of a ma... |
mat2pmatscmxcl 20364 | A transformed matrix multi... |
m2cpm 20365 | The result of a matrix tra... |
m2cpmf 20366 | The matrix transformation ... |
m2cpmf1 20367 | The matrix transformation ... |
m2cpmghm 20368 | The transformation of matr... |
m2cpmmhm 20369 | The transformation of matr... |
m2cpmrhm 20370 | The transformation of matr... |
m2pmfzmap 20371 | The transformed values of ... |
m2pmfzgsumcl 20372 | Closure of the sum of scal... |
cpm2mfval 20373 | Value of the inverse matri... |
cpm2mval 20374 | The result of an inverse m... |
cpm2mvalel 20375 | A (matrix) element of the ... |
cpm2mf 20376 | The inverse matrix transfo... |
m2cpminvid 20377 | The inverse transformation... |
m2cpminvid2lem 20378 | Lemma for ~ m2cpminvid2 . ... |
m2cpminvid2 20379 | The transformation applied... |
m2cpmfo 20380 | The matrix transformation ... |
m2cpmf1o 20381 | The matrix transformation ... |
m2cpmrngiso 20382 | The transformation of matr... |
matcpmric 20383 | The ring of matrices over ... |
m2cpminv 20384 | The inverse matrix transfo... |
m2cpminv0 20385 | The inverse matrix transfo... |
decpmatval0 20388 | The matrix consisting of t... |
decpmatval 20389 | The matrix consisting of t... |
decpmate 20390 | An entry of the matrix con... |
decpmatcl 20391 | Closure of the decompositi... |
decpmataa0 20392 | The matrix consisting of t... |
decpmatfsupp 20393 | The mapping to the matrice... |
decpmatid 20394 | The matrix consisting of t... |
decpmatmullem 20395 | Lemma for ~ decpmatmul . ... |
decpmatmul 20396 | The matrix consisting of t... |
decpmatmulsumfsupp 20397 | Lemma 0 for ~ pm2mpmhm . ... |
pmatcollpw1lem1 20398 | Lemma 1 for ~ pmatcollpw1 ... |
pmatcollpw1lem2 20399 | Lemma 2 for ~ pmatcollpw1 ... |
pmatcollpw1 20400 | Write a polynomial matrix ... |
pmatcollpw2lem 20401 | Lemma for ~ pmatcollpw2 . ... |
pmatcollpw2 20402 | Write a polynomial matrix ... |
monmatcollpw 20403 | The matrix consisting of t... |
pmatcollpwlem 20404 | Lemma for ~ pmatcollpw . ... |
pmatcollpw 20405 | Write a polynomial matrix ... |
pmatcollpwfi 20406 | Write a polynomial matrix ... |
pmatcollpw3lem 20407 | Lemma for ~ pmatcollpw3 an... |
pmatcollpw3 20408 | Write a polynomial matrix ... |
pmatcollpw3fi 20409 | Write a polynomial matrix ... |
pmatcollpw3fi1lem1 20410 | Lemma 1 for ~ pmatcollpw3f... |
pmatcollpw3fi1lem2 20411 | Lemma 2 for ~ pmatcollpw3f... |
pmatcollpw3fi1 20412 | Write a polynomial matrix ... |
pmatcollpwscmatlem1 20413 | Lemma 1 for ~ pmatcollpwsc... |
pmatcollpwscmatlem2 20414 | Lemma 2 for ~ pmatcollpwsc... |
pmatcollpwscmat 20415 | Write a scalar matrix over... |
pm2mpf1lem 20418 | Lemma for ~ pm2mpf1 . (Co... |
pm2mpval 20419 | Value of the transformatio... |
pm2mpfval 20420 | A polynomial matrix transf... |
pm2mpcl 20421 | The transformation of poly... |
pm2mpf 20422 | The transformation of poly... |
pm2mpf1 20423 | The transformation of poly... |
pm2mpcoe1 20424 | A coefficient of the polyn... |
idpm2idmp 20425 | The transformation of the ... |
mptcoe1matfsupp 20426 | The mapping extracting the... |
mply1topmatcllem 20427 | Lemma for ~ mply1topmatcl ... |
mply1topmatval 20428 | A polynomial over matrices... |
mply1topmatcl 20429 | A polynomial over matrices... |
mp2pm2mplem1 20430 | Lemma 1 for ~ mp2pm2mp . ... |
mp2pm2mplem2 20431 | Lemma 2 for ~ mp2pm2mp . ... |
mp2pm2mplem3 20432 | Lemma 3 for ~ mp2pm2mp . ... |
mp2pm2mplem4 20433 | Lemma 4 for ~ mp2pm2mp . ... |
mp2pm2mplem5 20434 | Lemma 5 for ~ mp2pm2mp . ... |
mp2pm2mp 20435 | A polynomial over matrices... |
pm2mpghmlem2 20436 | Lemma 2 for ~ pm2mpghm . ... |
pm2mpghmlem1 20437 | Lemma 1 for pm2mpghm . (C... |
pm2mpfo 20438 | The transformation of poly... |
pm2mpf1o 20439 | The transformation of poly... |
pm2mpghm 20440 | The transformation of poly... |
pm2mpgrpiso 20441 | The transformation of poly... |
pm2mpmhmlem1 20442 | Lemma 1 for ~ pm2mpmhm . ... |
pm2mpmhmlem2 20443 | Lemma 2 for ~ pm2mpmhm . ... |
pm2mpmhm 20444 | The transformation of poly... |
pm2mprhm 20445 | The transformation of poly... |
pm2mprngiso 20446 | The transformation of poly... |
pmmpric 20447 | The ring of polynomial mat... |
monmat2matmon 20448 | The transformation of a po... |
pm2mp 20449 | The transformation of a su... |
chmatcl 20452 | Closure of the characteris... |
chmatval 20453 | The entries of the charact... |
chpmatfval 20454 | Value of the characteristi... |
chpmatval 20455 | The characteristic polynom... |
chpmatply1 20456 | The characteristic polynom... |
chpmatval2 20457 | The characteristic polynom... |
chpmat0d 20458 | The characteristic polynom... |
chpmat1dlem 20459 | Lemma for ~ chpmat1d . (C... |
chpmat1d 20460 | The characteristic polynom... |
chpdmatlem0 20461 | Lemma 0 for ~ chpdmat . (... |
chpdmatlem1 20462 | Lemma 1 for ~ chpdmat . (... |
chpdmatlem2 20463 | Lemma 2 for ~ chpdmat . (... |
chpdmatlem3 20464 | Lemma 3 for ~ chpdmat . (... |
chpdmat 20465 | The characteristic polynom... |
chpscmat 20466 | The characteristic polynom... |
chpscmat0 20467 | The characteristic polynom... |
chpscmatgsumbin 20468 | The characteristic polynom... |
chpscmatgsummon 20469 | The characteristic polynom... |
chp0mat 20470 | The characteristic polynom... |
chpidmat 20471 | The characteristic polynom... |
chmaidscmat 20472 | The characteristic polynom... |
fvmptnn04if 20473 | The function values of a m... |
fvmptnn04ifa 20474 | The function value of a ma... |
fvmptnn04ifb 20475 | The function value of a ma... |
fvmptnn04ifc 20476 | The function value of a ma... |
fvmptnn04ifd 20477 | The function value of a ma... |
chfacfisf 20478 | The "characteristic factor... |
chfacfisfcpmat 20479 | The "characteristic factor... |
chfacffsupp 20480 | The "characteristic factor... |
chfacfscmulcl 20481 | Closure of a scaled value ... |
chfacfscmul0 20482 | A scaled value of the "cha... |
chfacfscmulfsupp 20483 | A mapping of scaled values... |
chfacfscmulgsum 20484 | Breaking up a sum of value... |
chfacfpmmulcl 20485 | Closure of the value of th... |
chfacfpmmul0 20486 | The value of the "characte... |
chfacfpmmulfsupp 20487 | A mapping of values of the... |
chfacfpmmulgsum 20488 | Breaking up a sum of value... |
chfacfpmmulgsum2 20489 | Breaking up a sum of value... |
cayhamlem1 20490 | Lemma 1 for ~ cayleyhamilt... |
cpmadurid 20491 | The right-hand fundamental... |
cpmidgsum 20492 | Representation of the iden... |
cpmidgsumm2pm 20493 | Representation of the iden... |
cpmidpmatlem1 20494 | Lemma 1 for ~ cpmidpmat . ... |
cpmidpmatlem2 20495 | Lemma 2 for ~ cpmidpmat . ... |
cpmidpmatlem3 20496 | Lemma 3 for ~ cpmidpmat . ... |
cpmidpmat 20497 | Representation of the iden... |
cpmadugsumlemB 20498 | Lemma B for ~ cpmadugsum .... |
cpmadugsumlemC 20499 | Lemma C for ~ cpmadugsum .... |
cpmadugsumlemF 20500 | Lemma F for ~ cpmadugsum .... |
cpmadugsumfi 20501 | The product of the charact... |
cpmadugsum 20502 | The product of the charact... |
cpmidgsum2 20503 | Representation of the iden... |
cpmidg2sum 20504 | Equality of two sums repre... |
cpmadumatpolylem1 20505 | Lemma 1 for ~ cpmadumatpol... |
cpmadumatpolylem2 20506 | Lemma 2 for ~ cpmadumatpol... |
cpmadumatpoly 20507 | The product of the charact... |
cayhamlem2 20508 | Lemma for ~ cayhamlem3 . ... |
chcoeffeqlem 20509 | Lemma for ~ chcoeffeq . (... |
chcoeffeq 20510 | The coefficients of the ch... |
cayhamlem3 20511 | Lemma for ~ cayhamlem4 . ... |
cayhamlem4 20512 | Lemma for ~ cayleyhamilton... |
cayleyhamilton0 20513 | The Cayley-Hamilton theore... |
cayleyhamilton 20514 | The Cayley-Hamilton theore... |
cayleyhamiltonALT 20515 | Alternate proof of ~ cayle... |
cayleyhamilton1 20516 | The Cayley-Hamilton theore... |
istopg 20525 | Express the predicate " ` ... |
istop2g 20526 | Express the predicate " ` ... |
uniopn 20527 | The union of a subset of a... |
iunopn 20528 | The indexed union of a sub... |
inopn 20529 | The intersection of two op... |
fitop 20530 | A topology is closed under... |
fiinopn 20531 | The intersection of a none... |
iinopn 20532 | The intersection of a none... |
unopn 20533 | The union of two open sets... |
0opn 20534 | The empty set is an open s... |
0ntop 20535 | The empty set is not a top... |
topopn 20536 | The underlying set of a to... |
eltopss 20537 | A member of a topology is ... |
riinopn 20538 | A finite indexed relative ... |
rintopn 20539 | A finite relative intersec... |
istopon 20540 | Property of being a topolo... |
topontop 20541 | A topology on a given base... |
toponuni 20542 | The base set of a topology... |
toponmax 20543 | The base set of a topology... |
toponss 20544 | A member of a topology is ... |
toponcom 20545 | If ` K ` is a topology on ... |
topontopi 20546 | A topology on a given base... |
toponunii 20547 | The base set of a topology... |
toptopon 20548 | Alternative definition of ... |
topgele 20549 | The topologies over the sa... |
topsn 20550 | The only topology on a sin... |
istps 20551 | Express the predicate "is ... |
istps2 20552 | Express the predicate "is ... |
tpsuni 20553 | The base set of a topologi... |
tpstop 20554 | The topology extractor on ... |
tpspropd 20555 | A topological space depend... |
tpsprop2d 20556 | A topological space depend... |
topontopn 20557 | Express the predicate "is ... |
tsettps 20558 | If the topology component ... |
istpsi 20559 | Properties that determine ... |
eltpsg 20560 | Properties that determine ... |
eltpsi 20561 | Properties that determine ... |
isbasisg 20562 | Express the predicate " ` ... |
isbasis2g 20563 | Express the predicate " ` ... |
isbasis3g 20564 | Express the predicate " ` ... |
basis1 20565 | Property of a basis. (Con... |
basis2 20566 | Property of a basis. (Con... |
fiinbas 20567 | If a set is closed under f... |
basdif0 20568 | A basis is not affected by... |
baspartn 20569 | A disjoint system of sets ... |
tgval 20570 | The topology generated by ... |
tgval2 20571 | Definition of a topology g... |
eltg 20572 | Membership in a topology g... |
eltg2 20573 | Membership in a topology g... |
eltg2b 20574 | Membership in a topology g... |
eltg4i 20575 | An open set in a topology ... |
eltg3i 20576 | The union of a set of basi... |
eltg3 20577 | Membership in a topology g... |
tgval3 20578 | Alternate expression for t... |
tg1 20579 | Property of a member of a ... |
tg2 20580 | Property of a member of a ... |
bastg 20581 | A member of a basis is a s... |
unitg 20582 | The topology generated by ... |
tgss 20583 | Subset relation for genera... |
tgcl 20584 | Show that a basis generate... |
tgclb 20585 | The property ~ tgcl can be... |
tgtopon 20586 | A basis generates a topolo... |
topbas 20587 | A topology is its own basi... |
tgtop 20588 | A topology is its own basi... |
eltop 20589 | Membership in a topology, ... |
eltop2 20590 | Membership in a topology. ... |
eltop3 20591 | Membership in a topology. ... |
fibas 20592 | A collection of finite int... |
tgdom 20593 | A space has no more open s... |
tgiun 20594 | The indexed union of a set... |
tgidm 20595 | The topology generator fun... |
bastop 20596 | Two ways to express that a... |
tgtop11 20597 | The topology generation fu... |
0top 20598 | The singleton of the empty... |
en1top 20599 | ` { (/) } ` is the only to... |
en2top 20600 | If a topology has two elem... |
tgss3 20601 | A criterion for determinin... |
tgss2 20602 | A criterion for determinin... |
basgen 20603 | Given a topology ` J ` , s... |
basgen2 20604 | Given a topology ` J ` , s... |
2basgen 20605 | Conditions that determine ... |
tgfiss 20606 | If a subbase is included i... |
tgdif0 20607 | A generated topology is no... |
bastop1 20608 | A subset of a topology is ... |
bastop2 20609 | A version of ~ bastop1 tha... |
distop 20610 | The discrete topology on a... |
distopon 20611 | The discrete topology on a... |
sn0topon 20612 | The singleton of the empty... |
sn0top 20613 | The singleton of the empty... |
indislem 20614 | A lemma to eliminate some ... |
indistopon 20615 | The indiscrete topology on... |
indistop 20616 | The indiscrete topology on... |
indisuni 20617 | The base set of the indisc... |
fctop 20618 | The finite complement topo... |
fctop2 20619 | The finite complement topo... |
cctop 20620 | The countable complement t... |
ppttop 20621 | The particular point topol... |
pptbas 20622 | The particular point topol... |
epttop 20623 | The excluded point topolog... |
indistpsx 20624 | The indiscrete topology on... |
indistps 20625 | The indiscrete topology on... |
indistps2 20626 | The indiscrete topology on... |
indistpsALT 20627 | The indiscrete topology on... |
indistps2ALT 20628 | The indiscrete topology on... |
distps 20629 | The discrete topology on a... |
fncld 20636 | The closed-set generator i... |
cldval 20637 | The set of closed sets of ... |
ntrfval 20638 | The interior function on t... |
clsfval 20639 | The closure function on th... |
cldrcl 20640 | Reverse closure of the clo... |
iscld 20641 | The predicate " ` S ` is a... |
iscld2 20642 | A subset of the underlying... |
cldss 20643 | A closed set is a subset o... |
cldss2 20644 | The set of closed sets is ... |
cldopn 20645 | The complement of a closed... |
isopn2 20646 | A subset of the underlying... |
opncld 20647 | The complement of an open ... |
difopn 20648 | The difference of a closed... |
topcld 20649 | The underlying set of a to... |
ntrval 20650 | The interior of a subset o... |
clsval 20651 | The closure of a subset of... |
0cld 20652 | The empty set is closed. ... |
iincld 20653 | The indexed intersection o... |
intcld 20654 | The intersection of a set ... |
uncld 20655 | The union of two closed se... |
cldcls 20656 | A closed subset equals its... |
incld 20657 | The intersection of two cl... |
riincld 20658 | An indexed relative inters... |
iuncld 20659 | A finite indexed union of ... |
unicld 20660 | A finite union of closed s... |
clscld 20661 | The closure of a subset of... |
clsf 20662 | The closure function is a ... |
ntropn 20663 | The interior of a subset o... |
clsval2 20664 | Express closure in terms o... |
ntrval2 20665 | Interior expressed in term... |
ntrdif 20666 | An interior of a complemen... |
clsdif 20667 | A closure of a complement ... |
clsss 20668 | Subset relationship for cl... |
ntrss 20669 | Subset relationship for in... |
sscls 20670 | A subset of a topology's u... |
ntrss2 20671 | A subset includes its inte... |
ssntr 20672 | An open subset of a set is... |
clsss3 20673 | The closure of a subset of... |
ntrss3 20674 | The interior of a subset o... |
ntrin 20675 | A pairwise intersection of... |
cmclsopn 20676 | The complement of a closur... |
cmntrcld 20677 | The complement of an inter... |
iscld3 20678 | A subset is closed iff it ... |
iscld4 20679 | A subset is closed iff it ... |
isopn3 20680 | A subset is open iff it eq... |
clsidm 20681 | The closure operation is i... |
ntridm 20682 | The interior operation is ... |
clstop 20683 | The closure of a topology'... |
ntrtop 20684 | The interior of a topology... |
0ntr 20685 | A subset with an empty int... |
clsss2 20686 | If a subset is included in... |
elcls 20687 | Membership in a closure. ... |
elcls2 20688 | Membership in a closure. ... |
clsndisj 20689 | Any open set containing a ... |
ntrcls0 20690 | A subset whose closure has... |
ntreq0 20691 | Two ways to say that a sub... |
cldmre 20692 | The closed sets of a topol... |
mrccls 20693 | Moore closure generalizes ... |
cls0 20694 | The closure of the empty s... |
ntr0 20695 | The interior of the empty ... |
isopn3i 20696 | An open subset equals its ... |
elcls3 20697 | Membership in a closure in... |
opncldf1 20698 | A bijection useful for con... |
opncldf2 20699 | The values of the open-clo... |
opncldf3 20700 | The values of the converse... |
isclo 20701 | A set ` A ` is clopen iff ... |
isclo2 20702 | A set ` A ` is clopen iff ... |
discld 20703 | The open sets of a discret... |
sn0cld 20704 | The closed sets of the top... |
indiscld 20705 | The closed sets of an indi... |
mretopd 20706 | A Moore collection which i... |
toponmre 20707 | The topologies over a give... |
cldmreon 20708 | The closed sets of a topol... |
iscldtop 20709 | A family is the closed set... |
mreclatdemoBAD 20710 | The closed subspaces of a ... |
neifval 20713 | The neighborhood function ... |
neif 20714 | The neighborhood function ... |
neiss2 20715 | A set with a neighborhood ... |
neival 20716 | The set of neighborhoods o... |
isnei 20717 | The predicate " ` N ` is a... |
neiint 20718 | An intuitive definition of... |
isneip 20719 | The predicate " ` N ` is a... |
neii1 20720 | A neighborhood is included... |
neisspw 20721 | The neighborhoods of any s... |
neii2 20722 | Property of a neighborhood... |
neiss 20723 | Any neighborhood of a set ... |
ssnei 20724 | A set is included in its n... |
elnei 20725 | A point belongs to any of ... |
0nnei 20726 | The empty set is not a nei... |
neips 20727 | A neighborhood of a set is... |
opnneissb 20728 | An open set is a neighborh... |
opnssneib 20729 | Any superset of an open se... |
ssnei2 20730 | Any subset of ` X ` contai... |
neindisj 20731 | Any neighborhood of an ele... |
opnneiss 20732 | An open set is a neighborh... |
opnneip 20733 | An open set is a neighborh... |
opnnei 20734 | A set is open iff it is a ... |
tpnei 20735 | The underlying set of a to... |
neiuni 20736 | The union of the neighborh... |
neindisj2 20737 | A point ` P ` belongs to t... |
topssnei 20738 | A finer topology has more ... |
innei 20739 | The intersection of two ne... |
opnneiid 20740 | Only an open set is a neig... |
neissex 20741 | For any neighborhood ` N `... |
0nei 20742 | The empty set is a neighbo... |
neipeltop 20743 | Lemma for ~ neiptopreu . ... |
neiptopuni 20744 | Lemma for ~ neiptopreu . ... |
neiptoptop 20745 | Lemma for ~ neiptopreu . ... |
neiptopnei 20746 | Lemma for ~ neiptopreu . ... |
neiptopreu 20747 | If, to each element ` P ` ... |
lpfval 20752 | The limit point function o... |
lpval 20753 | The set of limit points of... |
islp 20754 | The predicate " ` P ` is a... |
lpsscls 20755 | The limit points of a subs... |
lpss 20756 | The limit points of a subs... |
lpdifsn 20757 | ` P ` is a limit point of ... |
lpss3 20758 | Subset relationship for li... |
islp2 20759 | The predicate " ` P ` is a... |
islp3 20760 | The predicate " ` P ` is a... |
maxlp 20761 | A point is a limit point o... |
clslp 20762 | The closure of a subset of... |
islpi 20763 | A point belonging to a set... |
cldlp 20764 | A subset of a topological ... |
isperf 20765 | Definition of a perfect sp... |
isperf2 20766 | Definition of a perfect sp... |
isperf3 20767 | A perfect space is a topol... |
perflp 20768 | The limit points of a perf... |
perfi 20769 | Property of a perfect spac... |
perftop 20770 | A perfect space is a topol... |
restrcl 20771 | Reverse closure for the su... |
restbas 20772 | A subspace topology basis ... |
tgrest 20773 | A subspace can be generate... |
resttop 20774 | A subspace topology is a t... |
resttopon 20775 | A subspace topology is a t... |
restuni 20776 | The underlying set of a su... |
stoig 20777 | The topological space buil... |
restco 20778 | Composition of subspaces. ... |
restabs 20779 | Equivalence of being a sub... |
restin 20780 | When the subspace region i... |
restuni2 20781 | The underlying set of a su... |
resttopon2 20782 | The underlying set of a su... |
rest0 20783 | The subspace topology indu... |
restsn 20784 | The only subspace topology... |
restsn2 20785 | The subspace topology indu... |
restcld 20786 | A closed set of a subspace... |
restcldi 20787 | A closed set is closed in ... |
restcldr 20788 | A set which is closed in t... |
restopnb 20789 | If ` B ` is an open subset... |
ssrest 20790 | If ` K ` is a finer topolo... |
restopn2 20791 | The if ` A ` is open, then... |
restdis 20792 | A subspace of a discrete t... |
restfpw 20793 | The restriction of the set... |
neitr 20794 | The neighborhood of a trac... |
restcls 20795 | A closure in a subspace to... |
restntr 20796 | An interior in a subspace ... |
restlp 20797 | The limit points of a subs... |
restperf 20798 | Perfection of a subspace. ... |
perfopn 20799 | An open subset of a perfec... |
resstopn 20800 | The topology of a restrict... |
resstps 20801 | A restricted topological s... |
ordtbaslem 20802 | Lemma for ~ ordtbas . In ... |
ordtval 20803 | Value of the order topolog... |
ordtuni 20804 | Value of the order topolog... |
ordtbas2 20805 | Lemma for ~ ordtbas . (Co... |
ordtbas 20806 | In a total order, the fini... |
ordttopon 20807 | Value of the order topolog... |
ordtopn1 20808 | An upward ray ` ( P , +oo ... |
ordtopn2 20809 | A downward ray ` ( -oo , P... |
ordtopn3 20810 | An open interval ` ( A , B... |
ordtcld1 20811 | A downward ray ` ( -oo , P... |
ordtcld2 20812 | An upward ray ` [ P , +oo ... |
ordtcld3 20813 | A closed interval ` [ A , ... |
ordttop 20814 | The order topology is a to... |
ordtcnv 20815 | The order dual generates t... |
ordtrest 20816 | The subspace topology of a... |
ordtrest2lem 20817 | Lemma for ~ ordtrest2 . (... |
ordtrest2 20818 | An interval-closed set ` A... |
letopon 20819 | The topology of the extend... |
letop 20820 | The topology of the extend... |
letopuni 20821 | The topology of the extend... |
xrstopn 20822 | The topology component of ... |
xrstps 20823 | The extended real number s... |
leordtvallem1 20824 | Lemma for ~ leordtval . (... |
leordtvallem2 20825 | Lemma for ~ leordtval . (... |
leordtval2 20826 | The topology of the extend... |
leordtval 20827 | The topology of the extend... |
iccordt 20828 | A closed interval is close... |
iocpnfordt 20829 | An unbounded above open in... |
icomnfordt 20830 | An unbounded above open in... |
iooordt 20831 | An open interval is open i... |
reordt 20832 | The real numbers are an op... |
lecldbas 20833 | The set of closed interval... |
pnfnei 20834 | A neighborhood of ` +oo ` ... |
mnfnei 20835 | A neighborhood of ` -oo ` ... |
ordtrestixx 20836 | The restriction of the les... |
ordtresticc 20837 | The restriction of the les... |
lmrel 20844 | The topological space conv... |
lmrcl 20845 | Reverse closure for the co... |
lmfval 20846 | The relation "sequence ` f... |
cnfval 20847 | The set of all continuous ... |
cnpfval 20848 | The function mapping the p... |
iscn 20849 | The predicate " ` F ` is a... |
cnpval 20850 | The set of all functions f... |
iscnp 20851 | The predicate " ` F ` is a... |
iscn2 20852 | The predicate " ` F ` is a... |
iscnp2 20853 | The predicate " ` F ` is a... |
cntop1 20854 | Reverse closure for a cont... |
cntop2 20855 | Reverse closure for a cont... |
cnptop1 20856 | Reverse closure for a func... |
cnptop2 20857 | Reverse closure for a func... |
iscnp3 20858 | The predicate " ` F ` is a... |
cnprcl 20859 | Reverse closure for a func... |
cnf 20860 | A continuous function is a... |
cnpf 20861 | A continuous function at p... |
cnpcl 20862 | The value of a continuous ... |
cnf2 20863 | A continuous function is a... |
cnpf2 20864 | A continuous function at p... |
cnprcl2 20865 | Reverse closure for a func... |
tgcn 20866 | The continuity predicate w... |
tgcnp 20867 | The "continuous at a point... |
subbascn 20868 | The continuity predicate w... |
ssidcn 20869 | The identity function is a... |
cnpimaex 20870 | Property of a function con... |
idcn 20871 | A restricted identity func... |
lmbr 20872 | Express the binary relatio... |
lmbr2 20873 | Express the binary relatio... |
lmbrf 20874 | Express the binary relatio... |
lmconst 20875 | A constant sequence conver... |
lmcvg 20876 | Convergence property of a ... |
iscnp4 20877 | The predicate " ` F ` is a... |
cnpnei 20878 | A condition for continuity... |
cnima 20879 | An open subset of the codo... |
cnco 20880 | The composition of two con... |
cnpco 20881 | The composition of two con... |
cnclima 20882 | A closed subset of the cod... |
iscncl 20883 | A definition of a continuo... |
cncls2i 20884 | Property of the preimage o... |
cnntri 20885 | Property of the preimage o... |
cnclsi 20886 | Property of the image of a... |
cncls2 20887 | Continuity in terms of clo... |
cncls 20888 | Continuity in terms of clo... |
cnntr 20889 | Continuity in terms of int... |
cnss1 20890 | If the topology ` K ` is f... |
cnss2 20891 | If the topology ` K ` is f... |
cncnpi 20892 | A continuous function is c... |
cnsscnp 20893 | The set of continuous func... |
cncnp 20894 | A continuous function is c... |
cncnp2 20895 | A continuous function is c... |
cnnei 20896 | Continuity in terms of nei... |
cnconst2 20897 | A constant function is con... |
cnconst 20898 | A constant function is con... |
cnrest 20899 | Continuity of a restrictio... |
cnrest2 20900 | Equivalence of continuity ... |
cnrest2r 20901 | Equivalence of continuity ... |
cnpresti 20902 | One direction of ~ cnprest... |
cnprest 20903 | Equivalence of continuity ... |
cnprest2 20904 | Equivalence of point-conti... |
cndis 20905 | Every function is continuo... |
cnindis 20906 | Every function is continuo... |
cnpdis 20907 | If ` A ` is an isolated po... |
paste 20908 | Pasting lemma. If ` A ` a... |
lmfpm 20909 | If ` F ` converges, then `... |
lmfss 20910 | Inclusion of a function ha... |
lmcl 20911 | Closure of a limit. (Cont... |
lmss 20912 | Limit on a subspace. (Con... |
sslm 20913 | A finer topology has fewer... |
lmres 20914 | A function converges iff i... |
lmff 20915 | If ` F ` converges, there ... |
lmcls 20916 | Any convergent sequence of... |
lmcld 20917 | Any convergent sequence of... |
lmcnp 20918 | The image of a convergent ... |
lmcn 20919 | The image of a convergent ... |
ist0 20934 | The predicate "is a T_0 sp... |
ist1 20935 | The predicate ` J ` is T_1... |
ishaus 20936 | Express the predicate " ` ... |
iscnrm 20937 | The property of being comp... |
t0sep 20938 | Any two topologically indi... |
t0dist 20939 | Any two distinct points in... |
t1sncld 20940 | In a T_1 space, one-point ... |
t1ficld 20941 | In a T_1 space, finite set... |
hausnei 20942 | Neighborhood property of a... |
t0top 20943 | A T_0 space is a topologic... |
t1top 20944 | A T_1 space is a topologic... |
haustop 20945 | A Hausdorff space is a top... |
isreg 20946 | The predicate "is a regula... |
regtop 20947 | A regular space is a topol... |
regsep 20948 | In a regular space, every ... |
isnrm 20949 | The predicate "is a normal... |
nrmtop 20950 | A normal space is a topolo... |
cnrmtop 20951 | A completely normal space ... |
iscnrm2 20952 | The property of being comp... |
ispnrm 20953 | The property of being perf... |
pnrmnrm 20954 | A perfectly normal space i... |
pnrmtop 20955 | A perfectly normal space i... |
pnrmcld 20956 | A closed set in a perfectl... |
pnrmopn 20957 | An open set in a perfectly... |
ist0-2 20958 | The predicate "is a T_0 sp... |
ist0-3 20959 | The predicate "is a T_0 sp... |
cnt0 20960 | The preimage of a T_0 topo... |
ist1-2 20961 | An alternate characterizat... |
t1t0 20962 | A T_1 space is a T_0 space... |
ist1-3 20963 | A space is T_1 iff every p... |
cnt1 20964 | The preimage of a T_1 topo... |
ishaus2 20965 | Express the predicate " ` ... |
haust1 20966 | A Hausdorff space is a T_1... |
hausnei2 20967 | The Hausdorff condition st... |
cnhaus 20968 | The preimage of a Hausdorf... |
nrmsep3 20969 | In a normal space, given a... |
nrmsep2 20970 | In a normal space, any two... |
nrmsep 20971 | In a normal space, disjoin... |
isnrm2 20972 | An alternate characterizat... |
isnrm3 20973 | A topological space is nor... |
cnrmi 20974 | A subspace of a completely... |
cnrmnrm 20975 | A completely normal space ... |
restcnrm 20976 | A subspace of a completely... |
resthauslem 20977 | Lemma for ~ resthaus and s... |
lpcls 20978 | The limit points of the cl... |
perfcls 20979 | A subset of a perfect spac... |
restt0 20980 | A subspace of a T_0 topolo... |
restt1 20981 | A subspace of a T_1 topolo... |
resthaus 20982 | A subspace of a Hausdorff ... |
t1sep2 20983 | Any two points in a T_1 sp... |
t1sep 20984 | Any two distinct points in... |
sncld 20985 | A singleton is closed in a... |
sshauslem 20986 | Lemma for ~ sshaus and sim... |
sst0 20987 | A topology finer than a T_... |
sst1 20988 | A topology finer than a T_... |
sshaus 20989 | A topology finer than a Ha... |
regsep2 20990 | In a regular space, a clos... |
isreg2 20991 | A topological space is reg... |
dnsconst 20992 | If a continuous mapping to... |
ordtt1 20993 | The order topology is T_1 ... |
lmmo 20994 | A sequence in a Hausdorff ... |
lmfun 20995 | The convergence relation i... |
dishaus 20996 | A discrete topology is Hau... |
ordthauslem 20997 | Lemma for ~ ordthaus . (C... |
ordthaus 20998 | The order topology of a to... |
iscmp 21001 | The predicate "is a compac... |
cmpcov 21002 | An open cover of a compact... |
cmpcov2 21003 | Rewrite ~ cmpcov for the c... |
cmpcovf 21004 | Combine ~ cmpcov with ~ ac... |
cncmp 21005 | Compactness is respected b... |
fincmp 21006 | A finite topology is compa... |
0cmp 21007 | The singleton of the empty... |
cmptop 21008 | A compact topology is a to... |
rncmp 21009 | The image of a compact set... |
imacmp 21010 | The image of a compact set... |
discmp 21011 | A discrete topology is com... |
cmpsublem 21012 | Lemma for ~ cmpsub . (Con... |
cmpsub 21013 | Two equivalent ways of des... |
tgcmp 21014 | A topology generated by a ... |
cmpcld 21015 | A closed subset of a compa... |
uncmp 21016 | The union of two compact s... |
fiuncmp 21017 | A finite union of compact ... |
sscmp 21018 | A subset of a compact topo... |
hauscmplem 21019 | Lemma for ~ hauscmp . (Co... |
hauscmp 21020 | A compact subspace of a T2... |
cmpfi 21021 | If a topology is compact a... |
cmpfii 21022 | In a compact topology, a s... |
bwth 21023 | The glorious Bolzano-Weier... |
iscon 21026 | The predicate ` J ` is a c... |
iscon2 21027 | The predicate ` J ` is a c... |
conclo 21028 | The only nonempty clopen s... |
conndisj 21029 | If a topology is connected... |
contop 21030 | A connected topology is a ... |
indiscon 21031 | The indiscrete topology (o... |
dfcon2 21032 | An alternate definition of... |
consuba 21033 | Connectedness for a subspa... |
connsub 21034 | Two equivalent ways of say... |
cnconn 21035 | Connectedness is respected... |
nconsubb 21036 | Disconnectedness for a sub... |
consubclo 21037 | If a clopen set meets a co... |
conima 21038 | The image of a connected s... |
concn 21039 | A continuous function from... |
iunconlem 21040 | Lemma for ~ iuncon . (Con... |
iuncon 21041 | The indexed union of conne... |
uncon 21042 | The union of two connected... |
clscon 21043 | The closure of a connected... |
concompid 21044 | The connected component co... |
concompcon 21045 | The connected component co... |
concompss 21046 | The connected component co... |
concompcld 21047 | The connected component co... |
concompclo 21048 | The connected component co... |
t1conperf 21049 | A connected T_1 space is p... |
is1stc 21054 | The predicate "is a first-... |
is1stc2 21055 | An equivalent way of sayin... |
1stctop 21056 | A first-countable topology... |
1stcclb 21057 | A property of points in a ... |
1stcfb 21058 | For any point ` A ` in a f... |
is2ndc 21059 | The property of being seco... |
2ndctop 21060 | A second-countable topolog... |
2ndci 21061 | A countable basis generate... |
2ndcsb 21062 | Having a countable subbase... |
2ndcredom 21063 | A second-countable space h... |
2ndc1stc 21064 | A second-countable space i... |
1stcrestlem 21065 | Lemma for ~ 1stcrest . (C... |
1stcrest 21066 | A subspace of a first-coun... |
2ndcrest 21067 | A subspace of a second-cou... |
2ndcctbss 21068 | If a topology is second-co... |
2ndcdisj 21069 | Any disjoint family of ope... |
2ndcdisj2 21070 | Any disjoint collection of... |
2ndcomap 21071 | A surjective continuous op... |
2ndcsep 21072 | A second-countable topolog... |
dis2ndc 21073 | A discrete space is second... |
1stcelcls 21074 | A point belongs to the clo... |
1stccnp 21075 | A mapping is continuous at... |
1stccn 21076 | A mapping ` X --> Y ` , wh... |
islly 21081 | The property of being a lo... |
isnlly 21082 | The property of being an n... |
llyeq 21083 | Equality theorem for the `... |
nllyeq 21084 | Equality theorem for the `... |
llytop 21085 | A locally ` A ` space is a... |
nllytop 21086 | A locally ` A ` space is a... |
llyi 21087 | The property of a locally ... |
nllyi 21088 | The property of an n-local... |
nlly2i 21089 | Eliminate the neighborhood... |
llynlly 21090 | A locally ` A ` space is n... |
llyssnlly 21091 | A locally ` A ` space is n... |
llyss 21092 | The "locally" predicate re... |
nllyss 21093 | The "n-locally" predicate ... |
subislly 21094 | The property of a subspace... |
restnlly 21095 | If the property ` A ` pass... |
restlly 21096 | If the property ` A ` pass... |
islly2 21097 | An alternative expression ... |
llyrest 21098 | An open subspace of a loca... |
nllyrest 21099 | An open subspace of an n-l... |
loclly 21100 | If ` A ` is a local proper... |
llyidm 21101 | Idempotence of the "locall... |
nllyidm 21102 | Idempotence of the "n-loca... |
toplly 21103 | A topology is locally a to... |
topnlly 21104 | A topology is n-locally a ... |
hauslly 21105 | A Hausdorff space is local... |
hausnlly 21106 | A Hausdorff space is n-loc... |
hausllycmp 21107 | A compact Hausdorff space ... |
cldllycmp 21108 | A closed subspace of a loc... |
lly1stc 21109 | First-countability is a lo... |
dislly 21110 | The discrete space ` ~P X ... |
disllycmp 21111 | A discrete space is locall... |
dis1stc 21112 | A discrete space is first-... |
hausmapdom 21113 | If ` X ` is a first-counta... |
hauspwdom 21114 | Simplify the cardinal ` A ... |
refrel 21121 | Refinement is a relation. ... |
isref 21122 | The property of being a re... |
refbas 21123 | A refinement covers the sa... |
refssex 21124 | Every set in a refinement ... |
ssref 21125 | A subcover is a refinement... |
refref 21126 | Reflexivity of refinement.... |
reftr 21127 | Refinement is transitive. ... |
refun0 21128 | Adding the empty set prese... |
isptfin 21129 | The statement "is a point-... |
islocfin 21130 | The statement "is a locall... |
finptfin 21131 | A finite cover is a point-... |
ptfinfin 21132 | A point covered by a point... |
finlocfin 21133 | A finite cover of a topolo... |
locfintop 21134 | A locally finite cover cov... |
locfinbas 21135 | A locally finite cover mus... |
locfinnei 21136 | A point covered by a local... |
lfinpfin 21137 | A locally finite cover is ... |
lfinun 21138 | Adding a finite set preser... |
locfincmp 21139 | For a compact space, the l... |
unisngl 21140 | Taking the union of the se... |
dissnref 21141 | The set of singletons is a... |
dissnlocfin 21142 | The set of singletons is l... |
locfindis 21143 | The locally finite covers ... |
locfincf 21144 | A locally finite cover in ... |
comppfsc 21145 | A space where every open c... |
kgenval 21148 | Value of the compact gener... |
elkgen 21149 | Value of the compact gener... |
kgeni 21150 | Property of the open sets ... |
kgentopon 21151 | The compact generator gene... |
kgenuni 21152 | The base set of the compac... |
kgenftop 21153 | The compact generator gene... |
kgenf 21154 | The compact generator is a... |
kgentop 21155 | A compactly generated spac... |
kgenss 21156 | The compact generator gene... |
kgenhaus 21157 | The compact generator gene... |
kgencmp 21158 | The compact generator topo... |
kgencmp2 21159 | The compact generator topo... |
kgenidm 21160 | The compact generator is i... |
iskgen2 21161 | A space is compactly gener... |
iskgen3 21162 | Derive the usual definitio... |
llycmpkgen2 21163 | A locally compact space is... |
cmpkgen 21164 | A compact space is compact... |
llycmpkgen 21165 | A locally compact space is... |
1stckgenlem 21166 | The one-point compactifica... |
1stckgen 21167 | A first-countable space is... |
kgen2ss 21168 | The compact generator pres... |
kgencn 21169 | A function from a compactl... |
kgencn2 21170 | A function ` F : J --> K `... |
kgencn3 21171 | The set of continuous func... |
kgen2cn 21172 | A continuous function is a... |
txval 21177 | Value of the binary topolo... |
txuni2 21178 | The underlying set of the ... |
txbasex 21179 | The basis for the product ... |
txbas 21180 | The set of Cartesian produ... |
eltx 21181 | A set in a product is open... |
txtop 21182 | The product of two topolog... |
ptval 21183 | The value of the product t... |
ptpjpre1 21184 | The preimage of a projecti... |
elpt 21185 | Elementhood in the bases o... |
elptr 21186 | A basic open set in the pr... |
elptr2 21187 | A basic open set in the pr... |
ptbasid 21188 | The base set of the produc... |
ptuni2 21189 | The base set for the produ... |
ptbasin 21190 | The basis for a product to... |
ptbasin2 21191 | The basis for a product to... |
ptbas 21192 | The basis for a product to... |
ptpjpre2 21193 | The basis for a product to... |
ptbasfi 21194 | The basis for the product ... |
pttop 21195 | The product topology is a ... |
ptopn 21196 | A basic open set in the pr... |
ptopn2 21197 | A sub-basic open set in th... |
xkotf 21198 | Functionality of function ... |
xkobval 21199 | Alternative expression for... |
xkoval 21200 | Value of the compact-open ... |
xkotop 21201 | The compact-open topology ... |
xkoopn 21202 | A basic open set of the co... |
txtopi 21203 | The product of two topolog... |
txtopon 21204 | The underlying set of the ... |
txuni 21205 | The underlying set of the ... |
txunii 21206 | The underlying set of the ... |
ptuni 21207 | The base set for the produ... |
ptunimpt 21208 | Base set of a product topo... |
pttopon 21209 | The base set for the produ... |
pttoponconst 21210 | The base set for a product... |
ptuniconst 21211 | The base set for a product... |
xkouni 21212 | The base set of the compac... |
xkotopon 21213 | The base set of the compac... |
ptval2 21214 | The value of the product t... |
txopn 21215 | The product of two open se... |
txcld 21216 | The product of two closed ... |
txcls 21217 | Closure of a rectangle in ... |
txss12 21218 | Subset property of the top... |
txbasval 21219 | It is sufficient to consid... |
neitx 21220 | The Cartesian product of t... |
txcnpi 21221 | Continuity of a two-argume... |
tx1cn 21222 | Continuity of the first pr... |
tx2cn 21223 | Continuity of the second p... |
ptpjcn 21224 | Continuity of a projection... |
ptpjopn 21225 | The projection map is an o... |
ptcld 21226 | A closed box in the produc... |
ptcldmpt 21227 | A closed box in the produc... |
ptclsg 21228 | The closure of a box in th... |
ptcls 21229 | The closure of a box in th... |
dfac14lem 21230 | Lemma for ~ dfac14 . By e... |
dfac14 21231 | Theorem ~ ptcls is an equi... |
xkoccn 21232 | The "constant function" fu... |
txcnp 21233 | If two functions are conti... |
ptcnplem 21234 | Lemma for ~ ptcnp . (Cont... |
ptcnp 21235 | If every projection of a f... |
upxp 21236 | Universal property of the ... |
txcnmpt 21237 | A map into the product of ... |
uptx 21238 | Universal property of the ... |
txcn 21239 | A map into the product of ... |
ptcn 21240 | If every projection of a f... |
prdstopn 21241 | Topology of a structure pr... |
prdstps 21242 | A structure product of top... |
pwstps 21243 | A structure product of top... |
txrest 21244 | The subspace of a topologi... |
txdis 21245 | The topological product of... |
txindislem 21246 | Lemma for ~ txindis . (Co... |
txindis 21247 | The topological product of... |
txdis1cn 21248 | A function is jointly cont... |
txlly 21249 | If the property ` A ` is p... |
txnlly 21250 | If the property ` A ` is p... |
pthaus 21251 | The product of a collectio... |
ptrescn 21252 | Restriction is a continuou... |
txtube 21253 | The "tube lemma". If ` X ... |
txcmplem1 21254 | Lemma for ~ txcmp . (Cont... |
txcmplem2 21255 | Lemma for ~ txcmp . (Cont... |
txcmp 21256 | The topological product of... |
txcmpb 21257 | The topological product of... |
hausdiag 21258 | A topology is Hausdorff if... |
hauseqlcld 21259 | In a Hausdorff topology, t... |
txhaus 21260 | The topological product of... |
txlm 21261 | Two sequences converge iff... |
lmcn2 21262 | The image of a convergent ... |
tx1stc 21263 | The topological product of... |
tx2ndc 21264 | The topological product of... |
txkgen 21265 | The topological product of... |
xkohaus 21266 | If the codomain space is H... |
xkoptsub 21267 | The compact-open topology ... |
xkopt 21268 | The compact-open topology ... |
xkopjcn 21269 | Continuity of a projection... |
xkoco1cn 21270 | If ` F ` is a continuous f... |
xkoco2cn 21271 | If ` F ` is a continuous f... |
xkococnlem 21272 | Continuity of the composit... |
xkococn 21273 | Continuity of the composit... |
cnmptid 21274 | The identity function is c... |
cnmptc 21275 | A constant function is con... |
cnmpt11 21276 | The composition of continu... |
cnmpt11f 21277 | The composition of continu... |
cnmpt1t 21278 | The composition of continu... |
cnmpt12f 21279 | The composition of continu... |
cnmpt12 21280 | The composition of continu... |
cnmpt1st 21281 | The projection onto the fi... |
cnmpt2nd 21282 | The projection onto the se... |
cnmpt2c 21283 | A constant function is con... |
cnmpt21 21284 | The composition of continu... |
cnmpt21f 21285 | The composition of continu... |
cnmpt2t 21286 | The composition of continu... |
cnmpt22 21287 | The composition of continu... |
cnmpt22f 21288 | The composition of continu... |
cnmpt1res 21289 | The restriction of a conti... |
cnmpt2res 21290 | The restriction of a conti... |
cnmptcom 21291 | The argument converse of a... |
cnmptkc 21292 | The curried first projecti... |
cnmptkp 21293 | The evaluation of the inne... |
cnmptk1 21294 | The composition of a curri... |
cnmpt1k 21295 | The composition of a one-a... |
cnmptkk 21296 | The composition of two cur... |
xkofvcn 21297 | Joint continuity of the fu... |
cnmptk1p 21298 | The evaluation of a currie... |
cnmptk2 21299 | The uncurrying of a currie... |
xkoinjcn 21300 | Continuity of "injection",... |
cnmpt2k 21301 | The currying of a two-argu... |
txcon 21302 | The topological product of... |
imasnopn 21303 | If a relation graph is ope... |
imasncld 21304 | If a relation graph is clo... |
imasncls 21305 | If a relation graph is clo... |
qtopval 21308 | Value of the quotient topo... |
qtopval2 21309 | Value of the quotient topo... |
elqtop 21310 | Value of the quotient topo... |
qtopres 21311 | The quotient topology is u... |
qtoptop2 21312 | The quotient topology is a... |
qtoptop 21313 | The quotient topology is a... |
elqtop2 21314 | Value of the quotient topo... |
qtopuni 21315 | The base set of the quotie... |
elqtop3 21316 | Value of the quotient topo... |
qtoptopon 21317 | The base set of the quotie... |
qtopid 21318 | A quotient map is a contin... |
idqtop 21319 | The quotient topology indu... |
qtopcmplem 21320 | Lemma for ~ qtopcmp and ~ ... |
qtopcmp 21321 | A quotient of a compact sp... |
qtopcon 21322 | A quotient of a connected ... |
qtopkgen 21323 | A quotient of a compactly ... |
basqtop 21324 | An injection maps bases to... |
tgqtop 21325 | An injection maps generate... |
qtopcld 21326 | The property of being a cl... |
qtopcn 21327 | Universal property of a qu... |
qtopss 21328 | A surjective continuous fu... |
qtopeu 21329 | Universal property of the ... |
qtoprest 21330 | If ` A ` is a saturated op... |
qtopomap 21331 | If ` F ` is a surjective c... |
qtopcmap 21332 | If ` F ` is a surjective c... |
imastopn 21333 | The topology of an image s... |
imastps 21334 | The image of a topological... |
qustps 21335 | A quotient structure is a ... |
kqfval 21336 | Value of the function appe... |
kqfeq 21337 | Two points in the Kolmogor... |
kqffn 21338 | The topological indistingu... |
kqval 21339 | Value of the quotient topo... |
kqtopon 21340 | The Kolmogorov quotient is... |
kqid 21341 | The topological indistingu... |
ist0-4 21342 | The topological indistingu... |
kqfvima 21343 | When the image set is open... |
kqsat 21344 | Any open set is saturated ... |
kqdisj 21345 | A version of ~ imain for t... |
kqcldsat 21346 | Any closed set is saturate... |
kqopn 21347 | The topological indistingu... |
kqcld 21348 | The topological indistingu... |
kqt0lem 21349 | Lemma for ~ kqt0 . (Contr... |
isr0 21350 | The property " ` J ` is an... |
r0cld 21351 | The analogue of the T_1 ax... |
regr1lem 21352 | Lemma for ~ regr1 . (Cont... |
regr1lem2 21353 | A Kolmogorov quotient of a... |
kqreglem1 21354 | A Kolmogorov quotient of a... |
kqreglem2 21355 | If the Kolmogorov quotient... |
kqnrmlem1 21356 | A Kolmogorov quotient of a... |
kqnrmlem2 21357 | If the Kolmogorov quotient... |
kqtop 21358 | The Kolmogorov quotient is... |
kqt0 21359 | The Kolmogorov quotient is... |
kqf 21360 | The Kolmogorov quotient is... |
r0sep 21361 | The separation property of... |
nrmr0reg 21362 | A normal R_0 space is also... |
regr1 21363 | A regular space is R_1, wh... |
kqreg 21364 | The Kolmogorov quotient of... |
kqnrm 21365 | The Kolmogorov quotient of... |
hmeofn 21370 | The set of homeomorphisms ... |
hmeofval 21371 | The set of all the homeomo... |
ishmeo 21372 | The predicate F is a homeo... |
hmeocn 21373 | A homeomorphism is continu... |
hmeocnvcn 21374 | The converse of a homeomor... |
hmeocnv 21375 | The converse of a homeomor... |
hmeof1o2 21376 | A homeomorphism is a 1-1-o... |
hmeof1o 21377 | A homeomorphism is a 1-1-o... |
hmeoima 21378 | The image of an open set b... |
hmeoopn 21379 | Homeomorphisms preserve op... |
hmeocld 21380 | Homeomorphisms preserve cl... |
hmeocls 21381 | Homeomorphisms preserve cl... |
hmeontr 21382 | Homeomorphisms preserve in... |
hmeoimaf1o 21383 | The function mapping open ... |
hmeores 21384 | The restriction of a homeo... |
hmeoco 21385 | The composite of two homeo... |
idhmeo 21386 | The identity function is a... |
hmeocnvb 21387 | The converse of a homeomor... |
hmeoqtop 21388 | A homeomorphism is a quoti... |
hmph 21389 | Express the predicate ` J ... |
hmphi 21390 | If there is a homeomorphis... |
hmphtop 21391 | Reverse closure for the ho... |
hmphtop1 21392 | The relation "being homeom... |
hmphtop2 21393 | The relation "being homeom... |
hmphref 21394 | "Is homeomorphic to" is re... |
hmphsym 21395 | "Is homeomorphic to" is sy... |
hmphtr 21396 | "Is homeomorphic to" is tr... |
hmpher 21397 | "Is homeomorphic to" is an... |
hmphen 21398 | Homeomorphisms preserve th... |
hmphsymb 21399 | "Is homeomorphic to" is sy... |
haushmphlem 21400 | Lemma for ~ haushmph and s... |
cmphmph 21401 | Compactness is a topologic... |
conhmph 21402 | Connectedness is a topolog... |
t0hmph 21403 | T_0 is a topological prope... |
t1hmph 21404 | T_1 is a topological prope... |
haushmph 21405 | Hausdorff-ness is a topolo... |
reghmph 21406 | Regularity is a topologica... |
nrmhmph 21407 | Normality is a topological... |
hmph0 21408 | A topology homeomorphic to... |
hmphdis 21409 | Homeomorphisms preserve to... |
hmphindis 21410 | Homeomorphisms preserve to... |
indishmph 21411 | Equinumerous sets equipped... |
hmphen2 21412 | Homeomorphisms preserve th... |
cmphaushmeo 21413 | A continuous bijection fro... |
ordthmeolem 21414 | Lemma for ~ ordthmeo . (C... |
ordthmeo 21415 | An order isomorphism is a ... |
txhmeo 21416 | Lift a pair of homeomorphi... |
txswaphmeolem 21417 | Show inverse for the "swap... |
txswaphmeo 21418 | There is a homeomorphism f... |
pt1hmeo 21419 | The canonical homeomorphis... |
ptuncnv 21420 | Exhibit the converse funct... |
ptunhmeo 21421 | Define a homeomorphism fro... |
xpstopnlem1 21422 | The function ` F ` used in... |
xpstps 21423 | A binary product of topolo... |
xpstopnlem2 21424 | Lemma for ~ xpstopn . (Co... |
xpstopn 21425 | The topology on a binary p... |
ptcmpfi 21426 | A topological product of f... |
xkocnv 21427 | The inverse of the "curryi... |
xkohmeo 21428 | The Exponential Law for to... |
qtopf1 21429 | If a quotient map is injec... |
qtophmeo 21430 | If two functions on a base... |
t0kq 21431 | A topological space is T_0... |
kqhmph 21432 | A topological space is T_0... |
ist1-5lem 21433 | Lemma for ~ ist1-5 and sim... |
t1r0 21434 | A T_1 space is R_0. That ... |
ist1-5 21435 | A topological space is T_1... |
ishaus3 21436 | A topological space is Hau... |
nrmreg 21437 | A normal T_1 space is regu... |
reghaus 21438 | A regular T_0 space is Hau... |
nrmhaus 21439 | A T_1 normal space is Haus... |
elmptrab 21440 | Membership in a one-parame... |
elmptrab2OLD 21441 | Obsolete version of ~ elmp... |
elmptrab2 21442 | Membership in a one-parame... |
isfbas 21443 | The predicate " ` F ` is a... |
fbasne0 21444 | There are no empty filter ... |
0nelfb 21445 | No filter base contains th... |
fbsspw 21446 | A filter base on a set is ... |
fbelss 21447 | An element of the filter b... |
fbdmn0 21448 | The domain of a filter bas... |
isfbas2 21449 | The predicate " ` F ` is a... |
fbasssin 21450 | A filter base contains sub... |
fbssfi 21451 | A filter base contains sub... |
fbssint 21452 | A filter base contains sub... |
fbncp 21453 | A filter base does not con... |
fbun 21454 | A necessary and sufficient... |
fbfinnfr 21455 | No filter base containing ... |
opnfbas 21456 | The collection of open sup... |
trfbas2 21457 | Conditions for the trace o... |
trfbas 21458 | Conditions for the trace o... |
isfil 21461 | The predicate "is a filter... |
filfbas 21462 | A filter is a filter base.... |
0nelfil 21463 | The empty set doesn't belo... |
fileln0 21464 | An element of a filter is ... |
filsspw 21465 | A filter is a subset of th... |
filelss 21466 | An element of a filter is ... |
filss 21467 | A filter is closed under t... |
filin 21468 | A filter is closed under t... |
filtop 21469 | The underlying set belongs... |
isfil2 21470 | Derive the standard axioms... |
isfildlem 21471 | Lemma for ~ isfild . (Con... |
isfild 21472 | Sufficient condition for a... |
filfi 21473 | A filter is closed under t... |
filinn0 21474 | The intersection of two el... |
filintn0 21475 | A filter has the finite in... |
filn0 21476 | The empty set is not a fil... |
infil 21477 | The intersection of two fi... |
snfil 21478 | A singleton is a filter. ... |
fbasweak 21479 | A filter base on any set i... |
snfbas 21480 | Condition for a singleton ... |
fsubbas 21481 | A condition for a set to g... |
fbasfip 21482 | A filter base has the fini... |
fbunfip 21483 | A helpful lemma for showin... |
fgval 21484 | The filter generating clas... |
elfg 21485 | A condition for elements o... |
ssfg 21486 | A filter base is a subset ... |
fgss 21487 | A bigger base generates a ... |
fgss2 21488 | A condition for a filter t... |
fgfil 21489 | A filter generates itself.... |
elfilss 21490 | An element belongs to a fi... |
filfinnfr 21491 | No filter containing a fin... |
fgcl 21492 | A generated filter is a fi... |
fgabs 21493 | Absorption law for filter ... |
neifil 21494 | The neighborhoods of a non... |
filunibas 21495 | Recover the base set from ... |
filunirn 21496 | Two ways to express a filt... |
filcon 21497 | A filter gives rise to a c... |
fbasrn 21498 | Given a filter on a domain... |
filuni 21499 | The union of a nonempty se... |
trfil1 21500 | Conditions for the trace o... |
trfil2 21501 | Conditions for the trace o... |
trfil3 21502 | Conditions for the trace o... |
trfilss 21503 | If ` A ` is a member of th... |
fgtr 21504 | If ` A ` is a member of th... |
trfg 21505 | The trace operation and th... |
trnei 21506 | The trace, over a set ` A ... |
cfinfil 21507 | Relative complements of th... |
csdfil 21508 | The set of all elements wh... |
supfil 21509 | The supersets of a nonempt... |
zfbas 21510 | The set of upper sets of i... |
uzrest 21511 | The restriction of the set... |
uzfbas 21512 | The set of upper sets of i... |
isufil 21517 | The property of being an u... |
ufilfil 21518 | An ultrafilter is a filter... |
ufilss 21519 | For any subset of the base... |
ufilb 21520 | The complement is in an ul... |
ufilmax 21521 | Any filter finer than an u... |
isufil2 21522 | The maximal property of an... |
ufprim 21523 | An ultrafilter is a prime ... |
trufil 21524 | Conditions for the trace o... |
filssufilg 21525 | A filter is contained in s... |
filssufil 21526 | A filter is contained in s... |
isufl 21527 | Define the (strong) ultraf... |
ufli 21528 | Property of a set that sat... |
numufl 21529 | Consequence of ~ filssufil... |
fiufl 21530 | A finite set satisfies the... |
acufl 21531 | The axiom of choice implie... |
ssufl 21532 | If ` Y ` is a subset of ` ... |
ufileu 21533 | If the ultrafilter contain... |
filufint 21534 | A filter is equal to the i... |
uffix 21535 | Lemma for ~ fixufil and ~ ... |
fixufil 21536 | The condition describing a... |
uffixfr 21537 | An ultrafilter is either f... |
uffix2 21538 | A classification of fixed ... |
uffixsn 21539 | The singleton of the gener... |
ufildom1 21540 | An ultrafilter is generate... |
uffinfix 21541 | An ultrafilter containing ... |
cfinufil 21542 | An ultrafilter is free iff... |
ufinffr 21543 | An infinite subset is cont... |
ufilen 21544 | Any infinite set has an ul... |
ufildr 21545 | An ultrafilter gives rise ... |
fin1aufil 21546 | There are no definable fre... |
fmval 21557 | Introduce a function that ... |
fmfil 21558 | A mapping filter is a filt... |
fmf 21559 | Pushing-forward via a func... |
fmss 21560 | A finer filter produces a ... |
elfm 21561 | An element of a mapping fi... |
elfm2 21562 | An element of a mapping fi... |
fmfg 21563 | The image filter of a filt... |
elfm3 21564 | An alternate formulation o... |
imaelfm 21565 | An image of a filter eleme... |
rnelfmlem 21566 | Lemma for ~ rnelfm . (Con... |
rnelfm 21567 | A condition for a filter t... |
fmfnfmlem1 21568 | Lemma for ~ fmfnfm . (Con... |
fmfnfmlem2 21569 | Lemma for ~ fmfnfm . (Con... |
fmfnfmlem3 21570 | Lemma for ~ fmfnfm . (Con... |
fmfnfmlem4 21571 | Lemma for ~ fmfnfm . (Con... |
fmfnfm 21572 | A filter finer than an ima... |
fmufil 21573 | An image filter of an ultr... |
fmid 21574 | The filter map applied to ... |
fmco 21575 | Composition of image filte... |
ufldom 21576 | The ultrafilter lemma prop... |
flimval 21577 | The set of limit points of... |
elflim2 21578 | The predicate "is a limit ... |
flimtop 21579 | Reverse closure for the li... |
flimneiss 21580 | A filter contains the neig... |
flimnei 21581 | A filter contains all of t... |
flimelbas 21582 | A limit point of a filter ... |
flimfil 21583 | Reverse closure for the li... |
flimtopon 21584 | Reverse closure for the li... |
elflim 21585 | The predicate "is a limit ... |
flimss2 21586 | A limit point of a filter ... |
flimss1 21587 | A limit point of a filter ... |
neiflim 21588 | A point is a limit point o... |
flimopn 21589 | The condition for being a ... |
fbflim 21590 | A condition for a filter t... |
fbflim2 21591 | A condition for a filter b... |
flimclsi 21592 | The convergent points of a... |
hausflimlem 21593 | If ` A ` and ` B ` are bot... |
hausflimi 21594 | One direction of ~ hausfli... |
hausflim 21595 | A condition for a topology... |
flimcf 21596 | Fineness is properly chara... |
flimrest 21597 | The set of limit points in... |
flimclslem 21598 | Lemma for ~ flimcls . (Co... |
flimcls 21599 | Closure in terms of filter... |
flimsncls 21600 | If ` A ` is a limit point ... |
hauspwpwf1 21601 | Lemma for ~ hauspwpwdom . ... |
hauspwpwdom 21602 | If ` X ` is a Hausdorff sp... |
flffval 21603 | Given a topology and a fil... |
flfval 21604 | Given a function from a fi... |
flfnei 21605 | The property of being a li... |
flfneii 21606 | A neighborhood of a limit ... |
isflf 21607 | The property of being a li... |
flfelbas 21608 | A limit point of a functio... |
flffbas 21609 | Limit points of a function... |
flftg 21610 | Limit points of a function... |
hausflf 21611 | If a function has its valu... |
hausflf2 21612 | If a convergent function h... |
cnpflfi 21613 | Forward direction of ~ cnp... |
cnpflf2 21614 | ` F ` is continuous at poi... |
cnpflf 21615 | Continuity of a function a... |
cnflf 21616 | A function is continuous i... |
cnflf2 21617 | A function is continuous i... |
flfcnp 21618 | A continuous function pres... |
lmflf 21619 | The topological limit rela... |
txflf 21620 | Two sequences converge in ... |
flfcnp2 21621 | The image of a convergent ... |
fclsval 21622 | The set of all cluster poi... |
isfcls 21623 | A cluster point of a filte... |
fclsfil 21624 | Reverse closure for the cl... |
fclstop 21625 | Reverse closure for the cl... |
fclstopon 21626 | Reverse closure for the cl... |
isfcls2 21627 | A cluster point of a filte... |
fclsopn 21628 | Write the cluster point co... |
fclsopni 21629 | An open neighborhood of a ... |
fclselbas 21630 | A cluster point is in the ... |
fclsneii 21631 | A neighborhood of a cluste... |
fclssscls 21632 | The set of cluster points ... |
fclsnei 21633 | Cluster points in terms of... |
supnfcls 21634 | The filter of supersets of... |
fclsbas 21635 | Cluster points in terms of... |
fclsss1 21636 | A finer topology has fewer... |
fclsss2 21637 | A finer filter has fewer c... |
fclsrest 21638 | The set of cluster points ... |
fclscf 21639 | Characterization of finene... |
flimfcls 21640 | A limit point is a cluster... |
fclsfnflim 21641 | A filter clusters at a poi... |
flimfnfcls 21642 | A filter converges to a po... |
fclscmpi 21643 | Forward direction of ~ fcl... |
fclscmp 21644 | A space is compact iff eve... |
uffclsflim 21645 | The cluster points of an u... |
ufilcmp 21646 | A space is compact iff eve... |
fcfval 21647 | The set of cluster points ... |
isfcf 21648 | The property of being a cl... |
fcfnei 21649 | The property of being a cl... |
fcfelbas 21650 | A cluster point of a funct... |
fcfneii 21651 | A neighborhood of a cluste... |
flfssfcf 21652 | A limit point of a functio... |
uffcfflf 21653 | If the domain filter is an... |
cnpfcfi 21654 | Lemma for ~ cnpfcf . If a... |
cnpfcf 21655 | A function ` F ` is contin... |
cnfcf 21656 | Continuity of a function i... |
flfcntr 21657 | A continuous function's va... |
alexsublem 21658 | Lemma for ~ alexsub . (Co... |
alexsub 21659 | The Alexander Subbase Theo... |
alexsubb 21660 | Biconditional form of the ... |
alexsubALTlem1 21661 | Lemma for ~ alexsubALT . ... |
alexsubALTlem2 21662 | Lemma for ~ alexsubALT . ... |
alexsubALTlem3 21663 | Lemma for ~ alexsubALT . ... |
alexsubALTlem4 21664 | Lemma for ~ alexsubALT . ... |
alexsubALT 21665 | The Alexander Subbase Theo... |
ptcmplem1 21666 | Lemma for ~ ptcmp . (Cont... |
ptcmplem2 21667 | Lemma for ~ ptcmp . (Cont... |
ptcmplem3 21668 | Lemma for ~ ptcmp . (Cont... |
ptcmplem4 21669 | Lemma for ~ ptcmp . (Cont... |
ptcmplem5 21670 | Lemma for ~ ptcmp . (Cont... |
ptcmpg 21671 | Tychonoff's theorem: The ... |
ptcmp 21672 | Tychonoff's theorem: The ... |
cnextval 21675 | The function applying cont... |
cnextfval 21676 | The continuous extension o... |
cnextrel 21677 | In the general case, a con... |
cnextfun 21678 | If the target space is Hau... |
cnextfvval 21679 | The value of the continuou... |
cnextf 21680 | Extension by continuity. ... |
cnextcn 21681 | Extension by continuity. ... |
cnextfres1 21682 | ` F ` and its extension by... |
cnextfres 21683 | ` F ` and its extension by... |
istmd 21688 | The predicate "is a topolo... |
tmdmnd 21689 | A topological monoid is a ... |
tmdtps 21690 | A topological monoid is a ... |
istgp 21691 | The predicate "is a topolo... |
tgpgrp 21692 | A topological group is a g... |
tgptmd 21693 | A topological group is a t... |
tgptps 21694 | A topological group is a t... |
tmdtopon 21695 | The topology of a topologi... |
tgptopon 21696 | The topology of a topologi... |
tmdcn 21697 | In a topological monoid, t... |
tgpcn 21698 | In a topological group, th... |
tgpinv 21699 | In a topological group, th... |
grpinvhmeo 21700 | The inverse function in a ... |
cnmpt1plusg 21701 | Continuity of the group su... |
cnmpt2plusg 21702 | Continuity of the group su... |
tmdcn2 21703 | Write out the definition o... |
tgpsubcn 21704 | In a topological group, th... |
istgp2 21705 | A group with a topology is... |
tmdmulg 21706 | In a topological monoid, t... |
tgpmulg 21707 | In a topological group, th... |
tgpmulg2 21708 | In a topological monoid, t... |
tmdgsum 21709 | In a topological monoid, t... |
tmdgsum2 21710 | For any neighborhood ` U `... |
oppgtmd 21711 | The opposite of a topologi... |
oppgtgp 21712 | The opposite of a topologi... |
distgp 21713 | Any group equipped with th... |
indistgp 21714 | Any group equipped with th... |
symgtgp 21715 | The symmetric group is a t... |
tmdlactcn 21716 | The left group action of e... |
tgplacthmeo 21717 | The left group action of e... |
submtmd 21718 | A submonoid of a topologic... |
subgtgp 21719 | A subgroup of a topologica... |
subgntr 21720 | A subgroup of a topologica... |
opnsubg 21721 | An open subgroup of a topo... |
clssubg 21722 | The closure of a subgroup ... |
clsnsg 21723 | The closure of a normal su... |
cldsubg 21724 | A subgroup of finite index... |
tgpconcompeqg 21725 | The connected component co... |
tgpconcomp 21726 | The identity component, th... |
tgpconcompss 21727 | The identity component is ... |
ghmcnp 21728 | A group homomorphism on to... |
snclseqg 21729 | The coset of the closure o... |
tgphaus 21730 | A topological group is Hau... |
tgpt1 21731 | Hausdorff and T1 are equiv... |
tgpt0 21732 | Hausdorff and T0 are equiv... |
qustgpopn 21733 | A quotient map in a topolo... |
qustgplem 21734 | Lemma for ~ qustgp . (Con... |
qustgp 21735 | The quotient of a topologi... |
qustgphaus 21736 | The quotient of a topologi... |
prdstmdd 21737 | The product of a family of... |
prdstgpd 21738 | The product of a family of... |
tsmsfbas 21741 | The collection of all sets... |
tsmslem1 21742 | The finite partial sums of... |
tsmsval2 21743 | Definition of the topologi... |
tsmsval 21744 | Definition of the topologi... |
tsmspropd 21745 | The group sum depends only... |
eltsms 21746 | The property of being a su... |
tsmsi 21747 | The property of being a su... |
tsmscl 21748 | A sum in a topological gro... |
haustsms 21749 | In a Hausdorff topological... |
haustsms2 21750 | In a Hausdorff topological... |
tsmscls 21751 | One half of ~ tgptsmscls ,... |
tsmsgsum 21752 | The convergent points of a... |
tsmsid 21753 | If a sum is finite, the us... |
haustsmsid 21754 | In a Hausdorff topological... |
tsms0 21755 | The sum of zero is zero. ... |
tsmssubm 21756 | Evaluate an infinite group... |
tsmsres 21757 | Extend an infinite group s... |
tsmsf1o 21758 | Re-index an infinite group... |
tsmsmhm 21759 | Apply a continuous group h... |
tsmsadd 21760 | The sum of two infinite gr... |
tsmsinv 21761 | Inverse of an infinite gro... |
tsmssub 21762 | The difference of two infi... |
tgptsmscls 21763 | A sum in a topological gro... |
tgptsmscld 21764 | The set of limit points to... |
tsmssplit 21765 | Split a topological group ... |
tsmsxplem1 21766 | Lemma for ~ tsmsxp . (Con... |
tsmsxplem2 21767 | Lemma for ~ tsmsxp . (Con... |
tsmsxp 21768 | Write a sum over a two-dim... |
istrg 21777 | Express the predicate " ` ... |
trgtmd 21778 | The multiplicative monoid ... |
istdrg 21779 | Express the predicate " ` ... |
tdrgunit 21780 | The unit group of a topolo... |
trgtgp 21781 | A topological ring is a to... |
trgtmd2 21782 | A topological ring is a to... |
trgtps 21783 | A topological ring is a to... |
trgring 21784 | A topological ring is a ri... |
trggrp 21785 | A topological ring is a gr... |
tdrgtrg 21786 | A topological division rin... |
tdrgdrng 21787 | A topological division rin... |
tdrgring 21788 | A topological division rin... |
tdrgtmd 21789 | A topological division rin... |
tdrgtps 21790 | A topological division rin... |
istdrg2 21791 | A topological-ring divisio... |
mulrcn 21792 | The functionalization of t... |
invrcn2 21793 | The multiplicative inverse... |
invrcn 21794 | The multiplicative inverse... |
cnmpt1mulr 21795 | Continuity of ring multipl... |
cnmpt2mulr 21796 | Continuity of ring multipl... |
dvrcn 21797 | The division function is c... |
istlm 21798 | The predicate " ` W ` is a... |
vscacn 21799 | The scalar multiplication ... |
tlmtmd 21800 | A topological module is a ... |
tlmtps 21801 | A topological module is a ... |
tlmlmod 21802 | A topological module is a ... |
tlmtrg 21803 | The scalar ring of a topol... |
tlmscatps 21804 | The scalar ring of a topol... |
istvc 21805 | A topological vector space... |
tvctdrg 21806 | The scalar field of a topo... |
cnmpt1vsca 21807 | Continuity of scalar multi... |
cnmpt2vsca 21808 | Continuity of scalar multi... |
tlmtgp 21809 | A topological vector space... |
tvctlm 21810 | A topological vector space... |
tvclmod 21811 | A topological vector space... |
tvclvec 21812 | A topological vector space... |
ustfn 21815 | The defined uniform struct... |
ustval 21816 | The class of all uniform s... |
isust 21817 | The predicate " ` U ` is a... |
ustssxp 21818 | Entourages are subsets of ... |
ustssel 21819 | A uniform structure is upw... |
ustbasel 21820 | The full set is always an ... |
ustincl 21821 | A uniform structure is clo... |
ustdiag 21822 | The diagonal set is includ... |
ustinvel 21823 | If ` V ` is an entourage, ... |
ustexhalf 21824 | For each entourage ` V ` t... |
ustrel 21825 | The elements of uniform st... |
ustfilxp 21826 | A uniform structure on a n... |
ustne0 21827 | A uniform structure cannot... |
ustssco 21828 | In an uniform structure, a... |
ustexsym 21829 | In an uniform structure, f... |
ustex2sym 21830 | In an uniform structure, f... |
ustex3sym 21831 | In an uniform structure, f... |
ustref 21832 | Any element of the base se... |
ust0 21833 | The unique uniform structu... |
ustn0 21834 | The empty set is not an un... |
ustund 21835 | If two intersecting sets `... |
ustelimasn 21836 | Any point ` A ` is near en... |
ustneism 21837 | For a point ` A ` in ` X `... |
elrnust 21838 | First direction for ~ ustb... |
ustbas2 21839 | Second direction for ~ ust... |
ustuni 21840 | The set union of a uniform... |
ustbas 21841 | Recover the base of an uni... |
ustimasn 21842 | Lemma for ~ ustuqtop . (C... |
trust 21843 | The trace of a uniform str... |
utopval 21846 | The topology induced by a ... |
elutop 21847 | Open sets in the topology ... |
utoptop 21848 | The topology induced by a ... |
utopbas 21849 | The base of the topology i... |
utoptopon 21850 | Topology induced by a unif... |
restutop 21851 | Restriction of a topology ... |
restutopopn 21852 | The restriction of the top... |
ustuqtoplem 21853 | Lemma for ~ ustuqtop . (C... |
ustuqtop0 21854 | Lemma for ~ ustuqtop . (C... |
ustuqtop1 21855 | Lemma for ~ ustuqtop , sim... |
ustuqtop2 21856 | Lemma for ~ ustuqtop . (C... |
ustuqtop3 21857 | Lemma for ~ ustuqtop , sim... |
ustuqtop4 21858 | Lemma for ~ ustuqtop . (C... |
ustuqtop5 21859 | Lemma for ~ ustuqtop . (C... |
ustuqtop 21860 | For a given uniform struct... |
utopsnneiplem 21861 | The neighborhoods of a poi... |
utopsnneip 21862 | The neighborhoods of a poi... |
utopsnnei 21863 | Images of singletons by en... |
utop2nei 21864 | For any symmetrical entour... |
utop3cls 21865 | Relation between a topolog... |
utopreg 21866 | All Hausdorff uniform spac... |
ussval 21873 | The uniform structure on u... |
ussid 21874 | In case the base of the ` ... |
isusp 21875 | The predicate ` W ` is a u... |
ressunif 21876 | ` UnifSet ` is unaffected ... |
ressuss 21877 | Value of the uniform struc... |
ressust 21878 | The uniform structure of a... |
ressusp 21879 | The restriction of a unifo... |
tusval 21880 | The value of the uniform s... |
tuslem 21881 | Lemma for ~ tusbas , ~ tus... |
tusbas 21882 | The base set of a construc... |
tusunif 21883 | The uniform structure of a... |
tususs 21884 | The uniform structure of a... |
tustopn 21885 | The topology induced by a ... |
tususp 21886 | A constructed uniform spac... |
tustps 21887 | A constructed uniform spac... |
uspreg 21888 | If a uniform space is Haus... |
ucnval 21891 | The set of all uniformly c... |
isucn 21892 | The predicate " ` F ` is a... |
isucn2 21893 | The predicate " ` F ` is a... |
ucnimalem 21894 | Reformulate the ` G ` func... |
ucnima 21895 | An equivalent statement of... |
ucnprima 21896 | The preimage by a uniforml... |
iducn 21897 | The identity is uniformly ... |
cstucnd 21898 | A constant function is uni... |
ucncn 21899 | Uniform continuity implies... |
iscfilu 21902 | The predicate " ` F ` is a... |
cfilufbas 21903 | A Cauchy filter base is a ... |
cfiluexsm 21904 | For a Cauchy filter base a... |
fmucndlem 21905 | Lemma for ~ fmucnd . (Con... |
fmucnd 21906 | The image of a Cauchy filt... |
cfilufg 21907 | The filter generated by a ... |
trcfilu 21908 | Condition for the trace of... |
cfiluweak 21909 | A Cauchy filter base is al... |
neipcfilu 21910 | In an uniform space, a nei... |
iscusp 21913 | The predicate " ` W ` is a... |
cuspusp 21914 | A complete uniform space i... |
cuspcvg 21915 | In a complete uniform spac... |
iscusp2 21916 | The predicate " ` W ` is a... |
cnextucn 21917 | Extension by continuity. ... |
ucnextcn 21918 | Extension by continuity. ... |
ispsmet 21919 | Express the predicate " ` ... |
psmetdmdm 21920 | Recover the base set from ... |
psmetf 21921 | The distance function of a... |
psmetcl 21922 | Closure of the distance fu... |
psmet0 21923 | The distance function of a... |
psmettri2 21924 | Triangle inequality for th... |
psmetsym 21925 | The distance function of a... |
psmettri 21926 | Triangle inequality for th... |
psmetge0 21927 | The distance function of a... |
psmetxrge0 21928 | The distance function of a... |
psmetres2 21929 | Restriction of a pseudomet... |
psmetlecl 21930 | Real closure of an extende... |
distspace 21931 | A structure ` G ` with a d... |
ismet 21938 | Express the predicate " ` ... |
isxmet 21939 | Express the predicate " ` ... |
ismeti 21940 | Properties that determine ... |
isxmetd 21941 | Properties that determine ... |
isxmet2d 21942 | It is safe to only require... |
metflem 21943 | Lemma for ~ metf and other... |
xmetf 21944 | Mapping of the distance fu... |
metf 21945 | Mapping of the distance fu... |
xmetcl 21946 | Closure of the distance fu... |
metcl 21947 | Closure of the distance fu... |
ismet2 21948 | An extended metric is a me... |
metxmet 21949 | A metric is an extended me... |
xmetdmdm 21950 | Recover the base set from ... |
metdmdm 21951 | Recover the base set from ... |
xmetunirn 21952 | Two ways to express an ext... |
xmeteq0 21953 | The value of an extended m... |
meteq0 21954 | The value of a metric is z... |
xmettri2 21955 | Triangle inequality for th... |
mettri2 21956 | Triangle inequality for th... |
xmet0 21957 | The distance function of a... |
met0 21958 | The distance function of a... |
xmetge0 21959 | The distance function of a... |
metge0 21960 | The distance function of a... |
xmetlecl 21961 | Real closure of an extende... |
xmetsym 21962 | The distance function of a... |
xmetpsmet 21963 | An extended metric is a ps... |
xmettpos 21964 | The distance function of a... |
metsym 21965 | The distance function of a... |
xmettri 21966 | Triangle inequality for th... |
mettri 21967 | Triangle inequality for th... |
xmettri3 21968 | Triangle inequality for th... |
mettri3 21969 | Triangle inequality for th... |
xmetrtri 21970 | One half of the reverse tr... |
xmetrtri2 21971 | The reverse triangle inequ... |
metrtri 21972 | Reverse triangle inequalit... |
xmetgt0 21973 | The distance function of a... |
metgt0 21974 | The distance function of a... |
metn0 21975 | A metric space is nonempty... |
xmetres2 21976 | Restriction of an extended... |
metreslem 21977 | Lemma for ~ metres . (Con... |
metres2 21978 | Lemma for ~ metres . (Con... |
xmetres 21979 | A restriction of an extend... |
metres 21980 | A restriction of a metric ... |
0met 21981 | The empty metric. (Contri... |
prdsdsf 21982 | The product metric is a fu... |
prdsxmetlem 21983 | The product metric is an e... |
prdsxmet 21984 | The product metric is an e... |
prdsmet 21985 | The product metric is a me... |
ressprdsds 21986 | Restriction of a product m... |
resspwsds 21987 | Restriction of a product m... |
imasdsf1olem 21988 | Lemma for ~ imasdsf1o . (... |
imasdsf1o 21989 | The distance function is t... |
imasf1oxmet 21990 | The image of an extended m... |
imasf1omet 21991 | The image of a metric is a... |
xpsdsfn 21992 | Closure of the metric in a... |
xpsdsfn2 21993 | Closure of the metric in a... |
xpsxmetlem 21994 | Lemma for ~ xpsxmet . (Co... |
xpsxmet 21995 | A product metric of extend... |
xpsdsval 21996 | Value of the metric in a b... |
xpsmet 21997 | The direct product of two ... |
blfvalps 21998 | The value of the ball func... |
blfval 21999 | The value of the ball func... |
blvalps 22000 | The ball around a point ` ... |
blval 22001 | The ball around a point ` ... |
elblps 22002 | Membership in a ball. (Co... |
elbl 22003 | Membership in a ball. (Co... |
elbl2ps 22004 | Membership in a ball. (Co... |
elbl2 22005 | Membership in a ball. (Co... |
elbl3ps 22006 | Membership in a ball, with... |
elbl3 22007 | Membership in a ball, with... |
blcomps 22008 | Commute the arguments to t... |
blcom 22009 | Commute the arguments to t... |
xblpnfps 22010 | The infinity ball in an ex... |
xblpnf 22011 | The infinity ball in an ex... |
blpnf 22012 | The infinity ball in a sta... |
bldisj 22013 | Two balls are disjoint if ... |
blgt0 22014 | A nonempty ball implies th... |
bl2in 22015 | Two balls are disjoint if ... |
xblss2ps 22016 | One ball is contained in a... |
xblss2 22017 | One ball is contained in a... |
blss2ps 22018 | One ball is contained in a... |
blss2 22019 | One ball is contained in a... |
blhalf 22020 | A ball of radius ` R / 2 `... |
blfps 22021 | Mapping of a ball. (Contr... |
blf 22022 | Mapping of a ball. (Contr... |
blrnps 22023 | Membership in the range of... |
blrn 22024 | Membership in the range of... |
xblcntrps 22025 | A ball contains its center... |
xblcntr 22026 | A ball contains its center... |
blcntrps 22027 | A ball contains its center... |
blcntr 22028 | A ball contains its center... |
xbln0 22029 | A ball is nonempty iff the... |
bln0 22030 | A ball is not empty. (Con... |
blelrnps 22031 | A ball belongs to the set ... |
blelrn 22032 | A ball belongs to the set ... |
blssm 22033 | A ball is a subset of the ... |
unirnblps 22034 | The union of the set of ba... |
unirnbl 22035 | The union of the set of ba... |
blin 22036 | The intersection of two ba... |
ssblps 22037 | The size of a ball increas... |
ssbl 22038 | The size of a ball increas... |
blssps 22039 | Any point ` P ` in a ball ... |
blss 22040 | Any point ` P ` in a ball ... |
blssexps 22041 | Two ways to express the ex... |
blssex 22042 | Two ways to express the ex... |
ssblex 22043 | A nested ball exists whose... |
blin2 22044 | Given any two balls and a ... |
blbas 22045 | The balls of a metric spac... |
blres 22046 | A ball in a restricted met... |
xmeterval 22047 | Value of the "finitely sep... |
xmeter 22048 | The "finitely separated" r... |
xmetec 22049 | The equivalence classes un... |
blssec 22050 | A ball centered at ` P ` i... |
blpnfctr 22051 | The infinity ball in an ex... |
xmetresbl 22052 | An extended metric restric... |
mopnval 22053 | An open set is a subset of... |
mopntopon 22054 | The set of open sets of a ... |
mopntop 22055 | The set of open sets of a ... |
mopnuni 22056 | The union of all open sets... |
elmopn 22057 | The defining property of a... |
mopnfss 22058 | The family of open sets of... |
mopnm 22059 | The base set of a metric s... |
elmopn2 22060 | A defining property of an ... |
mopnss 22061 | An open set of a metric sp... |
isxms 22062 | Express the predicate " ` ... |
isxms2 22063 | Express the predicate " ` ... |
isms 22064 | Express the predicate " ` ... |
isms2 22065 | Express the predicate " ` ... |
xmstopn 22066 | The topology component of ... |
mstopn 22067 | The topology component of ... |
xmstps 22068 | A metric space is a topolo... |
msxms 22069 | A metric space is a topolo... |
mstps 22070 | A metric space is a topolo... |
xmsxmet 22071 | The distance function, sui... |
msmet 22072 | The distance function, sui... |
msf 22073 | Mapping of the distance fu... |
xmsxmet2 22074 | The distance function, sui... |
msmet2 22075 | The distance function, sui... |
mscl 22076 | Closure of the distance fu... |
xmscl 22077 | Closure of the distance fu... |
xmsge0 22078 | The distance function in a... |
xmseq0 22079 | The distance function in a... |
xmssym 22080 | The distance function in a... |
xmstri2 22081 | Triangle inequality for th... |
mstri2 22082 | Triangle inequality for th... |
xmstri 22083 | Triangle inequality for th... |
mstri 22084 | Triangle inequality for th... |
xmstri3 22085 | Triangle inequality for th... |
mstri3 22086 | Triangle inequality for th... |
msrtri 22087 | Reverse triangle inequalit... |
xmspropd 22088 | Property deduction for an ... |
mspropd 22089 | Property deduction for a m... |
setsmsbas 22090 | The base set of a construc... |
setsmsds 22091 | The distance function of a... |
setsmstset 22092 | The topology of a construc... |
setsmstopn 22093 | The topology of a construc... |
setsxms 22094 | The constructed metric spa... |
setsms 22095 | The constructed metric spa... |
tmsval 22096 | For any metric there is an... |
tmslem 22097 | Lemma for ~ tmsbas , ~ tms... |
tmsbas 22098 | The base set of a construc... |
tmsds 22099 | The metric of a constructe... |
tmstopn 22100 | The topology of a construc... |
tmsxms 22101 | The constructed metric spa... |
tmsms 22102 | The constructed metric spa... |
imasf1obl 22103 | The image of a metric spac... |
imasf1oxms 22104 | The image of a metric spac... |
imasf1oms 22105 | The image of a metric spac... |
prdsbl 22106 | A ball in the product metr... |
mopni 22107 | An open set of a metric sp... |
mopni2 22108 | An open set of a metric sp... |
mopni3 22109 | An open set of a metric sp... |
blssopn 22110 | The balls of a metric spac... |
unimopn 22111 | The union of a collection ... |
mopnin 22112 | The intersection of two op... |
mopn0 22113 | The empty set is an open s... |
rnblopn 22114 | A ball of a metric space i... |
blopn 22115 | A ball of a metric space i... |
neibl 22116 | The neighborhoods around a... |
blnei 22117 | A ball around a point is a... |
lpbl 22118 | Every ball around a limit ... |
blsscls2 22119 | A smaller closed ball is c... |
blcld 22120 | A "closed ball" in a metri... |
blcls 22121 | The closure of an open bal... |
blsscls 22122 | If two concentric balls ha... |
metss 22123 | Two ways of saying that me... |
metequiv 22124 | Two ways of saying that tw... |
metequiv2 22125 | If there is a sequence of ... |
metss2lem 22126 | Lemma for ~ metss2 . (Con... |
metss2 22127 | If the metric ` D ` is "st... |
comet 22128 | The composition of an exte... |
stdbdmetval 22129 | Value of the standard boun... |
stdbdxmet 22130 | The standard bounded metri... |
stdbdmet 22131 | The standard bounded metri... |
stdbdbl 22132 | The standard bounded metri... |
stdbdmopn 22133 | The standard bounded metri... |
mopnex 22134 | The topology generated by ... |
methaus 22135 | The topology generated by ... |
met1stc 22136 | The topology generated by ... |
met2ndci 22137 | A separable metric space (... |
met2ndc 22138 | A metric space is second-c... |
metrest 22139 | Two alternate formulations... |
ressxms 22140 | The restriction of a metri... |
ressms 22141 | The restriction of a metri... |
prdsmslem1 22142 | Lemma for ~ prdsms . The ... |
prdsxmslem1 22143 | Lemma for ~ prdsms . The ... |
prdsxmslem2 22144 | Lemma for ~ prdsxms . The... |
prdsxms 22145 | The indexed product struct... |
prdsms 22146 | The indexed product struct... |
pwsxms 22147 | The product of a finite fa... |
pwsms 22148 | The product of a finite fa... |
xpsxms 22149 | A binary product of metric... |
xpsms 22150 | A binary product of metric... |
tmsxps 22151 | Express the product of two... |
tmsxpsmopn 22152 | Express the product of two... |
tmsxpsval 22153 | Value of the product of tw... |
tmsxpsval2 22154 | Value of the product of tw... |
metcnp3 22155 | Two ways to express that `... |
metcnp 22156 | Two ways to say a mapping ... |
metcnp2 22157 | Two ways to say a mapping ... |
metcn 22158 | Two ways to say a mapping ... |
metcnpi 22159 | Epsilon-delta property of ... |
metcnpi2 22160 | Epsilon-delta property of ... |
metcnpi3 22161 | Epsilon-delta property of ... |
txmetcnp 22162 | Continuity of a binary ope... |
txmetcn 22163 | Continuity of a binary ope... |
metuval 22164 | Value of the uniform struc... |
metustel 22165 | Define a filter base ` F `... |
metustss 22166 | Range of the elements of t... |
metustrel 22167 | Elements of the filter bas... |
metustto 22168 | Any two elements of the fi... |
metustid 22169 | The identity diagonal is i... |
metustsym 22170 | Elements of the filter bas... |
metustexhalf 22171 | For any element ` A ` of t... |
metustfbas 22172 | The filter base generated ... |
metust 22173 | The uniform structure gene... |
cfilucfil 22174 | Given a metric ` D ` and a... |
metuust 22175 | The uniform structure gene... |
cfilucfil2 22176 | Given a metric ` D ` and a... |
blval2 22177 | The ball around a point ` ... |
elbl4 22178 | Membership in a ball, alte... |
metuel 22179 | Elementhood in the uniform... |
metuel2 22180 | Elementhood in the uniform... |
metustbl 22181 | The "section" image of an ... |
psmetutop 22182 | The topology induced by a ... |
xmetutop 22183 | The topology induced by a ... |
xmsusp 22184 | If the uniform set of a me... |
restmetu 22185 | The uniform structure gene... |
metucn 22186 | Uniform continuity in metr... |
dscmet 22187 | The discrete metric on any... |
dscopn 22188 | The discrete metric genera... |
nrmmetd 22189 | Show that a group norm gen... |
abvmet 22190 | An absolute value ` F ` ge... |
nmfval 22203 | The value of the norm func... |
nmval 22204 | The value of the norm func... |
nmfval2 22205 | The value of the norm func... |
nmval2 22206 | The value of the norm func... |
nmf2 22207 | The norm is a function fro... |
nmpropd 22208 | Weak property deduction fo... |
nmpropd2 22209 | Strong property deduction ... |
isngp 22210 | The property of being a no... |
isngp2 22211 | The property of being a no... |
isngp3 22212 | The property of being a no... |
ngpgrp 22213 | A normed group is a group.... |
ngpms 22214 | A normed group is a metric... |
ngpxms 22215 | A normed group is a metric... |
ngptps 22216 | A normed group is a topolo... |
ngpmet 22217 | The (induced) metric of a ... |
ngpds 22218 | Value of the distance func... |
ngpdsr 22219 | Value of the distance func... |
ngpds2 22220 | Write the distance between... |
ngpds2r 22221 | Write the distance between... |
ngpds3 22222 | Write the distance between... |
ngpds3r 22223 | Write the distance between... |
ngprcan 22224 | Cancel right addition insi... |
ngplcan 22225 | Cancel left addition insid... |
isngp4 22226 | Express the property of be... |
ngpinvds 22227 | Two elements are the same ... |
ngpsubcan 22228 | Cancel right subtraction i... |
nmf 22229 | The norm on a normed group... |
nmcl 22230 | The norm of a normed group... |
nmge0 22231 | The norm of a normed group... |
nmeq0 22232 | The identity is the only e... |
nmne0 22233 | The norm of a nonzero elem... |
nmrpcl 22234 | The norm of a nonzero elem... |
nminv 22235 | The norm of a negated elem... |
nmmtri 22236 | The triangle inequality fo... |
nmsub 22237 | The norm of the difference... |
nmrtri 22238 | Reverse triangle inequalit... |
nm2dif 22239 | Inequality for the differe... |
nmtri 22240 | The triangle inequality fo... |
nmtri2 22241 | Triangle inequality for th... |
ngpi 22242 | The properties of a normed... |
nm0 22243 | Norm of the identity eleme... |
nmgt0 22244 | The norm of a nonzero elem... |
sgrim 22245 | The induced metric on a su... |
sgrimval 22246 | The induced metric on a su... |
subgnm 22247 | The norm in a subgroup. (... |
subgnm2 22248 | A substructure assigns the... |
subgngp 22249 | A normed group restricted ... |
ngptgp 22250 | A normed abelian group is ... |
ngppropd 22251 | Property deduction for a n... |
reldmtng 22252 | The function ` toNrmGrp ` ... |
tngval 22253 | Value of the function whic... |
tnglem 22254 | Lemma for ~ tngbas and sim... |
tngbas 22255 | The base set of a structur... |
tngplusg 22256 | The group addition of a st... |
tng0 22257 | The group identity of a st... |
tngmulr 22258 | The ring multiplication of... |
tngsca 22259 | The scalar ring of a struc... |
tngvsca 22260 | The scalar multiplication ... |
tngip 22261 | The inner product operatio... |
tngds 22262 | The metric function of a s... |
tngtset 22263 | The topology generated by ... |
tngtopn 22264 | The topology generated by ... |
tngnm 22265 | The topology generated by ... |
tngngp2 22266 | A norm turns a group into ... |
tngngpd 22267 | Derive the axioms for a no... |
tngngp 22268 | Derive the axioms for a no... |
tnggrpr 22269 | If a structure equipped wi... |
tngngp3 22270 | Alternate definition of a ... |
nrmtngdist 22271 | The augmentation of a norm... |
nrmtngnrm 22272 | The augmentation of a norm... |
tngngpim 22273 | The induced metric of a no... |
isnrg 22274 | A normed ring is a ring wi... |
nrgabv 22275 | The norm of a normed ring ... |
nrgngp 22276 | A normed ring is a normed ... |
nrgring 22277 | A normed ring is a ring. ... |
nmmul 22278 | The norm of a product in a... |
nrgdsdi 22279 | Distribute a distance calc... |
nrgdsdir 22280 | Distribute a distance calc... |
nm1 22281 | The norm of one in a nonze... |
unitnmn0 22282 | The norm of a unit is nonz... |
nminvr 22283 | The norm of an inverse in ... |
nmdvr 22284 | The norm of a division in ... |
nrgdomn 22285 | A nonzero normed ring is a... |
nrgtgp 22286 | A normed ring is a topolog... |
subrgnrg 22287 | A normed ring restricted t... |
tngnrg 22288 | Given any absolute value o... |
isnlm 22289 | A normed (left) module is ... |
nmvs 22290 | Defining property of a nor... |
nlmngp 22291 | A normed module is a norme... |
nlmlmod 22292 | A normed module is a left ... |
nlmnrg 22293 | The scalar component of a ... |
nlmngp2 22294 | The scalar component of a ... |
nlmdsdi 22295 | Distribute a distance calc... |
nlmdsdir 22296 | Distribute a distance calc... |
nlmmul0or 22297 | If a scalar product is zer... |
sranlm 22298 | The subring algebra over a... |
nlmvscnlem2 22299 | Lemma for ~ nlmvscn . Com... |
nlmvscnlem1 22300 | Lemma for ~ nlmvscn . (Co... |
nlmvscn 22301 | The scalar multiplication ... |
rlmnlm 22302 | The ring module over a nor... |
rlmnm 22303 | The norm function in the r... |
nrgtrg 22304 | A normed ring is a topolog... |
nrginvrcnlem 22305 | Lemma for ~ nrginvrcn . C... |
nrginvrcn 22306 | The ring inverse function ... |
nrgtdrg 22307 | A normed division ring is ... |
nlmtlm 22308 | A normed module is a topol... |
isnvc 22309 | A normed vector space is j... |
nvcnlm 22310 | A normed vector space is a... |
nvclvec 22311 | A normed vector space is a... |
nvclmod 22312 | A normed vector space is a... |
isnvc2 22313 | A normed vector space is j... |
nvctvc 22314 | A normed vector space is a... |
lssnlm 22315 | A subspace of a normed mod... |
lssnvc 22316 | A subspace of a normed vec... |
rlmnvc 22317 | The ring module over a nor... |
ngpocelbl 22318 | Membership of an off-cente... |
nmoffn 22325 | The function producing ope... |
reldmnghm 22326 | Lemma for normed group hom... |
reldmnmhm 22327 | Lemma for module homomorph... |
nmofval 22328 | Value of the operator norm... |
nmoval 22329 | Value of the operator norm... |
nmogelb 22330 | Property of the operator n... |
nmolb 22331 | Any upper bound on the val... |
nmolb2d 22332 | Any upper bound on the val... |
nmof 22333 | The operator norm is a fun... |
nmocl 22334 | The operator norm of an op... |
nmoge0 22335 | The operator norm of an op... |
nghmfval 22336 | A normed group homomorphis... |
isnghm 22337 | A normed group homomorphis... |
isnghm2 22338 | A normed group homomorphis... |
isnghm3 22339 | A normed group homomorphis... |
bddnghm 22340 | A bounded group homomorphi... |
nghmcl 22341 | A normed group homomorphis... |
nmoi 22342 | The operator norm achieves... |
nmoix 22343 | The operator norm is a bou... |
nmoi2 22344 | The operator norm is a bou... |
nmoleub 22345 | The operator norm, defined... |
nghmrcl1 22346 | Reverse closure for a norm... |
nghmrcl2 22347 | Reverse closure for a norm... |
nghmghm 22348 | A normed group homomorphis... |
nmo0 22349 | The operator norm of the z... |
nmoeq0 22350 | The operator norm is zero ... |
nmoco 22351 | An upper bound on the oper... |
nghmco 22352 | The composition of normed ... |
nmotri 22353 | Triangle inequality for th... |
nghmplusg 22354 | The sum of two bounded lin... |
0nghm 22355 | The zero operator is a nor... |
nmoid 22356 | The operator norm of the i... |
idnghm 22357 | The identity operator is a... |
nmods 22358 | Upper bound for the distan... |
nghmcn 22359 | A normed group homomorphis... |
isnmhm 22360 | A normed module homomorphi... |
nmhmrcl1 22361 | Reverse closure for a norm... |
nmhmrcl2 22362 | Reverse closure for a norm... |
nmhmlmhm 22363 | A normed module homomorphi... |
nmhmnghm 22364 | A normed module homomorphi... |
nmhmghm 22365 | A normed module homomorphi... |
isnmhm2 22366 | A normed module homomorphi... |
nmhmcl 22367 | A normed module homomorphi... |
idnmhm 22368 | The identity operator is a... |
0nmhm 22369 | The zero operator is a bou... |
nmhmco 22370 | The composition of bounded... |
nmhmplusg 22371 | The sum of two bounded lin... |
qtopbaslem 22372 | The set of open intervals ... |
qtopbas 22373 | The set of open intervals ... |
retopbas 22374 | A basis for the standard t... |
retop 22375 | The standard topology on t... |
uniretop 22376 | The underlying set of the ... |
retopon 22377 | The standard topology on t... |
retps 22378 | The standard topological s... |
iooretop 22379 | Open intervals are open se... |
icccld 22380 | Closed intervals are close... |
icopnfcld 22381 | Right-unbounded closed int... |
iocmnfcld 22382 | Left-unbounded closed inte... |
qdensere 22383 | ` QQ ` is dense in the sta... |
cnmetdval 22384 | Value of the distance func... |
cnmet 22385 | The absolute value metric ... |
cnxmet 22386 | The absolute value metric ... |
cnbl0 22387 | Two ways to write the open... |
cnblcld 22388 | Two ways to write the clos... |
cnfldms 22389 | The complex number field i... |
cnfldxms 22390 | The complex number field i... |
cnfldtps 22391 | The complex number field i... |
cnfldnm 22392 | The norm of the field of c... |
cnngp 22393 | The complex numbers form a... |
cnnrg 22394 | The complex numbers form a... |
cnfldtopn 22395 | The topology of the comple... |
cnfldtopon 22396 | The topology of the comple... |
cnfldtop 22397 | The topology of the comple... |
cnfldhaus 22398 | The topology of the comple... |
zringnrg 22399 | The ring of integers is a ... |
remetdval 22400 | Value of the distance func... |
remet 22401 | The absolute value metric ... |
rexmet 22402 | The absolute value metric ... |
bl2ioo 22403 | A ball in terms of an open... |
ioo2bl 22404 | An open interval of reals ... |
ioo2blex 22405 | An open interval of reals ... |
blssioo 22406 | The balls of the standard ... |
tgioo 22407 | The topology generated by ... |
qdensere2 22408 | ` QQ ` is dense in ` RR ` ... |
blcvx 22409 | An open ball in the comple... |
rehaus 22410 | The standard topology on t... |
tgqioo 22411 | The topology generated by ... |
re2ndc 22412 | The standard topology on t... |
resubmet 22413 | The subspace topology indu... |
tgioo2 22414 | The standard topology on t... |
rerest 22415 | The subspace topology indu... |
tgioo3 22416 | The standard topology on t... |
xrtgioo 22417 | The topology on the extend... |
xrrest 22418 | The subspace topology indu... |
xrrest2 22419 | The subspace topology indu... |
xrsxmet 22420 | The metric on the extended... |
xrsdsre 22421 | The metric on the extended... |
xrsblre 22422 | Any ball of the metric of ... |
xrsmopn 22423 | The metric on the extended... |
zcld 22424 | The integers are a closed ... |
recld2 22425 | The real numbers are a clo... |
zcld2 22426 | The integers are a closed ... |
zdis 22427 | The integers are a discret... |
sszcld 22428 | Every subset of the intege... |
reperflem 22429 | A subset of the real numbe... |
reperf 22430 | The real numbers are a per... |
cnperf 22431 | The complex numbers are a ... |
iccntr 22432 | The interior of a closed i... |
icccmplem1 22433 | Lemma for ~ icccmp . (Con... |
icccmplem2 22434 | Lemma for ~ icccmp . (Con... |
icccmplem3 22435 | Lemma for ~ icccmp . (Con... |
icccmp 22436 | A closed interval in ` RR ... |
reconnlem1 22437 | Lemma for ~ reconn . Conn... |
reconnlem2 22438 | Lemma for ~ reconn . (Con... |
reconn 22439 | A subset of the reals is c... |
retopcon 22440 | Corollary of ~ reconn . T... |
iccconn 22441 | A closed interval is conne... |
opnreen 22442 | Every nonempty open set is... |
rectbntr0 22443 | A countable subset of the ... |
xrge0gsumle 22444 | A finite sum in the nonneg... |
xrge0tsms 22445 | Any finite or infinite sum... |
xrge0tsms2 22446 | Any finite or infinite sum... |
metdcnlem 22447 | The metric function of a m... |
xmetdcn2 22448 | The metric function of an ... |
xmetdcn 22449 | The metric function of an ... |
metdcn2 22450 | The metric function of a m... |
metdcn 22451 | The metric function of a m... |
msdcn 22452 | The metric function of a m... |
cnmpt1ds 22453 | Continuity of the metric f... |
cnmpt2ds 22454 | Continuity of the metric f... |
nmcn 22455 | The norm of a normed group... |
ngnmcncn 22456 | The norm of a normed group... |
abscn 22457 | The absolute value functio... |
metdsval 22458 | Value of the "distance to ... |
metdsf 22459 | The distance from a point ... |
metdsge 22460 | The distance from the poin... |
metds0 22461 | If a point is in a set, it... |
metdstri 22462 | A generalization of the tr... |
metdsle 22463 | The distance from a point ... |
metdsre 22464 | The distance from a point ... |
metdseq0 22465 | The distance from a point ... |
metdscnlem 22466 | Lemma for ~ metdscn . (Co... |
metdscn 22467 | The function ` F ` which g... |
metdscn2 22468 | The function ` F ` which g... |
metnrmlem1a 22469 | Lemma for ~ metnrm . (Con... |
metnrmlem1 22470 | Lemma for ~ metnrm . (Con... |
metnrmlem2 22471 | Lemma for ~ metnrm . (Con... |
metnrmlem3 22472 | Lemma for ~ metnrm . (Con... |
metnrm 22473 | A metric space is normal. ... |
metreg 22474 | A metric space is regular.... |
addcnlem 22475 | Lemma for ~ addcn , ~ subc... |
addcn 22476 | Complex number addition is... |
subcn 22477 | Complex number subtraction... |
mulcn 22478 | Complex number multiplicat... |
divcn 22479 | Complex number division is... |
cnfldtgp 22480 | The complex numbers form a... |
fsumcn 22481 | A finite sum of functions ... |
fsum2cn 22482 | Version of ~ fsumcn for tw... |
expcn 22483 | The power function on comp... |
divccn 22484 | Division by a nonzero cons... |
sqcn 22485 | The square function on com... |
iitopon 22490 | The unit interval is a top... |
iitop 22491 | The unit interval is a top... |
iiuni 22492 | The base set of the unit i... |
dfii2 22493 | Alternate definition of th... |
dfii3 22494 | Alternate definition of th... |
dfii4 22495 | Alternate definition of th... |
dfii5 22496 | The unit interval expresse... |
iicmp 22497 | The unit interval is compa... |
iicon 22498 | The unit interval is conne... |
cncfval 22499 | The value of the continuou... |
elcncf 22500 | Membership in the set of c... |
elcncf2 22501 | Version of ~ elcncf with a... |
cncfrss 22502 | Reverse closure of the con... |
cncfrss2 22503 | Reverse closure of the con... |
cncff 22504 | A continuous complex funct... |
cncfi 22505 | Defining property of a con... |
elcncf1di 22506 | Membership in the set of c... |
elcncf1ii 22507 | Membership in the set of c... |
rescncf 22508 | A continuous complex funct... |
cncffvrn 22509 | Change the codomain of a c... |
cncfss 22510 | The set of continuous func... |
climcncf 22511 | Image of a limit under a c... |
abscncf 22512 | Absolute value is continuo... |
recncf 22513 | Real part is continuous. ... |
imcncf 22514 | Imaginary part is continuo... |
cjcncf 22515 | Complex conjugate is conti... |
mulc1cncf 22516 | Multiplication by a consta... |
divccncf 22517 | Division by a constant is ... |
cncfco 22518 | The composition of two con... |
cncfmet 22519 | Relate complex function co... |
cncfcn 22520 | Relate complex function co... |
cncfcn1 22521 | Relate complex function co... |
cncfmptc 22522 | A constant function is a c... |
cncfmptid 22523 | The identity function is a... |
cncfmpt1f 22524 | Composition of continuous ... |
cncfmpt2f 22525 | Composition of continuous ... |
cncfmpt2ss 22526 | Composition of continuous ... |
addccncf 22527 | Adding a constant is a con... |
cdivcncf 22528 | Division with a constant n... |
negcncf 22529 | The negative function is c... |
negfcncf 22530 | The negative of a continuo... |
abscncfALT 22531 | Absolute value is continuo... |
cncfcnvcn 22532 | Rewrite ~ cmphaushmeo for ... |
expcncf 22533 | The power function on comp... |
cnmptre 22534 | Lemma for ~ iirevcn and re... |
cnmpt2pc 22535 | Piecewise definition of a ... |
iirev 22536 | Reverse the unit interval.... |
iirevcn 22537 | The reversion function is ... |
iihalf1 22538 | Map the first half of ` II... |
iihalf1cn 22539 | The first half function is... |
iihalf2 22540 | Map the second half of ` I... |
iihalf2cn 22541 | The second half function i... |
elii1 22542 | Divide the unit interval i... |
elii2 22543 | Divide the unit interval i... |
iimulcl 22544 | The unit interval is close... |
iimulcn 22545 | Multiplication is a contin... |
icoopnst 22546 | A half-open interval start... |
iocopnst 22547 | A half-open interval endin... |
icchmeo 22548 | The natural bijection from... |
icopnfcnv 22549 | Define a bijection from ` ... |
icopnfhmeo 22550 | The defined bijection from... |
iccpnfcnv 22551 | Define a bijection from ` ... |
iccpnfhmeo 22552 | The defined bijection from... |
xrhmeo 22553 | The bijection from ` [ -u ... |
xrhmph 22554 | The extended reals are hom... |
xrcmp 22555 | The topology of the extend... |
xrcon 22556 | The topology of the extend... |
icccvx 22557 | A linear combination of tw... |
oprpiece1res1 22558 | Restriction to the first p... |
oprpiece1res2 22559 | Restriction to the second ... |
cnrehmeo 22560 | The canonical bijection fr... |
cnheiborlem 22561 | Lemma for ~ cnheibor . (C... |
cnheibor 22562 | Heine-Borel theorem for co... |
cnllycmp 22563 | The topology on the comple... |
rellycmp 22564 | The topology on the reals ... |
bndth 22565 | The Boundedness Theorem. ... |
evth 22566 | The Extreme Value Theorem.... |
evth2 22567 | The Extreme Value Theorem,... |
lebnumlem1 22568 | Lemma for ~ lebnum . The ... |
lebnumlem2 22569 | Lemma for ~ lebnum . As a... |
lebnumlem3 22570 | Lemma for ~ lebnum . By t... |
lebnum 22571 | The Lebesgue number lemma,... |
xlebnum 22572 | Generalize ~ lebnum to ext... |
lebnumii 22573 | Specialize the Lebesgue nu... |
ishtpy 22579 | Membership in the class of... |
htpycn 22580 | A homotopy is a continuous... |
htpyi 22581 | A homotopy evaluated at it... |
ishtpyd 22582 | Deduction for membership i... |
htpycom 22583 | Given a homotopy from ` F ... |
htpyid 22584 | A homotopy from a function... |
htpyco1 22585 | Compose a homotopy with a ... |
htpyco2 22586 | Compose a homotopy with a ... |
htpycc 22587 | Concatenate two homotopies... |
isphtpy 22588 | Membership in the class of... |
phtpyhtpy 22589 | A path homotopy is a homot... |
phtpycn 22590 | A path homotopy is a conti... |
phtpyi 22591 | Membership in the class of... |
phtpy01 22592 | Two path-homotopic paths h... |
isphtpyd 22593 | Deduction for membership i... |
isphtpy2d 22594 | Deduction for membership i... |
phtpycom 22595 | Given a homotopy from ` F ... |
phtpyid 22596 | A homotopy from a path to ... |
phtpyco2 22597 | Compose a path homotopy wi... |
phtpycc 22598 | Concatenate two path homot... |
phtpcrel 22600 | The path homotopy relation... |
isphtpc 22601 | The relation "is path homo... |
phtpcer 22602 | Path homotopy is an equiva... |
phtpcerOLD 22603 | Obsolete proof of ~ phtpce... |
phtpc01 22604 | Path homotopic paths have ... |
reparphti 22605 | Lemma for ~ reparpht . (C... |
reparpht 22606 | Reparametrization lemma. ... |
phtpcco2 22607 | Compose a path homotopy wi... |
pcofval 22618 | The value of the path conc... |
pcoval 22619 | The concatenation of two p... |
pcovalg 22620 | Evaluate the concatenation... |
pcoval1 22621 | Evaluate the concatenation... |
pco0 22622 | The starting point of a pa... |
pco1 22623 | The ending point of a path... |
pcoval2 22624 | Evaluate the concatenation... |
pcocn 22625 | The concatenation of two p... |
copco 22626 | The composition of a conca... |
pcohtpylem 22627 | Lemma for ~ pcohtpy . (Co... |
pcohtpy 22628 | Homotopy invariance of pat... |
pcoptcl 22629 | A constant function is a p... |
pcopt 22630 | Concatenation with a point... |
pcopt2 22631 | Concatenation with a point... |
pcoass 22632 | Order of concatenation doe... |
pcorevcl 22633 | Closure for a reversed pat... |
pcorevlem 22634 | Lemma for ~ pcorev . Prov... |
pcorev 22635 | Concatenation with the rev... |
pcorev2 22636 | Concatenation with the rev... |
pcophtb 22637 | The path homotopy equivale... |
om1val 22638 | The definition of the loop... |
om1bas 22639 | The base set of the loop s... |
om1elbas 22640 | Elementhood in the base se... |
om1addcl 22641 | Closure of the group opera... |
om1plusg 22642 | The group operation (which... |
om1tset 22643 | The topology of the loop s... |
om1opn 22644 | The topology of the loop s... |
pi1val 22645 | The definition of the fund... |
pi1bas 22646 | The base set of the fundam... |
pi1blem 22647 | Lemma for ~ pi1buni . (Co... |
pi1buni 22648 | Another way to write the l... |
pi1bas2 22649 | The base set of the fundam... |
pi1eluni 22650 | Elementhood in the base se... |
pi1bas3 22651 | The base set of the fundam... |
pi1cpbl 22652 | The group operation, loop ... |
elpi1 22653 | The elements of the fundam... |
elpi1i 22654 | The elements of the fundam... |
pi1addf 22655 | The group operation of ` p... |
pi1addval 22656 | The concatenation of two p... |
pi1grplem 22657 | Lemma for ~ pi1grp . (Con... |
pi1grp 22658 | The fundamental group is a... |
pi1id 22659 | The identity element of th... |
pi1inv 22660 | An inverse in the fundamen... |
pi1xfrf 22661 | Functionality of the loop ... |
pi1xfrval 22662 | The value of the loop tran... |
pi1xfr 22663 | Given a path ` F ` and its... |
pi1xfrcnvlem 22664 | Given a path ` F ` between... |
pi1xfrcnv 22665 | Given a path ` F ` between... |
pi1xfrgim 22666 | The mapping ` G ` between ... |
pi1cof 22667 | Functionality of the loop ... |
pi1coval 22668 | The value of the loop tran... |
pi1coghm 22669 | The mapping ` G ` between ... |
isclm 22672 | A complex module is a left... |
clmsca 22673 | A complex module is a left... |
clmsubrg 22674 | A complex module is a left... |
clmlmod 22675 | A complex module is a left... |
clmgrp 22676 | A complex module is an add... |
clmabl 22677 | A complex module is an abe... |
clmring 22678 | The scalar ring of a compl... |
clmfgrp 22679 | The scalar ring of a compl... |
clm0 22680 | The zero of the scalar rin... |
clm1 22681 | The identity of the scalar... |
clmadd 22682 | The addition of the scalar... |
clmmul 22683 | The multiplication of the ... |
clmcj 22684 | The conjugation of the sca... |
isclmi 22685 | Reverse direction of ~ isc... |
clmzss 22686 | The scalar ring of a compl... |
clmsscn 22687 | The scalar ring of a compl... |
clmsub 22688 | Subtraction in the scalar ... |
clmneg 22689 | Negation in the scalar rin... |
clmneg1 22690 | Minus one is in the scalar... |
clmabs 22691 | Norm in the scalar ring of... |
clmacl 22692 | Closure of ring addition f... |
clmmcl 22693 | Closure of ring multiplica... |
clmsubcl 22694 | Closure of ring subtractio... |
lmhmclm 22695 | The domain of a linear ope... |
clmvscl 22696 | Closure of scalar product ... |
clmvsass 22697 | Associative law for scalar... |
clmvscom 22698 | Commutative law for the sc... |
clmvsdir 22699 | Distributive law for scala... |
clmvsdi 22700 | Distributive law for scala... |
clmvs1 22701 | Scalar product with ring u... |
clmvs2 22702 | A vector plus itself is tw... |
clm0vs 22703 | Zero times a vector is the... |
clmopfne 22704 | The (functionalized) opera... |
isclmp 22705 | The predicate "is a comple... |
isclmi0 22706 | Properties that determine ... |
clmvneg1 22707 | Minus 1 times a vector is ... |
clmvsneg 22708 | Multiplication of a vector... |
clmmulg 22709 | The group multiple functio... |
clmsubdir 22710 | Scalar multiplication dist... |
clmpm1dir 22711 | Subtractive distributive l... |
clmnegneg 22712 | Double negative of a vecto... |
clmnegsubdi2 22713 | Distribution of negative o... |
clmsub4 22714 | Rearrangement of 4 terms i... |
clmvsrinv 22715 | A vector minus itself. (C... |
clmvslinv 22716 | Minus a vector plus itself... |
clmvsubval 22717 | Value of vector subtractio... |
clmvsubval2 22718 | Value of vector subtractio... |
clmvz 22719 | Two ways to express the ne... |
zlmclm 22720 | The ` ZZ ` -module operati... |
clmzlmvsca 22721 | The scalar product of a co... |
nmoleub2lem 22722 | Lemma for ~ nmoleub2a and ... |
nmoleub2lem3 22723 | Lemma for ~ nmoleub2a and ... |
nmoleub2lem2 22724 | Lemma for ~ nmoleub2a and ... |
nmoleub2a 22725 | The operator norm is the s... |
nmoleub2b 22726 | The operator norm is the s... |
nmoleub3 22727 | The operator norm is the s... |
nmhmcn 22728 | A linear operator over a n... |
cmodscexp 22729 | The powers of ` _i ` belon... |
cmodscmulexp 22730 | The scalar product of a ve... |
cvslvec 22733 | A complex vector space is ... |
cvsclm 22734 | A complex vector space is ... |
iscvs 22735 | A complex vector space is ... |
iscvsp 22736 | The predicate "is a comple... |
iscvsi 22737 | Properties that determine ... |
cvsi 22738 | The properties of a comple... |
cvsunit 22739 | Unit group of the scalar r... |
cvsdiv 22740 | Division of the scalar rin... |
cvsdivcl 22741 | The scalar field of a comp... |
cvsmuleqdivd 22742 | An equality involving rati... |
cvsdiveqd 22743 | An equality involving rati... |
cnlmodlem1 22744 | Lemma 1 for ~ cnlmod . (C... |
cnlmodlem2 22745 | Lemma 2 for ~ cnlmod . (C... |
cnlmodlem3 22746 | Lemma 3 for ~ cnlmod . (C... |
cnlmod4 22747 | Lemma 4 for ~ cnlmod . (C... |
cnlmod 22748 | The set of complex numbers... |
cnstrcvs 22749 | The set of complex numbers... |
cnrbas 22750 | The set of complex numbers... |
cnrlmod 22751 | The set of complex numbers... |
cnrlvec 22752 | The set of complex numbers... |
cncvs 22753 | The set of complex numbers... |
recvs 22754 | The set of real numbers (a... |
qcvs 22755 | The set of rational number... |
zclmncvs 22756 | The set of integers (as a ... |
isncvsngp 22757 | The predicate "is a normed... |
isncvsngpd 22758 | Properties that determine ... |
ncvsi 22759 | The properties of a normed... |
ncvsprp 22760 | Proportionality property o... |
ncvsge0 22761 | The norm of a scalar produ... |
ncvsm1 22762 | The norm of the negative o... |
ncvsdif 22763 | The norm of the difference... |
ncvspi 22764 | The norm of a vector plus ... |
ncvs1 22765 | From any nonzero vector, c... |
cnrnvc 22766 | The set of complex numbers... |
cnncvs 22767 | The set of complex numbers... |
cnnm 22768 | The norm operation of the ... |
ncvspds 22769 | Value of the distance func... |
cnindmet 22770 | The metric induced on the ... |
cnncvsaddassdemo 22771 | Derive the associative law... |
cnncvsmulassdemo 22772 | Derive the associative law... |
cnncvsabsnegdemo 22773 | Derive the absolute value ... |
iscph 22778 | A complex pre-Hilbert spac... |
cphphl 22779 | A complex pre-Hilbert spac... |
cphnlm 22780 | A complex pre-Hilbert spac... |
cphngp 22781 | A complex pre-Hilbert spac... |
cphlmod 22782 | A complex pre-Hilbert spac... |
cphlvec 22783 | A complex pre-Hilbert spac... |
cphnvc 22784 | A complex pre-Hilbert spac... |
cphsubrglem 22785 | Lemma for ~ cphsubrg . (C... |
cphreccllem 22786 | Lemma for ~ cphreccl . (C... |
cphsca 22787 | A complex pre-Hilbert spac... |
cphsubrg 22788 | The scalar field of a comp... |
cphreccl 22789 | The scalar field of a comp... |
cphdivcl 22790 | The scalar field of a comp... |
cphcjcl 22791 | The scalar field of a comp... |
cphsqrtcl 22792 | The scalar field of a comp... |
cphabscl 22793 | The scalar field of a comp... |
cphsqrtcl2 22794 | The scalar field of a comp... |
cphsqrtcl3 22795 | If the scalar field contai... |
cphqss 22796 | The scalar field of a comp... |
cphclm 22797 | A complex pre-Hilbert spac... |
cphnmvs 22798 | Norm of a scalar product. ... |
cphipcl 22799 | An inner product is a memb... |
cphnmfval 22800 | The value of the norm in a... |
cphnm 22801 | The square of the norm is ... |
nmsq 22802 | The square of the norm is ... |
cphnmf 22803 | The norm of a vector is a ... |
cphnmcl 22804 | The norm of a vector is a ... |
reipcl 22805 | An inner product of an ele... |
ipge0 22806 | The inner product in a com... |
cphipcj 22807 | Conjugate of an inner prod... |
cphipipcj 22808 | An inner product times its... |
cphorthcom 22809 | Orthogonality (meaning inn... |
cphip0l 22810 | Inner product with a zero ... |
cphip0r 22811 | Inner product with a zero ... |
cphipeq0 22812 | The inner product of a vec... |
cphdir 22813 | Distributive law for inner... |
cphdi 22814 | Distributive law for inner... |
cph2di 22815 | Distributive law for inner... |
cphsubdir 22816 | Distributive law for inner... |
cphsubdi 22817 | Distributive law for inner... |
cph2subdi 22818 | Distributive law for inner... |
cphass 22819 | Associative law for inner ... |
cphassr 22820 | "Associative" law for seco... |
cph2ass 22821 | Move scalar multiplication... |
cphassi 22822 | Associative law for the fi... |
cphassir 22823 | "Associative" law for the ... |
tchex 22824 | Lemma for ~ tchbas and sim... |
tchval 22825 | Define a function to augme... |
tchbas 22826 | The base set of a pre-Hilb... |
tchplusg 22827 | The addition operation of ... |
tchsub 22828 | The subtraction operation ... |
tchmulr 22829 | The ring operation of a pr... |
tchsca 22830 | The scalar field of a pre-... |
tchvsca 22831 | The scalar multiplication ... |
tchip 22832 | The inner product of a pre... |
tchtopn 22833 | The topology of a pre-Hilb... |
tchphl 22834 | Augmentation of a pre-Hilb... |
tchnmfval 22835 | The norm of a pre-Hilbert ... |
tchnmval 22836 | The norm of a pre-Hilbert ... |
cphtchnm 22837 | The norm of a norm-augment... |
tchds 22838 | The distance of a pre-Hilb... |
tchclm 22839 | Lemma for ~ tchcph . (Con... |
tchcphlem3 22840 | Lemma for ~ tchcph : real ... |
ipcau2 22841 | The Cauchy-Schwarz inequal... |
tchcphlem1 22842 | Lemma for ~ tchcph : the t... |
tchcphlem2 22843 | Lemma for ~ tchcph : homog... |
tchcph 22844 | The standard definition of... |
ipcau 22845 | The Cauchy-Schwarz inequal... |
nmparlem 22846 | Lemma for ~ nmpar . (Cont... |
nmpar 22847 | A complex pre-Hilbert spac... |
cphipval2 22848 | Value of the inner product... |
4cphipval2 22849 | Four times the inner produ... |
cphipval 22850 | Value of the inner product... |
ipcnlem2 22851 | The inner product operatio... |
ipcnlem1 22852 | The inner product operatio... |
ipcn 22853 | The inner product operatio... |
cnmpt1ip 22854 | Continuity of inner produc... |
cnmpt2ip 22855 | Continuity of inner produc... |
csscld 22856 | A "closed subspace" in a c... |
clsocv 22857 | The orthogonal complement ... |
lmmbr 22864 | Express the binary relatio... |
lmmbr2 22865 | Express the binary relatio... |
lmmbr3 22866 | Express the binary relatio... |
lmmcvg 22867 | Convergence property of a ... |
lmmbrf 22868 | Express the binary relatio... |
lmnn 22869 | A condition that implies c... |
cfilfval 22870 | The set of Cauchy filters ... |
iscfil 22871 | The property of being a Ca... |
iscfil2 22872 | The property of being a Ca... |
cfilfil 22873 | A Cauchy filter is a filte... |
cfili 22874 | Property of a Cauchy filte... |
cfil3i 22875 | A Cauchy filter contains b... |
cfilss 22876 | A filter finer than a Cauc... |
fgcfil 22877 | The Cauchy filter conditio... |
fmcfil 22878 | The Cauchy filter conditio... |
iscfil3 22879 | A filter is Cauchy iff it ... |
cfilfcls 22880 | Similar to ultrafilters ( ... |
caufval 22881 | The set of Cauchy sequence... |
iscau 22882 | Express the property " ` F... |
iscau2 22883 | Express the property " ` F... |
iscau3 22884 | Express the Cauchy sequenc... |
iscau4 22885 | Express the property " ` F... |
iscauf 22886 | Express the property " ` F... |
caun0 22887 | A metric with a Cauchy seq... |
caufpm 22888 | Inclusion of a Cauchy sequ... |
caucfil 22889 | A Cauchy sequence predicat... |
iscmet 22890 | The property " ` D ` is a ... |
cmetcvg 22891 | The convergence of a Cauch... |
cmetmet 22892 | A complete metric space is... |
cmetmeti 22893 | A complete metric space is... |
cmetcaulem 22894 | Lemma for ~ cmetcau . (Co... |
cmetcau 22895 | The convergence of a Cauch... |
iscmet3lem3 22896 | Lemma for ~ iscmet3 . (Co... |
iscmet3lem1 22897 | Lemma for ~ iscmet3 . (Co... |
iscmet3lem2 22898 | Lemma for ~ iscmet3 . (Co... |
iscmet3 22899 | The property " ` D ` is a ... |
iscmet2 22900 | A metric ` D ` is complete... |
cfilresi 22901 | A Cauchy filter on a metri... |
cfilres 22902 | Cauchy filter on a metric ... |
caussi 22903 | Cauchy sequence on a metri... |
causs 22904 | Cauchy sequence on a metri... |
equivcfil 22905 | If the metric ` D ` is "st... |
equivcau 22906 | If the metric ` D ` is "st... |
lmle 22907 | If the distance from each ... |
nglmle 22908 | If the norm of each member... |
lmclim 22909 | Relate a limit on the metr... |
lmclimf 22910 | Relate a limit on the metr... |
metelcls 22911 | A point belongs to the clo... |
metcld 22912 | A subset of a metric space... |
metcld2 22913 | A subset of a metric space... |
caubl 22914 | Sufficient condition to en... |
caublcls 22915 | The convergent point of a ... |
metcnp4 22916 | Two ways to say a mapping ... |
metcn4 22917 | Two ways to say a mapping ... |
iscmet3i 22918 | Properties that determine ... |
lmcau 22919 | Every convergent sequence ... |
flimcfil 22920 | Every convergent filter in... |
cmetss 22921 | A subspace of a complete m... |
equivcmet 22922 | If two metrics are strongl... |
relcmpcmet 22923 | If ` D ` is a metric space... |
cmpcmet 22924 | A compact metric space is ... |
cfilucfil3 22925 | Given a metric ` D ` and a... |
cfilucfil4 22926 | Given a metric ` D ` and a... |
cncmet 22927 | The set of complex numbers... |
recmet 22928 | The real numbers are a com... |
bcthlem1 22929 | Lemma for ~ bcth . Substi... |
bcthlem2 22930 | Lemma for ~ bcth . The ba... |
bcthlem3 22931 | Lemma for ~ bcth . The li... |
bcthlem4 22932 | Lemma for ~ bcth . Given ... |
bcthlem5 22933 | Lemma for ~ bcth . The pr... |
bcth 22934 | Baire's Category Theorem. ... |
bcth2 22935 | Baire's Category Theorem, ... |
bcth3 22936 | Baire's Category Theorem, ... |
isbn 22943 | A Banach space is a normed... |
bnsca 22944 | The scalar field of a comp... |
bnnvc 22945 | A Banach space is a normed... |
bnnlm 22946 | A Banach space is a normed... |
bnngp 22947 | A Banach space is a normed... |
bnlmod 22948 | A Banach space is a left m... |
bncms 22949 | A Banach space is a comple... |
iscms 22950 | A complete metric space is... |
cmscmet 22951 | The induced metric on a co... |
bncmet 22952 | The induced metric on Bana... |
cmsms 22953 | A complete metric space is... |
cmspropd 22954 | Property deduction for a c... |
cmsss 22955 | The restriction of a compl... |
lssbn 22956 | A subspace of a Banach spa... |
cmetcusp1 22957 | If the uniform set of a co... |
cmetcusp 22958 | The uniform space generate... |
cncms 22959 | The field of complex numbe... |
cnflduss 22960 | The uniform structure of t... |
cnfldcusp 22961 | The field of complex numbe... |
resscdrg 22962 | The real numbers are a sub... |
cncdrg 22963 | The only complete subfield... |
srabn 22964 | The subring algebra over a... |
rlmbn 22965 | The ring module over a com... |
ishl 22966 | The predicate "is a comple... |
hlbn 22967 | Every complex Hilbert spac... |
hlcph 22968 | Every complex Hilbert spac... |
hlphl 22969 | Every complex Hilbert spac... |
hlcms 22970 | Every complex Hilbert spac... |
hlprlem 22971 | Lemma for ~ hlpr . (Contr... |
hlress 22972 | The scalar field of a comp... |
hlpr 22973 | The scalar field of a comp... |
ishl2 22974 | A Hilbert space is a compl... |
retopn 22975 | The topology of the real n... |
recms 22976 | The real numbers form a co... |
reust 22977 | The Uniform structure of t... |
recusp 22978 | The real numbers form a co... |
rrxval 22983 | Value of the generalized E... |
rrxbase 22984 | The base of the generalize... |
rrxprds 22985 | Expand the definition of t... |
rrxip 22986 | The inner product of the g... |
rrxnm 22987 | The norm of the generalize... |
rrxcph 22988 | Generalized Euclidean real... |
rrxds 22989 | The distance over generali... |
csbren 22990 | Cauchy-Schwarz-Bunjakovsky... |
trirn 22991 | Triangle inequality in R^n... |
rrxf 22992 | Euclidean vectors as funct... |
rrxfsupp 22993 | Euclidean vectors are of f... |
rrxsuppss 22994 | Support of Euclidean vecto... |
rrxmvallem 22995 | Support of the function us... |
rrxmval 22996 | The value of the Euclidean... |
rrxmfval 22997 | The value of the Euclidean... |
rrxmetlem 22998 | Lemma for ~ rrxmet . (Con... |
rrxmet 22999 | Euclidean space is a metri... |
rrxdstprj1 23000 | The distance between two p... |
ehlval 23001 | Value of the Euclidean spa... |
ehlbase 23002 | The base of the Euclidean ... |
minveclem1 23003 | Lemma for ~ minvec . The ... |
minveclem4c 23004 | Lemma for ~ minvec . The ... |
minveclem2 23005 | Lemma for ~ minvec . Any ... |
minveclem3a 23006 | Lemma for ~ minvec . ` D `... |
minveclem3b 23007 | Lemma for ~ minvec . The ... |
minveclem3 23008 | Lemma for ~ minvec . The ... |
minveclem4a 23009 | Lemma for ~ minvec . ` F `... |
minveclem4b 23010 | Lemma for ~ minvec . The ... |
minveclem4 23011 | Lemma for ~ minvec . The ... |
minveclem5 23012 | Lemma for ~ minvec . Disc... |
minveclem6 23013 | Lemma for ~ minvec . Any ... |
minveclem7 23014 | Lemma for ~ minvec . Sinc... |
minvec 23015 | Minimizing vector theorem,... |
pjthlem1 23016 | Lemma for ~ pjth . (Contr... |
pjthlem2 23017 | Lemma for ~ pjth . (Contr... |
pjth 23018 | Projection Theorem: Any H... |
pjth2 23019 | Projection Theorem with ab... |
cldcss 23020 | Corollary of the Projectio... |
cldcss2 23021 | Corollary of the Projectio... |
hlhil 23022 | Corollary of the Projectio... |
mulcncf 23023 | The multiplication of two ... |
pmltpclem1 23024 | Lemma for ~ pmltpc . (Con... |
pmltpclem2 23025 | Lemma for ~ pmltpc . (Con... |
pmltpc 23026 | Any function on the reals ... |
ivthlem1 23027 | Lemma for ~ ivth . The se... |
ivthlem2 23028 | Lemma for ~ ivth . Show t... |
ivthlem3 23029 | Lemma for ~ ivth , the int... |
ivth 23030 | The intermediate value the... |
ivth2 23031 | The intermediate value the... |
ivthle 23032 | The intermediate value the... |
ivthle2 23033 | The intermediate value the... |
ivthicc 23034 | The interval between any t... |
evthicc 23035 | Specialization of the Extr... |
evthicc2 23036 | Combine ~ ivthicc with ~ e... |
cniccbdd 23037 | A continuous function on a... |
ovolfcl 23042 | Closure for the interval e... |
ovolfioo 23043 | Unpack the interval coveri... |
ovolficc 23044 | Unpack the interval coveri... |
ovolficcss 23045 | Any (closed) interval cove... |
ovolfsval 23046 | The value of the interval ... |
ovolfsf 23047 | Closure for the interval l... |
ovolsf 23048 | Closure for the partial su... |
ovolval 23049 | The value of the outer mea... |
elovolm 23050 | Elementhood in the set ` M... |
elovolmr 23051 | Sufficient condition for e... |
ovolmge0 23052 | The set ` M ` is composed ... |
ovolcl 23053 | The volume of a set is an ... |
ovollb 23054 | The outer volume is a lowe... |
ovolgelb 23055 | The outer volume is the gr... |
ovolge0 23056 | The volume of a set is alw... |
ovolf 23057 | The domain and range of th... |
ovollecl 23058 | If an outer volume is boun... |
ovolsslem 23059 | Lemma for ~ ovolss . (Con... |
ovolss 23060 | The volume of a set is mon... |
ovolsscl 23061 | If a set is contained in a... |
ovolssnul 23062 | A subset of a nullset is n... |
ovollb2lem 23063 | Lemma for ~ ovollb2 . (Co... |
ovollb2 23064 | It is often more convenien... |
ovolctb 23065 | The volume of a denumerabl... |
ovolq 23066 | The rational numbers have ... |
ovolctb2 23067 | The volume of a countable ... |
ovol0 23068 | The empty set has 0 outer ... |
ovolfi 23069 | A finite set has 0 outer L... |
ovolsn 23070 | A singleton has 0 outer Le... |
ovolunlem1a 23071 | Lemma for ~ ovolun . (Con... |
ovolunlem1 23072 | Lemma for ~ ovolun . (Con... |
ovolunlem2 23073 | Lemma for ~ ovolun . (Con... |
ovolun 23074 | The Lebesgue outer measure... |
ovolunnul 23075 | Adding a nullset does not ... |
ovolfiniun 23076 | The Lebesgue outer measure... |
ovoliunlem1 23077 | Lemma for ~ ovoliun . (Co... |
ovoliunlem2 23078 | Lemma for ~ ovoliun . (Co... |
ovoliunlem3 23079 | Lemma for ~ ovoliun . (Co... |
ovoliun 23080 | The Lebesgue outer measure... |
ovoliun2 23081 | The Lebesgue outer measure... |
ovoliunnul 23082 | A countable union of nulls... |
shft2rab 23083 | If ` B ` is a shift of ` A... |
ovolshftlem1 23084 | Lemma for ~ ovolshft . (C... |
ovolshftlem2 23085 | Lemma for ~ ovolshft . (C... |
ovolshft 23086 | The Lebesgue outer measure... |
sca2rab 23087 | If ` B ` is a scale of ` A... |
ovolscalem1 23088 | Lemma for ~ ovolsca . (Co... |
ovolscalem2 23089 | Lemma for ~ ovolshft . (C... |
ovolsca 23090 | The Lebesgue outer measure... |
ovolicc1 23091 | The measure of a closed in... |
ovolicc2lem1 23092 | Lemma for ~ ovolicc2 . (C... |
ovolicc2lem2 23093 | Lemma for ~ ovolicc2 . (C... |
ovolicc2lem3 23094 | Lemma for ~ ovolicc2 . (C... |
ovolicc2lem4 23095 | Lemma for ~ ovolicc2 . (C... |
ovolicc2lem5 23096 | Lemma for ~ ovolicc2 . (C... |
ovolicc2 23097 | The measure of a closed in... |
ovolicc 23098 | The measure of a closed in... |
ovolicopnf 23099 | The measure of a right-unb... |
ovolre 23100 | The measure of the real nu... |
ismbl 23101 | The predicate " ` A ` is L... |
ismbl2 23102 | From ~ ovolun , it suffice... |
volres 23103 | A self-referencing abbrevi... |
volf 23104 | The domain and range of th... |
mblvol 23105 | The volume of a measurable... |
mblss 23106 | A measurable set is a subs... |
mblsplit 23107 | The defining property of m... |
volss 23108 | The Lebesgue measure is mo... |
cmmbl 23109 | The complement of a measur... |
nulmbl 23110 | A nullset is measurable. ... |
nulmbl2 23111 | A set of outer measure zer... |
unmbl 23112 | A union of measurable sets... |
shftmbl 23113 | A shift of a measurable se... |
0mbl 23114 | The empty set is measurabl... |
rembl 23115 | The set of all real number... |
unidmvol 23116 | The union of the Lebesgue ... |
inmbl 23117 | An intersection of measura... |
difmbl 23118 | A difference of measurable... |
finiunmbl 23119 | A finite union of measurab... |
volun 23120 | The Lebesgue measure funct... |
volinun 23121 | Addition of non-disjoint s... |
volfiniun 23122 | The volume of a disjoint f... |
iundisj 23123 | Rewrite a countable union ... |
iundisj2 23124 | A disjoint union is disjoi... |
voliunlem1 23125 | Lemma for ~ voliun . (Con... |
voliunlem2 23126 | Lemma for ~ voliun . (Con... |
voliunlem3 23127 | Lemma for ~ voliun . (Con... |
iunmbl 23128 | The measurable sets are cl... |
voliun 23129 | The Lebesgue measure funct... |
volsuplem 23130 | Lemma for ~ volsup . (Con... |
volsup 23131 | The volume of the limit of... |
iunmbl2 23132 | The measurable sets are cl... |
ioombl1lem1 23133 | Lemma for ~ ioombl1 . (Co... |
ioombl1lem2 23134 | Lemma for ~ ioombl1 . (Co... |
ioombl1lem3 23135 | Lemma for ~ ioombl1 . (Co... |
ioombl1lem4 23136 | Lemma for ~ ioombl1 . (Co... |
ioombl1 23137 | An open right-unbounded in... |
icombl1 23138 | A closed unbounded-above i... |
icombl 23139 | A closed-below, open-above... |
ioombl 23140 | An open real interval is m... |
iccmbl 23141 | A closed real interval is ... |
iccvolcl 23142 | A closed real interval has... |
ovolioo 23143 | The measure of an open int... |
ioovolcl 23144 | An open real interval has ... |
ovolfs2 23145 | Alternative expression for... |
ioorcl2 23146 | An open interval with fini... |
ioorf 23147 | Define a function from ope... |
ioorval 23148 | Define a function from ope... |
ioorinv2 23149 | The function ` F ` is an "... |
ioorinv 23150 | The function ` F ` is an "... |
ioorcl 23151 | The function ` F ` does no... |
uniiccdif 23152 | A union of closed interval... |
uniioovol 23153 | A disjoint union of open i... |
uniiccvol 23154 | An almost-disjoint union o... |
uniioombllem1 23155 | Lemma for ~ uniioombl . (... |
uniioombllem2a 23156 | Lemma for ~ uniioombl . (... |
uniioombllem2 23157 | Lemma for ~ uniioombl . (... |
uniioombllem3a 23158 | Lemma for ~ uniioombl . (... |
uniioombllem3 23159 | Lemma for ~ uniioombl . (... |
uniioombllem4 23160 | Lemma for ~ uniioombl . (... |
uniioombllem5 23161 | Lemma for ~ uniioombl . (... |
uniioombllem6 23162 | Lemma for ~ uniioombl . (... |
uniioombl 23163 | A disjoint union of open i... |
uniiccmbl 23164 | An almost-disjoint union o... |
dyadf 23165 | The function ` F ` returns... |
dyadval 23166 | Value of the dyadic ration... |
dyadovol 23167 | Volume of a dyadic rationa... |
dyadss 23168 | Two closed dyadic rational... |
dyaddisjlem 23169 | Lemma for ~ dyaddisj . (C... |
dyaddisj 23170 | Two closed dyadic rational... |
dyadmaxlem 23171 | Lemma for ~ dyadmax . (Co... |
dyadmax 23172 | Any nonempty set of dyadic... |
dyadmbllem 23173 | Lemma for ~ dyadmbl . (Co... |
dyadmbl 23174 | Any union of dyadic ration... |
opnmbllem 23175 | Lemma for ~ opnmbl . (Con... |
opnmbl 23176 | All open sets are measurab... |
opnmblALT 23177 | All open sets are measurab... |
subopnmbl 23178 | Sets which are open in a m... |
volsup2 23179 | The volume of ` A ` is the... |
volcn 23180 | The function formed by res... |
volivth 23181 | The Intermediate Value The... |
vitalilem1 23182 | Lemma for ~ vitali . (Con... |
vitalilem1OLD 23183 | Obsolete proof of ~ vitali... |
vitalilem2 23184 | Lemma for ~ vitali . (Con... |
vitalilem3 23185 | Lemma for ~ vitali . (Con... |
vitalilem4 23186 | Lemma for ~ vitali . (Con... |
vitalilem5 23187 | Lemma for ~ vitali . (Con... |
vitali 23188 | If the reals can be well-o... |
ismbf1 23199 | The predicate " ` F ` is a... |
mbff 23200 | A measurable function is a... |
mbfdm 23201 | The domain of a measurable... |
mbfconstlem 23202 | Lemma for ~ mbfconst . (C... |
ismbf 23203 | The predicate " ` F ` is a... |
ismbfcn 23204 | A complex function is meas... |
mbfima 23205 | Definitional property of a... |
mbfimaicc 23206 | The preimage of any closed... |
mbfimasn 23207 | The preimage of a point un... |
mbfconst 23208 | A constant function is mea... |
mbfid 23209 | The identity function is m... |
mbfmptcl 23210 | Lemma for the ` MblFn ` pr... |
mbfdm2 23211 | The domain of a measurable... |
ismbfcn2 23212 | A complex function is meas... |
ismbfd 23213 | Deduction to prove measura... |
ismbf2d 23214 | Deduction to prove measura... |
mbfeqalem 23215 | Lemma for ~ mbfeqa . (Con... |
mbfeqa 23216 | If two functions are equal... |
mbfres 23217 | The restriction of a measu... |
mbfres2 23218 | Measurability of a piecewi... |
mbfss 23219 | Change the domain of a mea... |
mbfmulc2lem 23220 | Multiplication by a consta... |
mbfmulc2re 23221 | Multiplication by a consta... |
mbfmax 23222 | The maximum of two functio... |
mbfneg 23223 | The negative of a measurab... |
mbfpos 23224 | The positive part of a mea... |
mbfposr 23225 | Converse to ~ mbfpos . (C... |
mbfposb 23226 | A function is measurable i... |
ismbf3d 23227 | Simplified form of ~ ismbf... |
mbfimaopnlem 23228 | Lemma for ~ mbfimaopn . (... |
mbfimaopn 23229 | The preimage of any open s... |
mbfimaopn2 23230 | The preimage of any set op... |
cncombf 23231 | The composition of a conti... |
cnmbf 23232 | A continuous function is m... |
mbfaddlem 23233 | The sum of two measurable ... |
mbfadd 23234 | The sum of two measurable ... |
mbfsub 23235 | The difference of two meas... |
mbfmulc2 23236 | A complex constant times a... |
mbfsup 23237 | The supremum of a sequence... |
mbfinf 23238 | The infimum of a sequence ... |
mbflimsup 23239 | The limit supremum of a se... |
mbflimlem 23240 | The pointwise limit of a s... |
mbflim 23241 | The pointwise limit of a s... |
0pval 23244 | The zero function evaluate... |
0plef 23245 | Two ways to say that the f... |
0pledm 23246 | Adjust the domain of the l... |
isi1f 23247 | The predicate " ` F ` is a... |
i1fmbf 23248 | Simple functions are measu... |
i1ff 23249 | A simple function is a fun... |
i1frn 23250 | A simple function has fini... |
i1fima 23251 | Any preimage of a simple f... |
i1fima2 23252 | Any preimage of a simple f... |
i1fima2sn 23253 | Preimage of a singleton. ... |
i1fd 23254 | A simplified set of assump... |
i1f0rn 23255 | Any simple function takes ... |
itg1val 23256 | The value of the integral ... |
itg1val2 23257 | The value of the integral ... |
itg1cl 23258 | Closure of the integral on... |
itg1ge0 23259 | Closure of the integral on... |
i1f0 23260 | The zero function is simpl... |
itg10 23261 | The zero function has zero... |
i1f1lem 23262 | Lemma for ~ i1f1 and ~ itg... |
i1f1 23263 | Base case simple functions... |
itg11 23264 | The integral of an indicat... |
itg1addlem1 23265 | Decompose a preimage, whic... |
i1faddlem 23266 | Decompose the preimage of ... |
i1fmullem 23267 | Decompose the preimage of ... |
i1fadd 23268 | The sum of two simple func... |
i1fmul 23269 | The pointwise product of t... |
itg1addlem2 23270 | Lemma for ~ itg1add . The... |
itg1addlem3 23271 | Lemma for ~ itg1add . (Co... |
itg1addlem4 23272 | Lemma for itg1add . (Cont... |
itg1addlem5 23273 | Lemma for itg1add . (Cont... |
itg1add 23274 | The integral of a sum of s... |
i1fmulclem 23275 | Decompose the preimage of ... |
i1fmulc 23276 | A nonnegative constant tim... |
itg1mulc 23277 | The integral of a constant... |
i1fres 23278 | The "restriction" of a sim... |
i1fpos 23279 | The positive part of a sim... |
i1fposd 23280 | Deduction form of ~ i1fpos... |
i1fsub 23281 | The difference of two simp... |
itg1sub 23282 | The integral of a differen... |
itg10a 23283 | The integral of a simple f... |
itg1ge0a 23284 | The integral of an almost ... |
itg1lea 23285 | Approximate version of ~ i... |
itg1le 23286 | If one simple function dom... |
itg1climres 23287 | Restricting the simple fun... |
mbfi1fseqlem1 23288 | Lemma for ~ mbfi1fseq . (... |
mbfi1fseqlem2 23289 | Lemma for ~ mbfi1fseq . (... |
mbfi1fseqlem3 23290 | Lemma for ~ mbfi1fseq . (... |
mbfi1fseqlem4 23291 | Lemma for ~ mbfi1fseq . T... |
mbfi1fseqlem5 23292 | Lemma for ~ mbfi1fseq . V... |
mbfi1fseqlem6 23293 | Lemma for ~ mbfi1fseq . V... |
mbfi1fseq 23294 | A characterization of meas... |
mbfi1flimlem 23295 | Lemma for ~ mbfi1flim . (... |
mbfi1flim 23296 | Any real measurable functi... |
mbfmullem2 23297 | Lemma for ~ mbfmul . (Con... |
mbfmullem 23298 | Lemma for ~ mbfmul . (Con... |
mbfmul 23299 | The product of two measura... |
itg2lcl 23300 | The set of lower sums is a... |
itg2val 23301 | Value of the integral on n... |
itg2l 23302 | Elementhood in the set ` L... |
itg2lr 23303 | Sufficient condition for e... |
xrge0f 23304 | A real function is a nonne... |
itg2cl 23305 | The integral of a nonnegat... |
itg2ub 23306 | The integral of a nonnegat... |
itg2leub 23307 | Any upper bound on the int... |
itg2ge0 23308 | The integral of a nonnegat... |
itg2itg1 23309 | The integral of a nonnegat... |
itg20 23310 | The integral of the zero f... |
itg2lecl 23311 | If an ` S.2 ` integral is ... |
itg2le 23312 | If one function dominates ... |
itg2const 23313 | Integral of a constant fun... |
itg2const2 23314 | When the base set of a con... |
itg2seq 23315 | Definitional property of t... |
itg2uba 23316 | Approximate version of ~ i... |
itg2lea 23317 | Approximate version of ~ i... |
itg2eqa 23318 | Approximate equality of in... |
itg2mulclem 23319 | Lemma for ~ itg2mulc . (C... |
itg2mulc 23320 | The integral of a nonnegat... |
itg2splitlem 23321 | Lemma for ~ itg2split . (... |
itg2split 23322 | The ` S.2 ` integral split... |
itg2monolem1 23323 | Lemma for ~ itg2mono . We... |
itg2monolem2 23324 | Lemma for ~ itg2mono . (C... |
itg2monolem3 23325 | Lemma for ~ itg2mono . (C... |
itg2mono 23326 | The Monotone Convergence T... |
itg2i1fseqle 23327 | Subject to the conditions ... |
itg2i1fseq 23328 | Subject to the conditions ... |
itg2i1fseq2 23329 | In an extension to the res... |
itg2i1fseq3 23330 | Special case of ~ itg2i1fs... |
itg2addlem 23331 | Lemma for ~ itg2add . (Co... |
itg2add 23332 | The ` S.2 ` integral is li... |
itg2gt0 23333 | If the function ` F ` is s... |
itg2cnlem1 23334 | Lemma for ~ itgcn . (Cont... |
itg2cnlem2 23335 | Lemma for ~ itgcn . (Cont... |
itg2cn 23336 | A sort of absolute continu... |
ibllem 23337 | Conditioned equality theor... |
isibl 23338 | The predicate " ` F ` is i... |
isibl2 23339 | The predicate " ` F ` is i... |
iblmbf 23340 | An integrable function is ... |
iblitg 23341 | If a function is integrabl... |
dfitg 23342 | Evaluate the class substit... |
itgex 23343 | An integral is a set. (Co... |
itgeq1f 23344 | Equality theorem for an in... |
itgeq1 23345 | Equality theorem for an in... |
nfitg1 23346 | Bound-variable hypothesis ... |
nfitg 23347 | Bound-variable hypothesis ... |
cbvitg 23348 | Change bound variable in a... |
cbvitgv 23349 | Change bound variable in a... |
itgeq2 23350 | Equality theorem for an in... |
itgresr 23351 | The domain of an integral ... |
itg0 23352 | The integral of anything o... |
itgz 23353 | The integral of zero on an... |
itgeq2dv 23354 | Equality theorem for an in... |
itgmpt 23355 | Change bound variable in a... |
itgcl 23356 | The integral of an integra... |
itgvallem 23357 | Substitution lemma. (Cont... |
itgvallem3 23358 | Lemma for ~ itgposval and ... |
ibl0 23359 | The zero function is integ... |
iblcnlem1 23360 | Lemma for ~ iblcnlem . (C... |
iblcnlem 23361 | Expand out the forall in ~... |
itgcnlem 23362 | Expand out the sum in ~ df... |
iblrelem 23363 | Integrability of a real fu... |
iblposlem 23364 | Lemma for ~ iblpos . (Con... |
iblpos 23365 | Integrability of a nonnega... |
iblre 23366 | Integrability of a real fu... |
itgrevallem1 23367 | Lemma for ~ itgposval and ... |
itgposval 23368 | The integral of a nonnegat... |
itgreval 23369 | Decompose the integral of ... |
itgrecl 23370 | Real closure of an integra... |
iblcn 23371 | Integrability of a complex... |
itgcnval 23372 | Decompose the integral of ... |
itgre 23373 | Real part of an integral. ... |
itgim 23374 | Imaginary part of an integ... |
iblneg 23375 | The negative of an integra... |
itgneg 23376 | Negation of an integral. ... |
iblss 23377 | A subset of an integrable ... |
iblss2 23378 | Change the domain of an in... |
itgitg2 23379 | Transfer an integral using... |
i1fibl 23380 | A simple function is integ... |
itgitg1 23381 | Transfer an integral using... |
itgle 23382 | Monotonicity of an integra... |
itgge0 23383 | The integral of a positive... |
itgss 23384 | Expand the set of an integ... |
itgss2 23385 | Expand the set of an integ... |
itgeqa 23386 | Approximate equality of in... |
itgss3 23387 | Expand the set of an integ... |
itgioo 23388 | Equality of integrals on o... |
itgless 23389 | Expand the integral of a n... |
iblconst 23390 | A constant function is int... |
itgconst 23391 | Integral of a constant fun... |
ibladdlem 23392 | Lemma for ~ ibladd . (Con... |
ibladd 23393 | Add two integrals over the... |
iblsub 23394 | Subtract two integrals ove... |
itgaddlem1 23395 | Lemma for ~ itgadd . (Con... |
itgaddlem2 23396 | Lemma for ~ itgadd . (Con... |
itgadd 23397 | Add two integrals over the... |
itgsub 23398 | Subtract two integrals ove... |
itgfsum 23399 | Take a finite sum of integ... |
iblabslem 23400 | Lemma for ~ iblabs . (Con... |
iblabs 23401 | The absolute value of an i... |
iblabsr 23402 | A measurable function is i... |
iblmulc2 23403 | Multiply an integral by a ... |
itgmulc2lem1 23404 | Lemma for ~ itgmulc2 : pos... |
itgmulc2lem2 23405 | Lemma for ~ itgmulc2 : rea... |
itgmulc2 23406 | Multiply an integral by a ... |
itgabs 23407 | The triangle inequality fo... |
itgsplit 23408 | The ` S. ` integral splits... |
itgspliticc 23409 | The ` S. ` integral splits... |
itgsplitioo 23410 | The ` S. ` integral splits... |
bddmulibl 23411 | A bounded function times a... |
bddibl 23412 | A bounded function is inte... |
cniccibl 23413 | A continuous function on a... |
itggt0 23414 | The integral of a strictly... |
itgcn 23415 | Transfer ~ itg2cn to the f... |
ditgeq1 23418 | Equality theorem for the d... |
ditgeq2 23419 | Equality theorem for the d... |
ditgeq3 23420 | Equality theorem for the d... |
ditgeq3dv 23421 | Equality theorem for the d... |
ditgex 23422 | A directed integral is a s... |
ditg0 23423 | Value of the directed inte... |
cbvditg 23424 | Change bound variable in a... |
cbvditgv 23425 | Change bound variable in a... |
ditgpos 23426 | Value of the directed inte... |
ditgneg 23427 | Value of the directed inte... |
ditgcl 23428 | Closure of a directed inte... |
ditgswap 23429 | Reverse a directed integra... |
ditgsplitlem 23430 | Lemma for ~ ditgsplit . (... |
ditgsplit 23431 | This theorem is the raison... |
reldv 23440 | The derivative function is... |
limcvallem 23441 | Lemma for ~ ellimc . (Con... |
limcfval 23442 | Value and set bounds on th... |
ellimc 23443 | Value of the limit predica... |
limcrcl 23444 | Reverse closure for the li... |
limccl 23445 | Closure of the limit opera... |
limcdif 23446 | It suffices to consider fu... |
ellimc2 23447 | Write the definition of a ... |
limcnlp 23448 | If ` B ` is not a limit po... |
ellimc3 23449 | Write the epsilon-delta de... |
limcflflem 23450 | Lemma for ~ limcflf . (Co... |
limcflf 23451 | The limit operator can be ... |
limcmo 23452 | If ` B ` is a limit point ... |
limcmpt 23453 | Express the limit operator... |
limcmpt2 23454 | Express the limit operator... |
limcresi 23455 | Any limit of ` F ` is also... |
limcres 23456 | If ` B ` is an interior po... |
cnplimc 23457 | A function is continuous a... |
cnlimc 23458 | ` F ` is a continuous func... |
cnlimci 23459 | If ` F ` is a continuous f... |
cnmptlimc 23460 | If ` F ` is a continuous f... |
limccnp 23461 | If the limit of ` F ` at `... |
limccnp2 23462 | The image of a convergent ... |
limcco 23463 | Composition of two limits.... |
limciun 23464 | A point is a limit of ` F ... |
limcun 23465 | A point is a limit of ` F ... |
dvlem 23466 | Closure for a difference q... |
dvfval 23467 | Value and set bounds on th... |
eldv 23468 | The differentiable predica... |
dvcl 23469 | The derivative function ta... |
dvbssntr 23470 | The set of differentiable ... |
dvbss 23471 | The set of differentiable ... |
dvbsss 23472 | The set of differentiable ... |
perfdvf 23473 | The derivative is a functi... |
recnprss 23474 | Both ` RR ` and ` CC ` are... |
recnperf 23475 | Both ` RR ` and ` CC ` are... |
dvfg 23476 | Explicitly write out the f... |
dvf 23477 | The derivative is a functi... |
dvfcn 23478 | The derivative is a functi... |
dvreslem 23479 | Lemma for ~ dvres . (Cont... |
dvres2lem 23480 | Lemma for ~ dvres2 . (Con... |
dvres 23481 | Restriction of a derivativ... |
dvres2 23482 | Restriction of the base se... |
dvres3 23483 | Restriction of a complex d... |
dvres3a 23484 | Restriction of a complex d... |
dvidlem 23485 | Lemma for ~ dvid and ~ dvc... |
dvconst 23486 | Derivative of a constant f... |
dvid 23487 | Derivative of the identity... |
dvcnp 23488 | The difference quotient is... |
dvcnp2 23489 | A function is continuous a... |
dvcn 23490 | A differentiable function ... |
dvnfval 23491 | Value of the iterated deri... |
dvnff 23492 | The iterated derivative is... |
dvn0 23493 | Zero times iterated deriva... |
dvnp1 23494 | Successor iterated derivat... |
dvn1 23495 | One times iterated derivat... |
dvnf 23496 | The N-times derivative is ... |
dvnbss 23497 | The set of N-times differe... |
dvnadd 23498 | The ` N ` -th derivative o... |
dvn2bss 23499 | An N-times differentiable ... |
dvnres 23500 | Multiple derivative versio... |
cpnfval 23501 | Condition for n-times cont... |
fncpn 23502 | The ` C^n ` object is a fu... |
elcpn 23503 | Condition for n-times cont... |
cpnord 23504 | ` C^n ` conditions are ord... |
cpncn 23505 | A ` C^n ` function is cont... |
cpnres 23506 | The restriction of a ` C^n... |
dvaddbr 23507 | The sum rule for derivativ... |
dvmulbr 23508 | The product rule for deriv... |
dvadd 23509 | The sum rule for derivativ... |
dvmul 23510 | The product rule for deriv... |
dvaddf 23511 | The sum rule for everywher... |
dvmulf 23512 | The product rule for every... |
dvcmul 23513 | The product rule when one ... |
dvcmulf 23514 | The product rule when one ... |
dvcobr 23515 | The chain rule for derivat... |
dvco 23516 | The chain rule for derivat... |
dvcof 23517 | The chain rule for everywh... |
dvcjbr 23518 | The derivative of the conj... |
dvcj 23519 | The derivative of the conj... |
dvfre 23520 | The derivative of a real f... |
dvnfre 23521 | The ` N ` -th derivative o... |
dvexp 23522 | Derivative of a power func... |
dvexp2 23523 | Derivative of an exponenti... |
dvrec 23524 | Derivative of the reciproc... |
dvmptres3 23525 | Function-builder for deriv... |
dvmptid 23526 | Function-builder for deriv... |
dvmptc 23527 | Function-builder for deriv... |
dvmptcl 23528 | Closure lemma for ~ dvmptc... |
dvmptadd 23529 | Function-builder for deriv... |
dvmptmul 23530 | Function-builder for deriv... |
dvmptres2 23531 | Function-builder for deriv... |
dvmptres 23532 | Function-builder for deriv... |
dvmptcmul 23533 | Function-builder for deriv... |
dvmptdivc 23534 | Function-builder for deriv... |
dvmptneg 23535 | Function-builder for deriv... |
dvmptsub 23536 | Function-builder for deriv... |
dvmptcj 23537 | Function-builder for deriv... |
dvmptre 23538 | Function-builder for deriv... |
dvmptim 23539 | Function-builder for deriv... |
dvmptntr 23540 | Function-builder for deriv... |
dvmptco 23541 | Function-builder for deriv... |
dvmptfsum 23542 | Function-builder for deriv... |
dvcnvlem 23543 | Lemma for ~ dvcnvre . (Co... |
dvcnv 23544 | A weak version of ~ dvcnvr... |
dvexp3 23545 | Derivative of an exponenti... |
dveflem 23546 | Derivative of the exponent... |
dvef 23547 | Derivative of the exponent... |
dvsincos 23548 | Derivative of the sine and... |
dvsin 23549 | Derivative of the sine fun... |
dvcos 23550 | Derivative of the cosine f... |
dvferm1lem 23551 | Lemma for ~ dvferm . (Con... |
dvferm1 23552 | One-sided version of ~ dvf... |
dvferm2lem 23553 | Lemma for ~ dvferm . (Con... |
dvferm2 23554 | One-sided version of ~ dvf... |
dvferm 23555 | Fermat's theorem on statio... |
rollelem 23556 | Lemma for ~ rolle . (Cont... |
rolle 23557 | Rolle's theorem. If ` F `... |
cmvth 23558 | Cauchy's Mean Value Theore... |
mvth 23559 | The Mean Value Theorem. I... |
dvlip 23560 | A function with derivative... |
dvlipcn 23561 | A complex function with de... |
dvlip2 23562 | Combine the results of ~ d... |
c1liplem1 23563 | Lemma for ~ c1lip1 . (Con... |
c1lip1 23564 | C1 functions are Lipschitz... |
c1lip2 23565 | C1 functions are Lipschitz... |
c1lip3 23566 | C1 functions are Lipschitz... |
dveq0 23567 | If a continuous function h... |
dv11cn 23568 | Two functions defined on a... |
dvgt0lem1 23569 | Lemma for ~ dvgt0 and ~ dv... |
dvgt0lem2 23570 | Lemma for ~ dvgt0 and ~ dv... |
dvgt0 23571 | A function on a closed int... |
dvlt0 23572 | A function on a closed int... |
dvge0 23573 | A function on a closed int... |
dvle 23574 | If ` A ( x ) , C ( x ) ` a... |
dvivthlem1 23575 | Lemma for ~ dvivth . (Con... |
dvivthlem2 23576 | Lemma for ~ dvivth . (Con... |
dvivth 23577 | Darboux' theorem, or the i... |
dvne0 23578 | A function on a closed int... |
dvne0f1 23579 | A function on a closed int... |
lhop1lem 23580 | Lemma for ~ lhop1 . (Cont... |
lhop1 23581 | L'Hôpital's Rule for... |
lhop2 23582 | L'Hôpital's Rule for... |
lhop 23583 | L'Hôpital's Rule. I... |
dvcnvrelem1 23584 | Lemma for ~ dvcnvre . (Co... |
dvcnvrelem2 23585 | Lemma for ~ dvcnvre . (Co... |
dvcnvre 23586 | The derivative rule for in... |
dvcvx 23587 | A real function with stric... |
dvfsumle 23588 | Compare a finite sum to an... |
dvfsumge 23589 | Compare a finite sum to an... |
dvfsumabs 23590 | Compare a finite sum to an... |
dvmptrecl 23591 | Real closure of a derivati... |
dvfsumrlimf 23592 | Lemma for ~ dvfsumrlim . ... |
dvfsumlem1 23593 | Lemma for ~ dvfsumrlim . ... |
dvfsumlem2 23594 | Lemma for ~ dvfsumrlim . ... |
dvfsumlem3 23595 | Lemma for ~ dvfsumrlim . ... |
dvfsumlem4 23596 | Lemma for ~ dvfsumrlim . ... |
dvfsumrlimge0 23597 | Lemma for ~ dvfsumrlim . ... |
dvfsumrlim 23598 | Compare a finite sum to an... |
dvfsumrlim2 23599 | Compare a finite sum to an... |
dvfsumrlim3 23600 | Conjoin the statements of ... |
dvfsum2 23601 | The reverse of ~ dvfsumrli... |
ftc1lem1 23602 | Lemma for ~ ftc1a and ~ ft... |
ftc1lem2 23603 | Lemma for ~ ftc1 . (Contr... |
ftc1a 23604 | The Fundamental Theorem of... |
ftc1lem3 23605 | Lemma for ~ ftc1 . (Contr... |
ftc1lem4 23606 | Lemma for ~ ftc1 . (Contr... |
ftc1lem5 23607 | Lemma for ~ ftc1 . (Contr... |
ftc1lem6 23608 | Lemma for ~ ftc1 . (Contr... |
ftc1 23609 | The Fundamental Theorem of... |
ftc1cn 23610 | Strengthen the assumptions... |
ftc2 23611 | The Fundamental Theorem of... |
ftc2ditglem 23612 | Lemma for ~ ftc2ditg . (C... |
ftc2ditg 23613 | Directed integral analogue... |
itgparts 23614 | Integration by parts. If ... |
itgsubstlem 23615 | Lemma for ~ itgsubst . (C... |
itgsubst 23616 | Integration by ` u ` -subs... |
reldmmdeg 23621 | Multivariate degree is a b... |
tdeglem1 23622 | Functionality of the total... |
tdeglem3 23623 | Additivity of the total de... |
tdeglem4 23624 | There is only one multi-in... |
tdeglem2 23625 | Simplification of total de... |
mdegfval 23626 | Value of the multivariate ... |
mdegval 23627 | Value of the multivariate ... |
mdegleb 23628 | Property of being of limit... |
mdeglt 23629 | If there is an upper limit... |
mdegldg 23630 | A nonzero polynomial has s... |
mdegxrcl 23631 | Closure of polynomial degr... |
mdegxrf 23632 | Functionality of polynomia... |
mdegcl 23633 | Sharp closure for multivar... |
mdeg0 23634 | Degree of the zero polynom... |
mdegnn0cl 23635 | Degree of a nonzero polyno... |
degltlem1 23636 | Theorem on arithmetic of e... |
degltp1le 23637 | Theorem on arithmetic of e... |
mdegaddle 23638 | The degree of a sum is at ... |
mdegvscale 23639 | The degree of a scalar mul... |
mdegvsca 23640 | The degree of a scalar mul... |
mdegle0 23641 | A polynomial has nonpositi... |
mdegmullem 23642 | Lemma for ~ mdegmulle2 . ... |
mdegmulle2 23643 | The multivariate degree of... |
deg1fval 23644 | Relate univariate polynomi... |
deg1xrf 23645 | Functionality of univariat... |
deg1xrcl 23646 | Closure of univariate poly... |
deg1cl 23647 | Sharp closure of univariat... |
mdegpropd 23648 | Property deduction for pol... |
deg1fvi 23649 | Univariate polynomial degr... |
deg1propd 23650 | Property deduction for pol... |
deg1z 23651 | Degree of the zero univari... |
deg1nn0cl 23652 | Degree of a nonzero univar... |
deg1n0ima 23653 | Degree image of a set of p... |
deg1nn0clb 23654 | A polynomial is nonzero if... |
deg1lt0 23655 | A polynomial is zero iff i... |
deg1ldg 23656 | A nonzero univariate polyn... |
deg1ldgn 23657 | An index at which a polyno... |
deg1ldgdomn 23658 | A nonzero univariate polyn... |
deg1leb 23659 | Property of being of limit... |
deg1val 23660 | Value of the univariate de... |
deg1lt 23661 | If the degree of a univari... |
deg1ge 23662 | Conversely, a nonzero coef... |
coe1mul3 23663 | The coefficient vector of ... |
coe1mul4 23664 | Value of the "leading" coe... |
deg1addle 23665 | The degree of a sum is at ... |
deg1addle2 23666 | If both factors have degre... |
deg1add 23667 | Exact degree of a sum of t... |
deg1vscale 23668 | The degree of a scalar tim... |
deg1vsca 23669 | The degree of a scalar tim... |
deg1invg 23670 | The degree of the negated ... |
deg1suble 23671 | The degree of a difference... |
deg1sub 23672 | Exact degree of a differen... |
deg1mulle2 23673 | Produce a bound on the pro... |
deg1sublt 23674 | Subtraction of two polynom... |
deg1le0 23675 | A polynomial has nonpositi... |
deg1sclle 23676 | A scalar polynomial has no... |
deg1scl 23677 | A nonzero scalar polynomia... |
deg1mul2 23678 | Degree of multiplication o... |
deg1mul3 23679 | Degree of multiplication o... |
deg1mul3le 23680 | Degree of multiplication o... |
deg1tmle 23681 | Limiting degree of a polyn... |
deg1tm 23682 | Exact degree of a polynomi... |
deg1pwle 23683 | Limiting degree of a varia... |
deg1pw 23684 | Exact degree of a variable... |
ply1nz 23685 | Univariate polynomials ove... |
ply1nzb 23686 | Univariate polynomials are... |
ply1domn 23687 | Corollary of ~ deg1mul2 : ... |
ply1idom 23688 | The ring of univariate pol... |
ply1divmo 23699 | Uniqueness of a quotient i... |
ply1divex 23700 | Lemma for ~ ply1divalg : e... |
ply1divalg 23701 | The division algorithm for... |
ply1divalg2 23702 | Reverse the order of multi... |
uc1pval 23703 | Value of the set of unitic... |
isuc1p 23704 | Being a unitic polynomial.... |
mon1pval 23705 | Value of the set of monic ... |
ismon1p 23706 | Being a monic polynomial. ... |
uc1pcl 23707 | Unitic polynomials are pol... |
mon1pcl 23708 | Monic polynomials are poly... |
uc1pn0 23709 | Unitic polynomials are not... |
mon1pn0 23710 | Monic polynomials are not ... |
uc1pdeg 23711 | Unitic polynomials have no... |
uc1pldg 23712 | Unitic polynomials have un... |
mon1pldg 23713 | Unitic polynomials have on... |
mon1puc1p 23714 | Monic polynomials are unit... |
uc1pmon1p 23715 | Make a unitic polynomial m... |
deg1submon1p 23716 | The difference of two moni... |
q1pval 23717 | Value of the univariate po... |
q1peqb 23718 | Characterizing property of... |
q1pcl 23719 | Closure of the quotient by... |
r1pval 23720 | Value of the polynomial re... |
r1pcl 23721 | Closure of remainder follo... |
r1pdeglt 23722 | The remainder has a degree... |
r1pid 23723 | Express the original polyn... |
dvdsq1p 23724 | Divisibility in a polynomi... |
dvdsr1p 23725 | Divisibility in a polynomi... |
ply1remlem 23726 | A term of the form ` x - N... |
ply1rem 23727 | The polynomial remainder t... |
facth1 23728 | The factor theorem and its... |
fta1glem1 23729 | Lemma for ~ fta1g . (Cont... |
fta1glem2 23730 | Lemma for ~ fta1g . (Cont... |
fta1g 23731 | The one-sided fundamental ... |
fta1blem 23732 | Lemma for ~ fta1b . (Cont... |
fta1b 23733 | The assumption that ` R ` ... |
drnguc1p 23734 | Over a division ring, all ... |
ig1peu 23735 | There is a unique monic po... |
ig1pval 23736 | Substitutions for the poly... |
ig1pval2 23737 | Generator of the zero idea... |
ig1pval3 23738 | Characterizing properties ... |
ig1pcl 23739 | The monic generator of an ... |
ig1pdvds 23740 | The monic generator of an ... |
ig1prsp 23741 | Any ideal of polynomials o... |
ply1lpir 23742 | The ring of polynomials ov... |
ply1pid 23743 | The polynomials over a fie... |
plyco0 23752 | Two ways to say that a fun... |
plyval 23753 | Value of the polynomial se... |
plybss 23754 | Reverse closure of the par... |
elply 23755 | Definition of a polynomial... |
elply2 23756 | The coefficient function c... |
plyun0 23757 | The set of polynomials is ... |
plyf 23758 | The polynomial is a functi... |
plyss 23759 | The polynomial set functio... |
plyssc 23760 | Every polynomial ring is c... |
elplyr 23761 | Sufficient condition for e... |
elplyd 23762 | Sufficient condition for e... |
ply1termlem 23763 | Lemma for ~ ply1term . (C... |
ply1term 23764 | A one-term polynomial. (C... |
plypow 23765 | A power is a polynomial. ... |
plyconst 23766 | A constant function is a p... |
ne0p 23767 | A test to show that a poly... |
ply0 23768 | The zero function is a pol... |
plyid 23769 | The identity function is a... |
plyeq0lem 23770 | Lemma for ~ plyeq0 . If `... |
plyeq0 23771 | If a polynomial is zero at... |
plypf1 23772 | Write the set of complex p... |
plyaddlem1 23773 | Derive the coefficient fun... |
plymullem1 23774 | Derive the coefficient fun... |
plyaddlem 23775 | Lemma for ~ plyadd . (Con... |
plymullem 23776 | Lemma for ~ plymul . (Con... |
plyadd 23777 | The sum of two polynomials... |
plymul 23778 | The product of two polynom... |
plysub 23779 | The difference of two poly... |
plyaddcl 23780 | The sum of two polynomials... |
plymulcl 23781 | The product of two polynom... |
plysubcl 23782 | The difference of two poly... |
coeval 23783 | Value of the coefficient f... |
coeeulem 23784 | Lemma for ~ coeeu . (Cont... |
coeeu 23785 | Uniqueness of the coeffici... |
coelem 23786 | Lemma for properties of th... |
coeeq 23787 | If ` A ` satisfies the pro... |
dgrval 23788 | Value of the degree functi... |
dgrlem 23789 | Lemma for ~ dgrcl and simi... |
coef 23790 | The domain and range of th... |
coef2 23791 | The domain and range of th... |
coef3 23792 | The domain and range of th... |
dgrcl 23793 | The degree of any polynomi... |
dgrub 23794 | If the ` M ` -th coefficie... |
dgrub2 23795 | All the coefficients above... |
dgrlb 23796 | If all the coefficients ab... |
coeidlem 23797 | Lemma for ~ coeid . (Cont... |
coeid 23798 | Reconstruct a polynomial a... |
coeid2 23799 | Reconstruct a polynomial a... |
coeid3 23800 | Reconstruct a polynomial a... |
plyco 23801 | The composition of two pol... |
coeeq2 23802 | Compute the coefficient fu... |
dgrle 23803 | Given an explicit expressi... |
dgreq 23804 | If the highest term in a p... |
0dgr 23805 | A constant function has de... |
0dgrb 23806 | A function has degree zero... |
dgrnznn 23807 | A nonzero polynomial with ... |
coefv0 23808 | The result of evaluating a... |
coeaddlem 23809 | Lemma for ~ coeadd and ~ d... |
coemullem 23810 | Lemma for ~ coemul and ~ d... |
coeadd 23811 | The coefficient function o... |
coemul 23812 | A coefficient of a product... |
coe11 23813 | The coefficient function i... |
coemulhi 23814 | The leading coefficient of... |
coemulc 23815 | The coefficient function i... |
coe0 23816 | The coefficients of the ze... |
coesub 23817 | The coefficient function o... |
coe1termlem 23818 | The coefficient function o... |
coe1term 23819 | The coefficient function o... |
dgr1term 23820 | The degree of a monomial. ... |
plycn 23821 | A polynomial is a continuo... |
dgr0 23822 | The degree of the zero pol... |
coeidp 23823 | The coefficients of the id... |
dgrid 23824 | The degree of the identity... |
dgreq0 23825 | The leading coefficient of... |
dgrlt 23826 | Two ways to say that the d... |
dgradd 23827 | The degree of a sum of pol... |
dgradd2 23828 | The degree of a sum of pol... |
dgrmul2 23829 | The degree of a product of... |
dgrmul 23830 | The degree of a product of... |
dgrmulc 23831 | Scalar multiplication by a... |
dgrsub 23832 | The degree of a difference... |
dgrcolem1 23833 | The degree of a compositio... |
dgrcolem2 23834 | Lemma for ~ dgrco . (Cont... |
dgrco 23835 | The degree of a compositio... |
plycjlem 23836 | Lemma for ~ plycj and ~ co... |
plycj 23837 | The double conjugation of ... |
coecj 23838 | Double conjugation of a po... |
plyrecj 23839 | A polynomial with real coe... |
plymul0or 23840 | Polynomial multiplication ... |
ofmulrt 23841 | The set of roots of a prod... |
plyreres 23842 | Real-coefficient polynomia... |
dvply1 23843 | Derivative of a polynomial... |
dvply2g 23844 | The derivative of a polyno... |
dvply2 23845 | The derivative of a polyno... |
dvnply2 23846 | Polynomials have polynomia... |
dvnply 23847 | Polynomials have polynomia... |
plycpn 23848 | Polynomials are smooth. (... |
quotval 23851 | Value of the quotient func... |
plydivlem1 23852 | Lemma for ~ plydivalg . (... |
plydivlem2 23853 | Lemma for ~ plydivalg . (... |
plydivlem3 23854 | Lemma for ~ plydivex . Ba... |
plydivlem4 23855 | Lemma for ~ plydivex . In... |
plydivex 23856 | Lemma for ~ plydivalg . (... |
plydiveu 23857 | Lemma for ~ plydivalg . (... |
plydivalg 23858 | The division algorithm on ... |
quotlem 23859 | Lemma for properties of th... |
quotcl 23860 | The quotient of two polyno... |
quotcl2 23861 | Closure of the quotient fu... |
quotdgr 23862 | Remainder property of the ... |
plyremlem 23863 | Closure of a linear factor... |
plyrem 23864 | The polynomial remainder t... |
facth 23865 | The factor theorem. If a ... |
fta1lem 23866 | Lemma for ~ fta1 . (Contr... |
fta1 23867 | The easy direction of the ... |
quotcan 23868 | Exact division with a mult... |
vieta1lem1 23869 | Lemma for ~ vieta1 . (Con... |
vieta1lem2 23870 | Lemma for ~ vieta1 : induc... |
vieta1 23871 | The first-order Vieta's fo... |
plyexmo 23872 | An infinite set of values ... |
elaa 23875 | Elementhood in the set of ... |
aacn 23876 | An algebraic number is a c... |
aasscn 23877 | The algebraic numbers are ... |
elqaalem1 23878 | Lemma for ~ elqaa . The f... |
elqaalem2 23879 | Lemma for ~ elqaa . (Cont... |
elqaalem3 23880 | Lemma for ~ elqaa . (Cont... |
elqaa 23881 | The set of numbers generat... |
qaa 23882 | Every rational number is a... |
qssaa 23883 | The rational numbers are c... |
iaa 23884 | The imaginary unit is alge... |
aareccl 23885 | The reciprocal of an algeb... |
aacjcl 23886 | The conjugate of an algebr... |
aannenlem1 23887 | Lemma for ~ aannen . (Con... |
aannenlem2 23888 | Lemma for ~ aannen . (Con... |
aannenlem3 23889 | The algebraic numbers are ... |
aannen 23890 | The algebraic numbers are ... |
aalioulem1 23891 | Lemma for ~ aaliou . An i... |
aalioulem2 23892 | Lemma for ~ aaliou . (Con... |
aalioulem3 23893 | Lemma for ~ aaliou . (Con... |
aalioulem4 23894 | Lemma for ~ aaliou . (Con... |
aalioulem5 23895 | Lemma for ~ aaliou . (Con... |
aalioulem6 23896 | Lemma for ~ aaliou . (Con... |
aaliou 23897 | Liouville's theorem on dio... |
geolim3 23898 | Geometric series convergen... |
aaliou2 23899 | Liouville's approximation ... |
aaliou2b 23900 | Liouville's approximation ... |
aaliou3lem1 23901 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem2 23902 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem3 23903 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem8 23904 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem4 23905 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem5 23906 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem6 23907 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem7 23908 | Lemma for ~ aaliou3 . (Co... |
aaliou3lem9 23909 | Example of a "Liouville nu... |
aaliou3 23910 | Example of a "Liouville nu... |
taylfvallem1 23915 | Lemma for ~ taylfval . (C... |
taylfvallem 23916 | Lemma for ~ taylfval . (C... |
taylfval 23917 | Define the Taylor polynomi... |
eltayl 23918 | Value of the Taylor series... |
taylf 23919 | The Taylor series defines ... |
tayl0 23920 | The Taylor series is alway... |
taylplem1 23921 | Lemma for ~ taylpfval and ... |
taylplem2 23922 | Lemma for ~ taylpfval and ... |
taylpfval 23923 | Define the Taylor polynomi... |
taylpf 23924 | The Taylor polynomial is a... |
taylpval 23925 | Value of the Taylor polyno... |
taylply2 23926 | The Taylor polynomial is a... |
taylply 23927 | The Taylor polynomial is a... |
dvtaylp 23928 | The derivative of the Tayl... |
dvntaylp 23929 | The ` M ` -th derivative o... |
dvntaylp0 23930 | The first ` N ` derivative... |
taylthlem1 23931 | Lemma for ~ taylth . This... |
taylthlem2 23932 | Lemma for ~ taylth . (Con... |
taylth 23933 | Taylor's theorem. The Tay... |
ulmrel 23936 | The uniform limit relation... |
ulmscl 23937 | Closure of the base set in... |
ulmval 23938 | Express the predicate: Th... |
ulmcl 23939 | Closure of a uniform limit... |
ulmf 23940 | Closure of a uniform limit... |
ulmpm 23941 | Closure of a uniform limit... |
ulmf2 23942 | Closure of a uniform limit... |
ulm2 23943 | Simplify ~ ulmval when ` F... |
ulmi 23944 | The uniform limit property... |
ulmclm 23945 | A uniform limit of functio... |
ulmres 23946 | A sequence of functions co... |
ulmshftlem 23947 | Lemma for ~ ulmshft . (Co... |
ulmshft 23948 | A sequence of functions co... |
ulm0 23949 | Every function converges u... |
ulmuni 23950 | An sequence of functions u... |
ulmdm 23951 | Two ways to express that a... |
ulmcaulem 23952 | Lemma for ~ ulmcau and ~ u... |
ulmcau 23953 | A sequence of functions co... |
ulmcau2 23954 | A sequence of functions co... |
ulmss 23955 | A uniform limit of functio... |
ulmbdd 23956 | A uniform limit of bounded... |
ulmcn 23957 | A uniform limit of continu... |
ulmdvlem1 23958 | Lemma for ~ ulmdv . (Cont... |
ulmdvlem2 23959 | Lemma for ~ ulmdv . (Cont... |
ulmdvlem3 23960 | Lemma for ~ ulmdv . (Cont... |
ulmdv 23961 | If ` F ` is a sequence of ... |
mtest 23962 | The Weierstrass M-test. I... |
mtestbdd 23963 | Given the hypotheses of th... |
mbfulm 23964 | A uniform limit of measura... |
iblulm 23965 | A uniform limit of integra... |
itgulm 23966 | A uniform limit of integra... |
itgulm2 23967 | A uniform limit of integra... |
pserval 23968 | Value of the function ` G ... |
pserval2 23969 | Value of the function ` G ... |
psergf 23970 | The sequence of terms in t... |
radcnvlem1 23971 | Lemma for ~ radcnvlt1 , ~ ... |
radcnvlem2 23972 | Lemma for ~ radcnvlt1 , ~ ... |
radcnvlem3 23973 | Lemma for ~ radcnvlt1 , ~ ... |
radcnv0 23974 | Zero is always a convergen... |
radcnvcl 23975 | The radius of convergence ... |
radcnvlt1 23976 | If ` X ` is within the ope... |
radcnvlt2 23977 | If ` X ` is within the ope... |
radcnvle 23978 | If ` X ` is a convergent p... |
dvradcnv 23979 | The radius of convergence ... |
pserulm 23980 | If ` S ` is a region conta... |
psercn2 23981 | Since by ~ pserulm the ser... |
psercnlem2 23982 | Lemma for ~ psercn . (Con... |
psercnlem1 23983 | Lemma for ~ psercn . (Con... |
psercn 23984 | An infinite series converg... |
pserdvlem1 23985 | Lemma for ~ pserdv . (Con... |
pserdvlem2 23986 | Lemma for ~ pserdv . (Con... |
pserdv 23987 | The derivative of a power ... |
pserdv2 23988 | The derivative of a power ... |
abelthlem1 23989 | Lemma for ~ abelth . (Con... |
abelthlem2 23990 | Lemma for ~ abelth . The ... |
abelthlem3 23991 | Lemma for ~ abelth . (Con... |
abelthlem4 23992 | Lemma for ~ abelth . (Con... |
abelthlem5 23993 | Lemma for ~ abelth . (Con... |
abelthlem6 23994 | Lemma for ~ abelth . (Con... |
abelthlem7a 23995 | Lemma for ~ abelth . (Con... |
abelthlem7 23996 | Lemma for ~ abelth . (Con... |
abelthlem8 23997 | Lemma for ~ abelth . (Con... |
abelthlem9 23998 | Lemma for ~ abelth . By a... |
abelth 23999 | Abel's theorem. If the po... |
abelth2 24000 | Abel's theorem, restricted... |
efcn 24001 | The exponential function i... |
sincn 24002 | Sine is continuous. (Cont... |
coscn 24003 | Cosine is continuous. (Co... |
reeff1olem 24004 | Lemma for ~ reeff1o . (Co... |
reeff1o 24005 | The real exponential funct... |
reefiso 24006 | The exponential function o... |
efcvx 24007 | The exponential function o... |
reefgim 24008 | The exponential function i... |
pilem1 24009 | Lemma for ~ pire , ~ pigt2... |
pilem2 24010 | Lemma for ~ pire , ~ pigt2... |
pilem3 24011 | Lemma for ~ pire , ~ pigt2... |
pigt2lt4 24012 | ` _pi ` is between 2 and 4... |
sinpi 24013 | The sine of ` _pi ` is 0. ... |
pire 24014 | ` _pi ` is a real number. ... |
picn 24015 | ` _pi ` is a complex numbe... |
pipos 24016 | ` _pi ` is positive. (Con... |
pirp 24017 | ` _pi ` is a positive real... |
negpicn 24018 | ` -u _pi ` is a real numbe... |
sinhalfpilem 24019 | Lemma for ~ sinhalfpi and ... |
halfpire 24020 | ` _pi / 2 ` is real. (Con... |
neghalfpire 24021 | ` -u _pi / 2 ` is real. (... |
neghalfpirx 24022 | ` -u _pi / 2 ` is an exten... |
pidiv2halves 24023 | Adding ` _pi / 2 ` to itse... |
sinhalfpi 24024 | The sine of ` _pi / 2 ` is... |
coshalfpi 24025 | The cosine of ` _pi / 2 ` ... |
cosneghalfpi 24026 | The cosine of ` -u _pi / 2... |
efhalfpi 24027 | The exponential of ` _i _p... |
cospi 24028 | The cosine of ` _pi ` is `... |
efipi 24029 | The exponential of ` _i x.... |
eulerid 24030 | Euler's identity. (Contri... |
sin2pi 24031 | The sine of ` 2 _pi ` is 0... |
cos2pi 24032 | The cosine of ` 2 _pi ` is... |
ef2pi 24033 | The exponential of ` 2 _pi... |
ef2kpi 24034 | The exponential of ` 2 K _... |
efper 24035 | The exponential function i... |
sinperlem 24036 | Lemma for ~ sinper and ~ c... |
sinper 24037 | The sine function is perio... |
cosper 24038 | The cosine function is per... |
sin2kpi 24039 | If ` K ` is an integer, th... |
cos2kpi 24040 | If ` K ` is an integer, th... |
sin2pim 24041 | Sine of a number subtracte... |
cos2pim 24042 | Cosine of a number subtrac... |
sinmpi 24043 | Sine of a number less ` _p... |
cosmpi 24044 | Cosine of a number less ` ... |
sinppi 24045 | Sine of a number plus ` _p... |
cosppi 24046 | Cosine of a complex number... |
efimpi 24047 | The exponential function o... |
sinhalfpip 24048 | The sine of ` _pi / 2 ` pl... |
sinhalfpim 24049 | The sine of ` _pi / 2 ` mi... |
coshalfpip 24050 | The cosine of ` _pi / 2 ` ... |
coshalfpim 24051 | The cosine of ` _pi / 2 ` ... |
ptolemy 24052 | Ptolemy's Theorem. This t... |
sincosq1lem 24053 | Lemma for ~ sincosq1sgn . ... |
sincosq1sgn 24054 | The signs of the sine and ... |
sincosq2sgn 24055 | The signs of the sine and ... |
sincosq3sgn 24056 | The signs of the sine and ... |
sincosq4sgn 24057 | The signs of the sine and ... |
coseq00topi 24058 | Location of the zeroes of ... |
coseq0negpitopi 24059 | Location of the zeroes of ... |
tanrpcl 24060 | Positive real closure of t... |
tangtx 24061 | The tangent function is gr... |
tanabsge 24062 | The tangent function is gr... |
sinq12gt0 24063 | The sine of a number stric... |
sinq12ge0 24064 | The sine of a number betwe... |
sinq34lt0t 24065 | The sine of a number stric... |
cosq14gt0 24066 | The cosine of a number str... |
cosq14ge0 24067 | The cosine of a number bet... |
sincosq1eq 24068 | Complementarity of the sin... |
sincos4thpi 24069 | The sine and cosine of ` _... |
tan4thpi 24070 | The tangent of ` _pi / 4 `... |
sincos6thpi 24071 | The sine and cosine of ` _... |
sincos3rdpi 24072 | The sine and cosine of ` _... |
pige3 24073 | ` _pi ` is greater or equa... |
abssinper 24074 | The absolute value of sine... |
sinkpi 24075 | The sine of an integer mul... |
coskpi 24076 | The absolute value of the ... |
sineq0 24077 | A complex number whose sin... |
coseq1 24078 | A complex number whose cos... |
efeq1 24079 | A complex number whose exp... |
cosne0 24080 | The cosine function has no... |
cosordlem 24081 | Lemma for ~ cosord . (Con... |
cosord 24082 | Cosine is decreasing over ... |
cos11 24083 | Cosine is one-to-one over ... |
sinord 24084 | Sine is increasing over th... |
recosf1o 24085 | The cosine function is a b... |
resinf1o 24086 | The sine function is a bij... |
tanord1 24087 | The tangent function is st... |
tanord 24088 | The tangent function is st... |
tanregt0 24089 | The positivity of ` tan ( ... |
negpitopissre 24090 | ` ( -u _pi (,] _pi ) ` is ... |
efgh 24091 | The exponential function o... |
efif1olem1 24092 | Lemma for ~ efif1o . (Con... |
efif1olem2 24093 | Lemma for ~ efif1o . (Con... |
efif1olem3 24094 | Lemma for ~ efif1o . (Con... |
efif1olem4 24095 | The exponential function o... |
efif1o 24096 | The exponential function o... |
efifo 24097 | The exponential function o... |
eff1olem 24098 | The exponential function m... |
eff1o 24099 | The exponential function m... |
efabl 24100 | The image of a subgroup of... |
efsubm 24101 | The image of a subgroup of... |
circgrp 24102 | The circle group ` T ` is ... |
circsubm 24103 | The circle group ` T ` is ... |
rzgrp 24104 | The quotient group R/Z is ... |
logrn 24109 | The range of the natural l... |
ellogrn 24110 | Write out the property ` A... |
dflog2 24111 | The natural logarithm func... |
relogrn 24112 | The range of the natural l... |
logrncn 24113 | The range of the natural l... |
eff1o2 24114 | The exponential function r... |
logf1o 24115 | The natural logarithm func... |
dfrelog 24116 | The natural logarithm func... |
relogf1o 24117 | The natural logarithm func... |
logrncl 24118 | Closure of the natural log... |
logcl 24119 | Closure of the natural log... |
logimcl 24120 | Closure of the imaginary p... |
logcld 24121 | The logarithm of a nonzero... |
logimcld 24122 | The imaginary part of the ... |
logimclad 24123 | The imaginary part of the ... |
abslogimle 24124 | The imaginary part of the ... |
logrnaddcl 24125 | The range of the natural l... |
relogcl 24126 | Closure of the natural log... |
eflog 24127 | Relationship between the n... |
logeq0im1 24128 | If the logarithm of a numb... |
logccne0 24129 | The logarithm isn't 0 if i... |
logne0 24130 | Logarithm of a non-1 posit... |
reeflog 24131 | Relationship between the n... |
logef 24132 | Relationship between the n... |
relogef 24133 | Relationship between the n... |
logeftb 24134 | Relationship between the n... |
relogeftb 24135 | Relationship between the n... |
log1 24136 | The natural logarithm of `... |
loge 24137 | The natural logarithm of `... |
logneg 24138 | The natural logarithm of a... |
logm1 24139 | The natural logarithm of n... |
lognegb 24140 | If a number has imaginary ... |
relogoprlem 24141 | Lemma for ~ relogmul and ~... |
relogmul 24142 | The natural logarithm of t... |
relogdiv 24143 | The natural logarithm of t... |
explog 24144 | Exponentiation of a nonzer... |
reexplog 24145 | Exponentiation of a positi... |
relogexp 24146 | The natural logarithm of p... |
relog 24147 | Real part of a logarithm. ... |
relogiso 24148 | The natural logarithm func... |
reloggim 24149 | The natural logarithm is a... |
logltb 24150 | The natural logarithm func... |
logfac 24151 | The logarithm of a factori... |
eflogeq 24152 | Solve an equation involvin... |
logleb 24153 | Natural logarithm preserve... |
rplogcl 24154 | Closure of the logarithm f... |
logge0 24155 | The logarithm of a number ... |
logcj 24156 | The natural logarithm dist... |
efiarg 24157 | The exponential of the "ar... |
cosargd 24158 | The cosine of the argument... |
cosarg0d 24159 | The cosine of the argument... |
argregt0 24160 | Closure of the argument of... |
argrege0 24161 | Closure of the argument of... |
argimgt0 24162 | Closure of the argument of... |
argimlt0 24163 | Closure of the argument of... |
logimul 24164 | Multiplying a number by ` ... |
logneg2 24165 | The logarithm of the negat... |
logmul2 24166 | Generalization of ~ relogm... |
logdiv2 24167 | Generalization of ~ relogd... |
abslogle 24168 | Bound on the magnitude of ... |
tanarg 24169 | The basic relation between... |
logdivlti 24170 | The ` log x / x ` function... |
logdivlt 24171 | The ` log x / x ` function... |
logdivle 24172 | The ` log x / x ` function... |
relogcld 24173 | Closure of the natural log... |
reeflogd 24174 | Relationship between the n... |
relogmuld 24175 | The natural logarithm of t... |
relogdivd 24176 | The natural logarithm of t... |
logled 24177 | Natural logarithm preserve... |
relogefd 24178 | Relationship between the n... |
rplogcld 24179 | Closure of the logarithm f... |
logge0d 24180 | The logarithm of a number ... |
divlogrlim 24181 | The inverse logarithm func... |
logno1 24182 | The logarithm function is ... |
dvrelog 24183 | The derivative of the real... |
relogcn 24184 | The real logarithm functio... |
ellogdm 24185 | Elementhood in the "contin... |
logdmn0 24186 | A number in the continuous... |
logdmnrp 24187 | A number in the continuous... |
logdmss 24188 | The continuity domain of `... |
logcnlem2 24189 | Lemma for ~ logcn . (Cont... |
logcnlem3 24190 | Lemma for ~ logcn . (Cont... |
logcnlem4 24191 | Lemma for ~ logcn . (Cont... |
logcnlem5 24192 | Lemma for ~ logcn . (Cont... |
logcn 24193 | The logarithm function is ... |
dvloglem 24194 | Lemma for ~ dvlog . (Cont... |
logdmopn 24195 | The "continuous domain" of... |
logf1o2 24196 | The logarithm maps its con... |
dvlog 24197 | The derivative of the comp... |
dvlog2lem 24198 | Lemma for ~ dvlog2 . (Con... |
dvlog2 24199 | The derivative of the comp... |
advlog 24200 | The antiderivative of the ... |
advlogexp 24201 | The antiderivative of a po... |
efopnlem1 24202 | Lemma for ~ efopn . (Cont... |
efopnlem2 24203 | Lemma for ~ efopn . (Cont... |
efopn 24204 | The exponential map is an ... |
logtayllem 24205 | Lemma for ~ logtayl . (Co... |
logtayl 24206 | The Taylor series for ` -u... |
logtaylsum 24207 | The Taylor series for ` -u... |
logtayl2 24208 | Power series expression fo... |
logccv 24209 | The natural logarithm func... |
cxpval 24210 | Value of the complex power... |
cxpef 24211 | Value of the complex power... |
0cxp 24212 | Value of the complex power... |
cxpexpz 24213 | Relate the complex power f... |
cxpexp 24214 | Relate the complex power f... |
logcxp 24215 | Logarithm of a complex pow... |
cxp0 24216 | Value of the complex power... |
cxp1 24217 | Value of the complex power... |
1cxp 24218 | Value of the complex power... |
ecxp 24219 | Write the exponential func... |
cxpcl 24220 | Closure of the complex pow... |
recxpcl 24221 | Real closure of the comple... |
rpcxpcl 24222 | Positive real closure of t... |
cxpne0 24223 | Complex exponentiation is ... |
cxpeq0 24224 | Complex exponentiation is ... |
cxpadd 24225 | Sum of exponents law for c... |
cxpp1 24226 | Value of a nonzero complex... |
cxpneg 24227 | Value of a complex number ... |
cxpsub 24228 | Exponent subtraction law f... |
cxpge0 24229 | Nonnegative exponentiation... |
mulcxplem 24230 | Lemma for ~ mulcxp . (Con... |
mulcxp 24231 | Complex exponentiation of ... |
cxprec 24232 | Complex exponentiation of ... |
divcxp 24233 | Complex exponentiation of ... |
cxpmul 24234 | Product of exponents law f... |
cxpmul2 24235 | Product of exponents law f... |
cxproot 24236 | The complex power function... |
cxpmul2z 24237 | Generalize ~ cxpmul2 to ne... |
abscxp 24238 | Absolute value of a power,... |
abscxp2 24239 | Absolute value of a power,... |
cxplt 24240 | Ordering property for comp... |
cxple 24241 | Ordering property for comp... |
cxplea 24242 | Ordering property for comp... |
cxple2 24243 | Ordering property for comp... |
cxplt2 24244 | Ordering property for comp... |
cxple2a 24245 | Ordering property for comp... |
cxplt3 24246 | Ordering property for comp... |
cxple3 24247 | Ordering property for comp... |
cxpsqrtlem 24248 | Lemma for ~ cxpsqrt . (Co... |
cxpsqrt 24249 | The complex exponential fu... |
logsqrt 24250 | Logarithm of a square root... |
cxp0d 24251 | Value of the complex power... |
cxp1d 24252 | Value of the complex power... |
1cxpd 24253 | Value of the complex power... |
cxpcld 24254 | Closure of the complex pow... |
cxpmul2d 24255 | Product of exponents law f... |
0cxpd 24256 | Value of the complex power... |
cxpexpzd 24257 | Relate the complex power f... |
cxpefd 24258 | Value of the complex power... |
cxpne0d 24259 | Complex exponentiation is ... |
cxpp1d 24260 | Value of a nonzero complex... |
cxpnegd 24261 | Value of a complex number ... |
cxpmul2zd 24262 | Generalize ~ cxpmul2 to ne... |
cxpaddd 24263 | Sum of exponents law for c... |
cxpsubd 24264 | Exponent subtraction law f... |
cxpltd 24265 | Ordering property for comp... |
cxpled 24266 | Ordering property for comp... |
cxplead 24267 | Ordering property for comp... |
divcxpd 24268 | Complex exponentiation of ... |
recxpcld 24269 | Positive real closure of t... |
cxpge0d 24270 | Nonnegative exponentiation... |
cxple2ad 24271 | Ordering property for comp... |
cxplt2d 24272 | Ordering property for comp... |
cxple2d 24273 | Ordering property for comp... |
mulcxpd 24274 | Complex exponentiation of ... |
cxprecd 24275 | Complex exponentiation of ... |
rpcxpcld 24276 | Positive real closure of t... |
logcxpd 24277 | Logarithm of a complex pow... |
cxplt3d 24278 | Ordering property for comp... |
cxple3d 24279 | Ordering property for comp... |
cxpmuld 24280 | Product of exponents law f... |
dvcxp1 24281 | The derivative of a comple... |
dvcxp2 24282 | The derivative of a comple... |
dvsqrt 24283 | The derivative of the real... |
dvcncxp1 24284 | Derivative of complex powe... |
dvcnsqrt 24285 | Derivative of square root ... |
cxpcn 24286 | Domain of continuity of th... |
cxpcn2 24287 | Continuity of the complex ... |
cxpcn3lem 24288 | Lemma for ~ cxpcn3 . (Con... |
cxpcn3 24289 | Extend continuity of the c... |
resqrtcn 24290 | Continuity of the real squ... |
sqrtcn 24291 | Continuity of the square r... |
cxpaddlelem 24292 | Lemma for ~ cxpaddle . (C... |
cxpaddle 24293 | Ordering property for comp... |
abscxpbnd 24294 | Bound on the absolute valu... |
root1id 24295 | Property of an ` N ` -th r... |
root1eq1 24296 | The only powers of an ` N ... |
root1cj 24297 | Within the ` N ` -th roots... |
cxpeq 24298 | Solve an equation involvin... |
loglesqrt 24299 | An upper bound on the loga... |
logreclem 24300 | Symmetry of the natural lo... |
logrec 24301 | Logarithm of a reciprocal ... |
logbval 24304 | Define the value of the ` ... |
logbcl 24305 | General logarithm closure.... |
logbid1 24306 | General logarithm is 1 whe... |
logb1 24307 | The logarithm of ` 1 ` to ... |
elogb 24308 | The general logarithm of a... |
logbchbase 24309 | Change of base for logarit... |
relogbval 24310 | Value of the general logar... |
relogbcl 24311 | Closure of the general log... |
relogbzcl 24312 | Closure of the general log... |
relogbreexp 24313 | Power law for the general ... |
relogbzexp 24314 | Power law for the general ... |
relogbmul 24315 | The logarithm of the produ... |
relogbmulexp 24316 | The logarithm of the produ... |
relogbdiv 24317 | The logarithm of the quoti... |
relogbexp 24318 | Identity law for general l... |
nnlogbexp 24319 | Identity law for general l... |
logbrec 24320 | Logarithm of a reciprocal ... |
logbleb 24321 | The general logarithm func... |
logblt 24322 | The general logarithm func... |
relogbcxp 24323 | Identity law for the gener... |
cxplogb 24324 | Identity law for the gener... |
relogbcxpb 24325 | The logarithm is the inver... |
logbmpt 24326 | The general logarithm to a... |
logbf 24327 | The general logarithm to a... |
logbfval 24328 | The general logarithm of a... |
relogbf 24329 | The general logarithm to a... |
logblog 24330 | The general logarithm to t... |
angval 24331 | Define the angle function,... |
angcan 24332 | Cancel a constant multipli... |
angneg 24333 | Cancel a negative sign in ... |
angvald 24334 | The (signed) angle between... |
angcld 24335 | The (signed) angle between... |
angrteqvd 24336 | Two vectors are at a right... |
cosangneg2d 24337 | The cosine of the angle be... |
angrtmuld 24338 | Perpendicularity of two ve... |
ang180lem1 24339 | Lemma for ~ ang180 . Show... |
ang180lem2 24340 | Lemma for ~ ang180 . Show... |
ang180lem3 24341 | Lemma for ~ ang180 . Sinc... |
ang180lem4 24342 | Lemma for ~ ang180 . Redu... |
ang180lem5 24343 | Lemma for ~ ang180 : Redu... |
ang180 24344 | The sum of angles ` m A B ... |
lawcoslem1 24345 | Lemma for ~ lawcos . Here... |
lawcos 24346 | Law of cosines (also known... |
pythag 24347 | Pythagorean theorem. Give... |
isosctrlem1 24348 | Lemma for ~ isosctr . (Co... |
isosctrlem2 24349 | Lemma for ~ isosctr . Cor... |
isosctrlem3 24350 | Lemma for ~ isosctr . Cor... |
isosctr 24351 | Isosceles triangle theorem... |
ssscongptld 24352 | If two triangles have equa... |
affineequiv 24353 | Equivalence between two wa... |
affineequiv2 24354 | Equivalence between two wa... |
angpieqvdlem 24355 | Equivalence used in the pr... |
angpieqvdlem2 24356 | Equivalence used in ~ angp... |
angpined 24357 | If the angle at ABC is ` _... |
angpieqvd 24358 | The angle ABC is ` _pi ` i... |
chordthmlem 24359 | If M is the midpoint of AB... |
chordthmlem2 24360 | If M is the midpoint of AB... |
chordthmlem3 24361 | If M is the midpoint of AB... |
chordthmlem4 24362 | If P is on the segment AB ... |
chordthmlem5 24363 | If P is on the segment AB ... |
chordthm 24364 | The intersecting chords th... |
heron 24365 | Heron's formula gives the ... |
quad2 24366 | The quadratic equation, wi... |
quad 24367 | The quadratic equation. (... |
1cubrlem 24368 | The cube roots of unity. ... |
1cubr 24369 | The cube roots of unity. ... |
dcubic1lem 24370 | Lemma for ~ dcubic1 and ~ ... |
dcubic2 24371 | Reverse direction of ~ dcu... |
dcubic1 24372 | Forward direction of ~ dcu... |
dcubic 24373 | Solutions to the depressed... |
mcubic 24374 | Solutions to a monic cubic... |
cubic2 24375 | The solution to the genera... |
cubic 24376 | The cubic equation, which ... |
binom4 24377 | Work out a quartic binomia... |
dquartlem1 24378 | Lemma for ~ dquart . (Con... |
dquartlem2 24379 | Lemma for ~ dquart . (Con... |
dquart 24380 | Solve a depressed quartic ... |
quart1cl 24381 | Closure lemmas for ~ quart... |
quart1lem 24382 | Lemma for ~ quart1 . (Con... |
quart1 24383 | Depress a quartic equation... |
quartlem1 24384 | Lemma for ~ quart . (Cont... |
quartlem2 24385 | Closure lemmas for ~ quart... |
quartlem3 24386 | Closure lemmas for ~ quart... |
quartlem4 24387 | Closure lemmas for ~ quart... |
quart 24388 | The quartic equation, writ... |
asinlem 24395 | The argument to the logari... |
asinlem2 24396 | The argument to the logari... |
asinlem3a 24397 | Lemma for ~ asinlem3 . (C... |
asinlem3 24398 | The argument to the logari... |
asinf 24399 | Domain and range of the ar... |
asincl 24400 | Closure for the arcsin fun... |
acosf 24401 | Domain and range of the ar... |
acoscl 24402 | Closure for the arccos fun... |
atandm 24403 | Since the property is a li... |
atandm2 24404 | This form of ~ atandm is a... |
atandm3 24405 | A compact form of ~ atandm... |
atandm4 24406 | A compact form of ~ atandm... |
atanf 24407 | Domain and range of the ar... |
atancl 24408 | Closure for the arctan fun... |
asinval 24409 | Value of the arcsin functi... |
acosval 24410 | Value of the arccos functi... |
atanval 24411 | Value of the arctan functi... |
atanre 24412 | A real number is in the do... |
asinneg 24413 | The arcsine function is od... |
acosneg 24414 | The negative symmetry rela... |
efiasin 24415 | The exponential of the arc... |
sinasin 24416 | The arcsine function is an... |
cosacos 24417 | The arccosine function is ... |
asinsinlem 24418 | Lemma for ~ asinsin . (Co... |
asinsin 24419 | The arcsine function compo... |
acoscos 24420 | The arccosine function is ... |
asin1 24421 | The arcsine of ` 1 ` is ` ... |
acos1 24422 | The arcsine of ` 1 ` is ` ... |
reasinsin 24423 | The arcsine function compo... |
asinsinb 24424 | Relationship between sine ... |
acoscosb 24425 | Relationship between sine ... |
asinbnd 24426 | The arcsine function has r... |
acosbnd 24427 | The arccosine function has... |
asinrebnd 24428 | Bounds on the arcsine func... |
asinrecl 24429 | The arcsine function is re... |
acosrecl 24430 | The arccosine function is ... |
cosasin 24431 | The cosine of the arcsine ... |
sinacos 24432 | The sine of the arccosine ... |
atandmneg 24433 | The domain of the arctange... |
atanneg 24434 | The arctangent function is... |
atan0 24435 | The arctangent of zero is ... |
atandmcj 24436 | The arctangent function di... |
atancj 24437 | The arctangent function di... |
atanrecl 24438 | The arctangent function is... |
efiatan 24439 | Value of the exponential o... |
atanlogaddlem 24440 | Lemma for ~ atanlogadd . ... |
atanlogadd 24441 | The rule ` sqrt ( z w ) = ... |
atanlogsublem 24442 | Lemma for ~ atanlogsub . ... |
atanlogsub 24443 | A variation on ~ atanlogad... |
efiatan2 24444 | Value of the exponential o... |
2efiatan 24445 | Value of the exponential o... |
tanatan 24446 | The arctangent function is... |
atandmtan 24447 | The tangent function has r... |
cosatan 24448 | The cosine of an arctangen... |
cosatanne0 24449 | The arctangent function ha... |
atantan 24450 | The arctangent function is... |
atantanb 24451 | Relationship between tange... |
atanbndlem 24452 | Lemma for ~ atanbnd . (Co... |
atanbnd 24453 | The arctangent function is... |
atanord 24454 | The arctangent function is... |
atan1 24455 | The arctangent of ` 1 ` is... |
bndatandm 24456 | A point in the open unit d... |
atans 24457 | The "domain of continuity"... |
atans2 24458 | It suffices to show that `... |
atansopn 24459 | The domain of continuity o... |
atansssdm 24460 | The domain of continuity o... |
ressatans 24461 | The real number line is a ... |
dvatan 24462 | The derivative of the arct... |
atancn 24463 | The arctangent is a contin... |
atantayl 24464 | The Taylor series for ` ar... |
atantayl2 24465 | The Taylor series for ` ar... |
atantayl3 24466 | The Taylor series for ` ar... |
leibpilem1 24467 | Lemma for ~ leibpi . (Con... |
leibpilem2 24468 | The Leibniz formula for ` ... |
leibpi 24469 | The Leibniz formula for ` ... |
leibpisum 24470 | The Leibniz formula for ` ... |
log2cnv 24471 | Using the Taylor series fo... |
log2tlbnd 24472 | Bound the error term in th... |
log2ublem1 24473 | Lemma for ~ log2ub . The ... |
log2ublem2 24474 | Lemma for ~ log2ub . (Con... |
log2ublem3 24475 | Lemma for ~ log2ub . In d... |
log2ub 24476 | ` log 2 ` is less than ` 2... |
log2le1 24477 | ` log 2 ` is less than ` 1... |
birthdaylem1 24478 | Lemma for ~ birthday . (C... |
birthdaylem2 24479 | For general ` N ` and ` K ... |
birthdaylem3 24480 | For general ` N ` and ` K ... |
birthday 24481 | The Birthday Problem. The... |
dmarea 24484 | The domain of the area fun... |
areambl 24485 | The fibers of a measurable... |
areass 24486 | A measurable region is a s... |
dfarea 24487 | Rewrite ~ df-area self-ref... |
areaf 24488 | Area measurement is a func... |
areacl 24489 | The area of a measurable r... |
areage0 24490 | The area of a measurable r... |
areaval 24491 | The area of a measurable r... |
rlimcnp 24492 | Relate a limit of a real-v... |
rlimcnp2 24493 | Relate a limit of a real-v... |
rlimcnp3 24494 | Relate a limit of a real-v... |
xrlimcnp 24495 | Relate a limit of a real-v... |
efrlim 24496 | The limit of the sequence ... |
dfef2 24497 | The limit of the sequence ... |
cxplim 24498 | A power to a negative expo... |
sqrtlim 24499 | The inverse square root fu... |
rlimcxp 24500 | Any power to a positive ex... |
o1cxp 24501 | An eventually bounded func... |
cxp2limlem 24502 | A linear factor grows slow... |
cxp2lim 24503 | Any power grows slower tha... |
cxploglim 24504 | The logarithm grows slower... |
cxploglim2 24505 | Every power of the logarit... |
divsqrtsumlem 24506 | Lemma for ~ divsqrsum and ... |
divsqrsumf 24507 | The function ` F ` used in... |
divsqrsum 24508 | The sum ` sum_ n <_ x ( 1 ... |
divsqrtsum2 24509 | A bound on the distance of... |
divsqrtsumo1 24510 | The sum ` sum_ n <_ x ( 1 ... |
cvxcl 24511 | Closure of a 0-1 linear co... |
scvxcvx 24512 | A strictly convex function... |
jensenlem1 24513 | Lemma for ~ jensen . (Con... |
jensenlem2 24514 | Lemma for ~ jensen . (Con... |
jensen 24515 | Jensen's inequality, a fin... |
amgmlem 24516 | Lemma for ~ amgm . (Contr... |
amgm 24517 | Inequality of arithmetic a... |
logdifbnd 24520 | Bound on the difference of... |
logdiflbnd 24521 | Lower bound on the differe... |
emcllem1 24522 | Lemma for ~ emcl . The se... |
emcllem2 24523 | Lemma for ~ emcl . ` F ` i... |
emcllem3 24524 | Lemma for ~ emcl . The fu... |
emcllem4 24525 | Lemma for ~ emcl . The di... |
emcllem5 24526 | Lemma for ~ emcl . The pa... |
emcllem6 24527 | Lemma for ~ emcl . By the... |
emcllem7 24528 | Lemma for ~ emcl and ~ har... |
emcl 24529 | Closure and bounds for the... |
harmonicbnd 24530 | A bound on the harmonic se... |
harmonicbnd2 24531 | A bound on the harmonic se... |
emre 24532 | The Euler-Mascheroni const... |
emgt0 24533 | The Euler-Mascheroni const... |
harmonicbnd3 24534 | A bound on the harmonic se... |
harmoniclbnd 24535 | A bound on the harmonic se... |
harmonicubnd 24536 | A bound on the harmonic se... |
harmonicbnd4 24537 | The asymptotic behavior of... |
fsumharmonic 24538 | Bound a finite sum based o... |
zetacvg 24541 | The zeta series is converg... |
eldmgm 24548 | Elementhood in the set of ... |
dmgmaddn0 24549 | If ` A ` is not a nonposit... |
dmlogdmgm 24550 | If ` A ` is in the continu... |
rpdmgm 24551 | A positive real number is ... |
dmgmn0 24552 | If ` A ` is not a nonposit... |
dmgmaddnn0 24553 | If ` A ` is not a nonposit... |
dmgmdivn0 24554 | Lemma for ~ lgamf . (Cont... |
lgamgulmlem1 24555 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem2 24556 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem3 24557 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem4 24558 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem5 24559 | Lemma for ~ lgamgulm . (C... |
lgamgulmlem6 24560 | The series ` G ` is unifor... |
lgamgulm 24561 | The series ` G ` is unifor... |
lgamgulm2 24562 | Rewrite the limit of the s... |
lgambdd 24563 | The log-Gamma function is ... |
lgamucov 24564 | The ` U ` regions used in ... |
lgamucov2 24565 | The ` U ` regions used in ... |
lgamcvglem 24566 | Lemma for ~ lgamf and ~ lg... |
lgamcl 24567 | The log-Gamma function is ... |
lgamf 24568 | The log-Gamma function is ... |
gamf 24569 | The Gamma function is a co... |
gamcl 24570 | The exponential of the log... |
eflgam 24571 | The exponential of the log... |
gamne0 24572 | The Gamma function is neve... |
igamval 24573 | Value of the inverse Gamma... |
igamz 24574 | Value of the inverse Gamma... |
igamgam 24575 | Value of the inverse Gamma... |
igamlgam 24576 | Value of the inverse Gamma... |
igamf 24577 | Closure of the inverse Gam... |
igamcl 24578 | Closure of the inverse Gam... |
gamigam 24579 | The Gamma function is the ... |
lgamcvg 24580 | The series ` G ` converges... |
lgamcvg2 24581 | The series ` G ` converges... |
gamcvg 24582 | The pointwise exponential ... |
lgamp1 24583 | The functional equation of... |
gamp1 24584 | The functional equation of... |
gamcvg2lem 24585 | Lemma for ~ gamcvg2 . (Co... |
gamcvg2 24586 | An infinite product expres... |
regamcl 24587 | The Gamma function is real... |
relgamcl 24588 | The log-Gamma function is ... |
rpgamcl 24589 | The log-Gamma function is ... |
lgam1 24590 | The log-Gamma function at ... |
gam1 24591 | The log-Gamma function at ... |
facgam 24592 | The Gamma function general... |
gamfac 24593 | The Gamma function general... |
wilthlem1 24594 | The only elements that are... |
wilthlem2 24595 | Lemma for ~ wilth : induct... |
wilthlem3 24596 | Lemma for ~ wilth . Here ... |
wilth 24597 | Wilson's theorem. A numbe... |
wilthimp 24598 | The forward implication of... |
ftalem1 24599 | Lemma for ~ fta : "growth... |
ftalem2 24600 | Lemma for ~ fta . There e... |
ftalem3 24601 | Lemma for ~ fta . There e... |
ftalem4 24602 | Lemma for ~ fta : Closure... |
ftalem5 24603 | Lemma for ~ fta : Main pr... |
ftalem6 24604 | Lemma for ~ fta : Dischar... |
ftalem7 24605 | Lemma for ~ fta . Shift t... |
fta 24606 | The Fundamental Theorem of... |
basellem1 24607 | Lemma for ~ basel . Closu... |
basellem2 24608 | Lemma for ~ basel . Show ... |
basellem3 24609 | Lemma for ~ basel . Using... |
basellem4 24610 | Lemma for ~ basel . By ~ ... |
basellem5 24611 | Lemma for ~ basel . Using... |
basellem6 24612 | Lemma for ~ basel . The f... |
basellem7 24613 | Lemma for ~ basel . The f... |
basellem8 24614 | Lemma for ~ basel . The f... |
basellem9 24615 | Lemma for ~ basel . Since... |
basel 24616 | The sum of the inverse squ... |
efnnfsumcl 24629 | Finite sum closure in the ... |
ppisval 24630 | The set of primes less tha... |
ppisval2 24631 | The set of primes less tha... |
ppifi 24632 | The set of primes less tha... |
prmdvdsfi 24633 | The set of prime divisors ... |
chtf 24634 | Domain and range of the Ch... |
chtcl 24635 | Real closure of the Chebys... |
chtval 24636 | Value of the Chebyshev fun... |
efchtcl 24637 | The Chebyshev function is ... |
chtge0 24638 | The Chebyshev function is ... |
vmaval 24639 | Value of the von Mangoldt ... |
isppw 24640 | Two ways to say that ` A `... |
isppw2 24641 | Two ways to say that ` A `... |
vmappw 24642 | Value of the von Mangoldt ... |
vmaprm 24643 | Value of the von Mangoldt ... |
vmacl 24644 | Closure for the von Mangol... |
vmaf 24645 | Functionality of the von M... |
efvmacl 24646 | The von Mangoldt is closed... |
vmage0 24647 | The von Mangoldt function ... |
chpval 24648 | Value of the second Chebys... |
chpf 24649 | Functionality of the secon... |
chpcl 24650 | Closure for the second Che... |
efchpcl 24651 | The second Chebyshev funct... |
chpge0 24652 | The second Chebyshev funct... |
ppival 24653 | Value of the prime-countin... |
ppival2 24654 | Value of the prime-countin... |
ppival2g 24655 | Value of the prime-countin... |
ppif 24656 | Domain and range of the pr... |
ppicl 24657 | Real closure of the prime-... |
muval 24658 | The value of the Möbi... |
muval1 24659 | The value of the Möbi... |
muval2 24660 | The value of the Möbi... |
isnsqf 24661 | Two ways to say that a num... |
issqf 24662 | Two ways to say that a num... |
sqfpc 24663 | The prime count of a squar... |
dvdssqf 24664 | A divisor of a squarefree ... |
sqf11 24665 | A squarefree number is com... |
muf 24666 | The Möbius function i... |
mucl 24667 | Closure of the Möbius... |
sgmval 24668 | The value of the divisor f... |
sgmval2 24669 | The value of the divisor f... |
0sgm 24670 | The value of the sum-of-di... |
sgmf 24671 | The divisor function is a ... |
sgmcl 24672 | Closure of the divisor fun... |
sgmnncl 24673 | Closure of the divisor fun... |
mule1 24674 | The Möbius function t... |
chtfl 24675 | The Chebyshev function doe... |
chpfl 24676 | The second Chebyshev funct... |
ppiprm 24677 | The prime-counting functio... |
ppinprm 24678 | The prime-counting functio... |
chtprm 24679 | The Chebyshev function at ... |
chtnprm 24680 | The Chebyshev function at ... |
chpp1 24681 | The second Chebyshev funct... |
chtwordi 24682 | The Chebyshev function is ... |
chpwordi 24683 | The second Chebyshev funct... |
chtdif 24684 | The difference of the Cheb... |
efchtdvds 24685 | The exponentiated Chebyshe... |
ppifl 24686 | The prime-counting functio... |
ppip1le 24687 | The prime-counting functio... |
ppiwordi 24688 | The prime-counting functio... |
ppidif 24689 | The difference of the prim... |
ppi1 24690 | The prime-counting functio... |
cht1 24691 | The Chebyshev function at ... |
vma1 24692 | The von Mangoldt function ... |
chp1 24693 | The second Chebyshev funct... |
ppi1i 24694 | Inference form of ~ ppiprm... |
ppi2i 24695 | Inference form of ~ ppinpr... |
ppi2 24696 | The prime-counting functio... |
ppi3 24697 | The prime-counting functio... |
cht2 24698 | The Chebyshev function at ... |
cht3 24699 | The Chebyshev function at ... |
ppinncl 24700 | Closure of the prime-count... |
chtrpcl 24701 | Closure of the Chebyshev f... |
ppieq0 24702 | The prime-counting functio... |
ppiltx 24703 | The prime-counting functio... |
prmorcht 24704 | Relate the primorial (prod... |
mumullem1 24705 | Lemma for ~ mumul . A mul... |
mumullem2 24706 | Lemma for ~ mumul . The p... |
mumul 24707 | The Möbius function i... |
sqff1o 24708 | There is a bijection from ... |
fsumdvdsdiaglem 24709 | A "diagonal commutation" o... |
fsumdvdsdiag 24710 | A "diagonal commutation" o... |
fsumdvdscom 24711 | A double commutation of di... |
dvdsppwf1o 24712 | A bijection from the divis... |
dvdsflf1o 24713 | A bijection from the numbe... |
dvdsflsumcom 24714 | A sum commutation from ` s... |
fsumfldivdiaglem 24715 | Lemma for ~ fsumfldivdiag ... |
fsumfldivdiag 24716 | The right-hand side of ~ d... |
musum 24717 | The sum of the Möbius... |
musumsum 24718 | Evaluate a collapsing sum ... |
muinv 24719 | The Möbius inversion ... |
dvdsmulf1o 24720 | If ` M ` and ` N ` are two... |
fsumdvdsmul 24721 | Product of two divisor sum... |
sgmppw 24722 | The value of the divisor f... |
0sgmppw 24723 | A prime power ` P ^ K ` ha... |
1sgmprm 24724 | The sum of divisors for a ... |
1sgm2ppw 24725 | The sum of the divisors of... |
sgmmul 24726 | The divisor function for f... |
ppiublem1 24727 | Lemma for ~ ppiub . (Cont... |
ppiublem2 24728 | A prime greater than ` 3 `... |
ppiub 24729 | An upper bound on the prim... |
vmalelog 24730 | The von Mangoldt function ... |
chtlepsi 24731 | The first Chebyshev functi... |
chprpcl 24732 | Closure of the second Cheb... |
chpeq0 24733 | The second Chebyshev funct... |
chteq0 24734 | The first Chebyshev functi... |
chtleppi 24735 | Upper bound on the ` theta... |
chtublem 24736 | Lemma for ~ chtub . (Cont... |
chtub 24737 | An upper bound on the Cheb... |
fsumvma 24738 | Rewrite a sum over the von... |
fsumvma2 24739 | Apply ~ fsumvma for the co... |
pclogsum 24740 | The logarithmic analogue o... |
vmasum 24741 | The sum of the von Mangold... |
logfac2 24742 | Another expression for the... |
chpval2 24743 | Express the second Chebysh... |
chpchtsum 24744 | The second Chebyshev funct... |
chpub 24745 | An upper bound on the seco... |
logfacubnd 24746 | A simple upper bound on th... |
logfaclbnd 24747 | A lower bound on the logar... |
logfacbnd3 24748 | Show the stronger statemen... |
logfacrlim 24749 | Combine the estimates ~ lo... |
logexprlim 24750 | The sum ` sum_ n <_ x , lo... |
logfacrlim2 24751 | Write out ~ logfacrlim as ... |
mersenne 24752 | A Mersenne prime is a prim... |
perfect1 24753 | Euclid's contribution to t... |
perfectlem1 24754 | Lemma for ~ perfect . (Co... |
perfectlem2 24755 | Lemma for ~ perfect . (Co... |
perfect 24756 | The Euclid-Euler theorem, ... |
dchrval 24759 | Value of the group of Diri... |
dchrbas 24760 | Base set of the group of D... |
dchrelbas 24761 | A Dirichlet character is a... |
dchrelbas2 24762 | A Dirichlet character is a... |
dchrelbas3 24763 | A Dirichlet character is a... |
dchrelbasd 24764 | A Dirichlet character is a... |
dchrrcl 24765 | Reverse closure for a Diri... |
dchrmhm 24766 | A Dirichlet character is a... |
dchrf 24767 | A Dirichlet character is a... |
dchrelbas4 24768 | A Dirichlet character is a... |
dchrzrh1 24769 | Value of a Dirichlet chara... |
dchrzrhcl 24770 | A Dirichlet character take... |
dchrzrhmul 24771 | A Dirichlet character is c... |
dchrplusg 24772 | Group operation on the gro... |
dchrmul 24773 | Group operation on the gro... |
dchrmulcl 24774 | Closure of the group opera... |
dchrn0 24775 | A Dirichlet character is n... |
dchr1cl 24776 | Closure of the principal D... |
dchrmulid2 24777 | Left identity for the prin... |
dchrinvcl 24778 | Closure of the group inver... |
dchrabl 24779 | The set of Dirichlet chara... |
dchrfi 24780 | The group of Dirichlet cha... |
dchrghm 24781 | A Dirichlet character rest... |
dchr1 24782 | Value of the principal Dir... |
dchreq 24783 | A Dirichlet character is d... |
dchrresb 24784 | A Dirichlet character is d... |
dchrabs 24785 | A Dirichlet character take... |
dchrinv 24786 | The inverse of a Dirichlet... |
dchrabs2 24787 | A Dirichlet character take... |
dchr1re 24788 | The principal Dirichlet ch... |
dchrptlem1 24789 | Lemma for ~ dchrpt . (Con... |
dchrptlem2 24790 | Lemma for ~ dchrpt . (Con... |
dchrptlem3 24791 | Lemma for ~ dchrpt . (Con... |
dchrpt 24792 | For any element other than... |
dchrsum2 24793 | An orthogonality relation ... |
dchrsum 24794 | An orthogonality relation ... |
sumdchr2 24795 | Lemma for ~ sumdchr . (Co... |
dchrhash 24796 | There are exactly ` phi ( ... |
sumdchr 24797 | An orthogonality relation ... |
dchr2sum 24798 | An orthogonality relation ... |
sum2dchr 24799 | An orthogonality relation ... |
bcctr 24800 | Value of the central binom... |
pcbcctr 24801 | Prime count of a central b... |
bcmono 24802 | The binomial coefficient i... |
bcmax 24803 | The binomial coefficient t... |
bcp1ctr 24804 | Ratio of two central binom... |
bclbnd 24805 | A bound on the binomial co... |
efexple 24806 | Convert a bound on a power... |
bpos1lem 24807 | Lemma for ~ bpos1 . (Cont... |
bpos1 24808 | Bertrand's postulate, chec... |
bposlem1 24809 | An upper bound on the prim... |
bposlem2 24810 | There are no odd primes in... |
bposlem3 24811 | Lemma for ~ bpos . Since ... |
bposlem4 24812 | Lemma for ~ bpos . (Contr... |
bposlem5 24813 | Lemma for ~ bpos . Bound ... |
bposlem6 24814 | Lemma for ~ bpos . By usi... |
bposlem7 24815 | Lemma for ~ bpos . The fu... |
bposlem8 24816 | Lemma for ~ bpos . Evalua... |
bposlem9 24817 | Lemma for ~ bpos . Derive... |
bpos 24818 | Bertrand's postulate: ther... |
zabsle1 24821 | ` { -u 1 , 0 , 1 } ` is th... |
lgslem1 24822 | When ` a ` is coprime to t... |
lgslem2 24823 | The set ` Z ` of all integ... |
lgslem3 24824 | The set ` Z ` of all integ... |
lgslem4 24825 | The function ` F ` is clos... |
lgsval 24826 | Value of the Legendre symb... |
lgsfval 24827 | Value of the function ` F ... |
lgsfcl2 24828 | The function ` F ` is clos... |
lgscllem 24829 | The Legendre symbol is an ... |
lgsfcl 24830 | Closure of the function ` ... |
lgsfle1 24831 | The function ` F ` has mag... |
lgsval2lem 24832 | Lemma for ~ lgsval2 . (Co... |
lgsval4lem 24833 | Lemma for ~ lgsval4 . (Co... |
lgscl2 24834 | The Legendre symbol is an ... |
lgs0 24835 | The Legendre symbol when t... |
lgscl 24836 | The Legendre symbol is an ... |
lgsle1 24837 | The Legendre symbol has ab... |
lgsval2 24838 | The Legendre symbol at a p... |
lgs2 24839 | The Legendre symbol at ` 2... |
lgsval3 24840 | The Legendre symbol at an ... |
lgsvalmod 24841 | The Legendre symbol is equ... |
lgsval4 24842 | Restate ~ lgsval for nonze... |
lgsfcl3 24843 | Closure of the function ` ... |
lgsval4a 24844 | Same as ~ lgsval4 for posi... |
lgscl1 24845 | The value of the Legendre ... |
lgsneg 24846 | The Legendre symbol is eit... |
lgsneg1 24847 | The Legendre symbol for no... |
lgsmod 24848 | The Legendre (Jacobi) symb... |
lgsdilem 24849 | Lemma for ~ lgsdi and ~ lg... |
lgsdir2lem1 24850 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2lem2 24851 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2lem3 24852 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2lem4 24853 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2lem5 24854 | Lemma for ~ lgsdir2 . (Co... |
lgsdir2 24855 | The Legendre symbol is com... |
lgsdirprm 24856 | The Legendre symbol is com... |
lgsdir 24857 | The Legendre symbol is com... |
lgsdilem2 24858 | Lemma for ~ lgsdi . (Cont... |
lgsdi 24859 | The Legendre symbol is com... |
lgsne0 24860 | The Legendre symbol is non... |
lgsabs1 24861 | The Legendre symbol is non... |
lgssq 24862 | The Legendre symbol at a s... |
lgssq2 24863 | The Legendre symbol at a s... |
lgsprme0 24864 | The Legendre symbol at any... |
1lgs 24865 | The Legendre symbol at ` 1... |
lgs1 24866 | The Legendre symbol at ` 1... |
lgsmodeq 24867 | The Legendre (Jacobi) symb... |
lgsmulsqcoprm 24868 | The Legendre (Jacobi) symb... |
lgsdirnn0 24869 | Variation on ~ lgsdir vali... |
lgsdinn0 24870 | Variation on ~ lgsdi valid... |
lgsqrlem1 24871 | Lemma for ~ lgsqr . (Cont... |
lgsqrlem2 24872 | Lemma for ~ lgsqr . (Cont... |
lgsqrlem3 24873 | Lemma for ~ lgsqr . (Cont... |
lgsqrlem4 24874 | Lemma for ~ lgsqr . (Cont... |
lgsqrlem5 24875 | Lemma for ~ lgsqr . (Cont... |
lgsqr 24876 | The Legendre symbol for od... |
lgsqrmod 24877 | If the Legendre symbol of ... |
lgsqrmodndvds 24878 | If the Legendre symbol of ... |
lgsdchrval 24879 | The Legendre symbol functi... |
lgsdchr 24880 | The Legendre symbol functi... |
gausslemma2dlem0a 24881 | Auxiliary lemma 1 for ~ ga... |
gausslemma2dlem0b 24882 | Auxiliary lemma 2 for ~ ga... |
gausslemma2dlem0c 24883 | Auxiliary lemma 3 for ~ ga... |
gausslemma2dlem0d 24884 | Auxiliary lemma 4 for ~ ga... |
gausslemma2dlem0e 24885 | Auxiliary lemma 5 for ~ ga... |
gausslemma2dlem0f 24886 | Auxiliary lemma 6 for ~ ga... |
gausslemma2dlem0g 24887 | Auxiliary lemma 7 for ~ ga... |
gausslemma2dlem0h 24888 | Auxiliary lemma 8 for ~ ga... |
gausslemma2dlem0i 24889 | Auxiliary lemma 9 for ~ ga... |
gausslemma2dlem1a 24890 | Lemma for ~ gausslemma2dle... |
gausslemma2dlem1 24891 | Lemma 1 for ~ gausslemma2d... |
gausslemma2dlem2 24892 | Lemma 2 for ~ gausslemma2d... |
gausslemma2dlem3 24893 | Lemma 3 for ~ gausslemma2d... |
gausslemma2dlem4 24894 | Lemma 4 for ~ gausslemma2d... |
gausslemma2dlem5a 24895 | Lemma for ~ gausslemma2dle... |
gausslemma2dlem5 24896 | Lemma 5 for ~ gausslemma2d... |
gausslemma2dlem6 24897 | Lemma 6 for ~ gausslemma2d... |
gausslemma2dlem7 24898 | Lemma 7 for ~ gausslemma2d... |
gausslemma2d 24899 | Gauss' Lemma (see also the... |
lgseisenlem1 24900 | Lemma for ~ lgseisen . If... |
lgseisenlem2 24901 | Lemma for ~ lgseisen . Th... |
lgseisenlem3 24902 | Lemma for ~ lgseisen . (C... |
lgseisenlem4 24903 | Lemma for ~ lgseisen . Th... |
lgseisen 24904 | Eisenstein's lemma, an exp... |
lgsquadlem1 24905 | Lemma for ~ lgsquad . Cou... |
lgsquadlem2 24906 | Lemma for ~ lgsquad . Cou... |
lgsquadlem3 24907 | Lemma for ~ lgsquad . (Co... |
lgsquad 24908 | The Law of Quadratic Recip... |
lgsquad2lem1 24909 | Lemma for ~ lgsquad2 . (C... |
lgsquad2lem2 24910 | Lemma for ~ lgsquad2 . (C... |
lgsquad2 24911 | Extend ~ lgsquad to coprim... |
lgsquad3 24912 | Extend ~ lgsquad2 to integ... |
m1lgs 24913 | The first supplement to th... |
2lgslem1a1 24914 | Lemma 1 for ~ 2lgslem1a . ... |
2lgslem1a2 24915 | Lemma 2 for ~ 2lgslem1a . ... |
2lgslem1a 24916 | Lemma 1 for ~ 2lgslem1 . ... |
2lgslem1b 24917 | Lemma 2 for ~ 2lgslem1 . ... |
2lgslem1c 24918 | Lemma 3 for ~ 2lgslem1 . ... |
2lgslem1 24919 | Lemma 1 for ~ 2lgs . (Con... |
2lgslem2 24920 | Lemma 2 for ~ 2lgs . (Con... |
2lgslem3a 24921 | Lemma for ~ 2lgslem3a1 . ... |
2lgslem3b 24922 | Lemma for ~ 2lgslem3b1 . ... |
2lgslem3c 24923 | Lemma for ~ 2lgslem3c1 . ... |
2lgslem3d 24924 | Lemma for ~ 2lgslem3d1 . ... |
2lgslem3a1 24925 | Lemma 1 for ~ 2lgslem3 . ... |
2lgslem3b1 24926 | Lemma 2 for ~ 2lgslem3 . ... |
2lgslem3c1 24927 | Lemma 3 for ~ 2lgslem3 . ... |
2lgslem3d1 24928 | Lemma 4 for ~ 2lgslem3 . ... |
2lgslem3 24929 | Lemma 3 for ~ 2lgs . (Con... |
2lgs2 24930 | The Legendre symbol for ` ... |
2lgslem4 24931 | Lemma 4 for ~ 2lgs : speci... |
2lgs 24932 | The second supplement to t... |
2lgsoddprmlem1 24933 | Lemma 1 for ~ 2lgsoddprm .... |
2lgsoddprmlem2 24934 | Lemma 2 for ~ 2lgsoddprm .... |
2lgsoddprmlem3a 24935 | Lemma 1 for ~ 2lgsoddprmle... |
2lgsoddprmlem3b 24936 | Lemma 2 for ~ 2lgsoddprmle... |
2lgsoddprmlem3c 24937 | Lemma 3 for ~ 2lgsoddprmle... |
2lgsoddprmlem3d 24938 | Lemma 4 for ~ 2lgsoddprmle... |
2lgsoddprmlem3 24939 | Lemma 3 for ~ 2lgsoddprm .... |
2lgsoddprmlem4 24940 | Lemma 4 for ~ 2lgsoddprm .... |
2lgsoddprm 24941 | The second supplement to t... |
2sqlem1 24942 | Lemma for ~ 2sq . (Contri... |
2sqlem2 24943 | Lemma for ~ 2sq . (Contri... |
mul2sq 24944 | Fibonacci's identity (actu... |
2sqlem3 24945 | Lemma for ~ 2sqlem5 . (Co... |
2sqlem4 24946 | Lemma for ~ 2sqlem5 . (Co... |
2sqlem5 24947 | Lemma for ~ 2sq . If a nu... |
2sqlem6 24948 | Lemma for ~ 2sq . If a nu... |
2sqlem7 24949 | Lemma for ~ 2sq . (Contri... |
2sqlem8a 24950 | Lemma for ~ 2sqlem8 . (Co... |
2sqlem8 24951 | Lemma for ~ 2sq . (Contri... |
2sqlem9 24952 | Lemma for ~ 2sq . (Contri... |
2sqlem10 24953 | Lemma for ~ 2sq . Every f... |
2sqlem11 24954 | Lemma for ~ 2sq . (Contri... |
2sq 24955 | All primes of the form ` 4... |
2sqblem 24956 | The converse to ~ 2sq . (... |
2sqb 24957 | The converse to ~ 2sq . (... |
chebbnd1lem1 24958 | Lemma for ~ chebbnd1 : sho... |
chebbnd1lem2 24959 | Lemma for ~ chebbnd1 : Sh... |
chebbnd1lem3 24960 | Lemma for ~ chebbnd1 : get... |
chebbnd1 24961 | The Chebyshev bound: The ... |
chtppilimlem1 24962 | Lemma for ~ chtppilim . (... |
chtppilimlem2 24963 | Lemma for ~ chtppilim . (... |
chtppilim 24964 | The ` theta ` function is ... |
chto1ub 24965 | The ` theta ` function is ... |
chebbnd2 24966 | The Chebyshev bound, part ... |
chto1lb 24967 | The ` theta ` function is ... |
chpchtlim 24968 | The ` psi ` and ` theta ` ... |
chpo1ub 24969 | The ` psi ` function is up... |
chpo1ubb 24970 | The ` psi ` function is up... |
vmadivsum 24971 | The sum of the von Mangold... |
vmadivsumb 24972 | Give a total bound on the ... |
rplogsumlem1 24973 | Lemma for ~ rplogsum . (C... |
rplogsumlem2 24974 | Lemma for ~ rplogsum . Eq... |
dchrisum0lem1a 24975 | Lemma for ~ dchrisum0lem1 ... |
rpvmasumlem 24976 | Lemma for ~ rpvmasum . Ca... |
dchrisumlema 24977 | Lemma for ~ dchrisum . Le... |
dchrisumlem1 24978 | Lemma for ~ dchrisum . Le... |
dchrisumlem2 24979 | Lemma for ~ dchrisum . Le... |
dchrisumlem3 24980 | Lemma for ~ dchrisum . Le... |
dchrisum 24981 | If ` n e. [ M , +oo ) |-> ... |
dchrmusumlema 24982 | Lemma for ~ dchrmusum and ... |
dchrmusum2 24983 | The sum of the Möbius... |
dchrvmasumlem1 24984 | An alternative expression ... |
dchrvmasum2lem 24985 | Give an expression for ` l... |
dchrvmasum2if 24986 | Combine the results of ~ d... |
dchrvmasumlem2 24987 | Lemma for ~ dchrvmasum . ... |
dchrvmasumlem3 24988 | Lemma for ~ dchrvmasum . ... |
dchrvmasumlema 24989 | Lemma for ~ dchrvmasum and... |
dchrvmasumiflem1 24990 | Lemma for ~ dchrvmasumif .... |
dchrvmasumiflem2 24991 | Lemma for ~ dchrvmasum . ... |
dchrvmasumif 24992 | An asymptotic approximatio... |
dchrvmaeq0 24993 | The set ` W ` is the colle... |
dchrisum0fval 24994 | Value of the function ` F ... |
dchrisum0fmul 24995 | The function ` F ` , the d... |
dchrisum0ff 24996 | The function ` F ` is a re... |
dchrisum0flblem1 24997 | Lemma for ~ dchrisum0flb .... |
dchrisum0flblem2 24998 | Lemma for ~ dchrisum0flb .... |
dchrisum0flb 24999 | The divisor sum of a real ... |
dchrisum0fno1 25000 | The sum ` sum_ k <_ x , F ... |
rpvmasum2 25001 | A partial result along the... |
dchrisum0re 25002 | Suppose ` X ` is a non-pri... |
dchrisum0lema 25003 | Lemma for ~ dchrisum0 . A... |
dchrisum0lem1b 25004 | Lemma for ~ dchrisum0lem1 ... |
dchrisum0lem1 25005 | Lemma for ~ dchrisum0 . (... |
dchrisum0lem2a 25006 | Lemma for ~ dchrisum0 . (... |
dchrisum0lem2 25007 | Lemma for ~ dchrisum0 . (... |
dchrisum0lem3 25008 | Lemma for ~ dchrisum0 . (... |
dchrisum0 25009 | The sum ` sum_ n e. NN , X... |
dchrisumn0 25010 | The sum ` sum_ n e. NN , X... |
dchrmusumlem 25011 | The sum of the Möbius... |
dchrvmasumlem 25012 | The sum of the Möbius... |
dchrmusum 25013 | The sum of the Möbius... |
dchrvmasum 25014 | The sum of the von Mangold... |
rpvmasum 25015 | The sum of the von Mangold... |
rplogsum 25016 | The sum of ` log p / p ` o... |
dirith2 25017 | Dirichlet's theorem: there... |
dirith 25018 | Dirichlet's theorem: there... |
mudivsum 25019 | Asymptotic formula for ` s... |
mulogsumlem 25020 | Lemma for ~ mulogsum . (C... |
mulogsum 25021 | Asymptotic formula for ... |
logdivsum 25022 | Asymptotic analysis of ... |
mulog2sumlem1 25023 | Asymptotic formula for ... |
mulog2sumlem2 25024 | Lemma for ~ mulog2sum . (... |
mulog2sumlem3 25025 | Lemma for ~ mulog2sum . (... |
mulog2sum 25026 | Asymptotic formula for ... |
vmalogdivsum2 25027 | The sum ` sum_ n <_ x , La... |
vmalogdivsum 25028 | The sum ` sum_ n <_ x , La... |
2vmadivsumlem 25029 | Lemma for ~ 2vmadivsum . ... |
2vmadivsum 25030 | The sum ` sum_ m n <_ x , ... |
logsqvma 25031 | A formula for ` log ^ 2 ( ... |
logsqvma2 25032 | The Möbius inverse of... |
log2sumbnd 25033 | Bound on the difference be... |
selberglem1 25034 | Lemma for ~ selberg . Est... |
selberglem2 25035 | Lemma for ~ selberg . (Co... |
selberglem3 25036 | Lemma for ~ selberg . Est... |
selberg 25037 | Selberg's symmetry formula... |
selbergb 25038 | Convert eventual boundedne... |
selberg2lem 25039 | Lemma for ~ selberg2 . Eq... |
selberg2 25040 | Selberg's symmetry formula... |
selberg2b 25041 | Convert eventual boundedne... |
chpdifbndlem1 25042 | Lemma for ~ chpdifbnd . (... |
chpdifbndlem2 25043 | Lemma for ~ chpdifbnd . (... |
chpdifbnd 25044 | A bound on the difference ... |
logdivbnd 25045 | A bound on a sum of logs, ... |
selberg3lem1 25046 | Introduce a log weighting ... |
selberg3lem2 25047 | Lemma for ~ selberg3 . Eq... |
selberg3 25048 | Introduce a log weighting ... |
selberg4lem1 25049 | Lemma for ~ selberg4 . Eq... |
selberg4 25050 | The Selberg symmetry formu... |
pntrval 25051 | Define the residual of the... |
pntrf 25052 | Functionality of the resid... |
pntrmax 25053 | There is a bound on the re... |
pntrsumo1 25054 | A bound on a sum over ` R ... |
pntrsumbnd 25055 | A bound on a sum over ` R ... |
pntrsumbnd2 25056 | A bound on a sum over ` R ... |
selbergr 25057 | Selberg's symmetry formula... |
selberg3r 25058 | Selberg's symmetry formula... |
selberg4r 25059 | Selberg's symmetry formula... |
selberg34r 25060 | The sum of ~ selberg3r and... |
pntsval 25061 | Define the "Selberg functi... |
pntsf 25062 | Functionality of the Selbe... |
selbergs 25063 | Selberg's symmetry formula... |
selbergsb 25064 | Selberg's symmetry formula... |
pntsval2 25065 | The Selberg function can b... |
pntrlog2bndlem1 25066 | The sum of ~ selberg3r and... |
pntrlog2bndlem2 25067 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bndlem3 25068 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bndlem4 25069 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bndlem5 25070 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bndlem6a 25071 | Lemma for ~ pntrlog2bndlem... |
pntrlog2bndlem6 25072 | Lemma for ~ pntrlog2bnd . ... |
pntrlog2bnd 25073 | A bound on ` R ( x ) log ^... |
pntpbnd1a 25074 | Lemma for ~ pntpbnd . (Co... |
pntpbnd1 25075 | Lemma for ~ pntpbnd . (Co... |
pntpbnd2 25076 | Lemma for ~ pntpbnd . (Co... |
pntpbnd 25077 | Lemma for ~ pnt . Establi... |
pntibndlem1 25078 | Lemma for ~ pntibnd . (Co... |
pntibndlem2a 25079 | Lemma for ~ pntibndlem2 . ... |
pntibndlem2 25080 | Lemma for ~ pntibnd . The... |
pntibndlem3 25081 | Lemma for ~ pntibnd . Pac... |
pntibnd 25082 | Lemma for ~ pnt . Establi... |
pntlemd 25083 | Lemma for ~ pnt . Closure... |
pntlemc 25084 | Lemma for ~ pnt . Closure... |
pntlema 25085 | Lemma for ~ pnt . Closure... |
pntlemb 25086 | Lemma for ~ pnt . Unpack ... |
pntlemg 25087 | Lemma for ~ pnt . Closure... |
pntlemh 25088 | Lemma for ~ pnt . Bounds ... |
pntlemn 25089 | Lemma for ~ pnt . The "na... |
pntlemq 25090 | Lemma for ~ pntlemj . (Co... |
pntlemr 25091 | Lemma for ~ pntlemj . (Co... |
pntlemj 25092 | Lemma for ~ pnt . The ind... |
pntlemi 25093 | Lemma for ~ pnt . Elimina... |
pntlemf 25094 | Lemma for ~ pnt . Add up ... |
pntlemk 25095 | Lemma for ~ pnt . Evaluat... |
pntlemo 25096 | Lemma for ~ pnt . Combine... |
pntleme 25097 | Lemma for ~ pnt . Package... |
pntlem3 25098 | Lemma for ~ pnt . Equatio... |
pntlemp 25099 | Lemma for ~ pnt . Wrappin... |
pntleml 25100 | Lemma for ~ pnt . Equatio... |
pnt3 25101 | The Prime Number Theorem, ... |
pnt2 25102 | The Prime Number Theorem, ... |
pnt 25103 | The Prime Number Theorem: ... |
abvcxp 25104 | Raising an absolute value ... |
padicfval 25105 | Value of the p-adic absolu... |
padicval 25106 | Value of the p-adic absolu... |
ostth2lem1 25107 | Lemma for ~ ostth2 , altho... |
qrngbas 25108 | The base set of the field ... |
qdrng 25109 | The rationals form a divis... |
qrng0 25110 | The zero element of the fi... |
qrng1 25111 | The unit element of the fi... |
qrngneg 25112 | The additive inverse in th... |
qrngdiv 25113 | The division operation in ... |
qabvle 25114 | By using induction on ` N ... |
qabvexp 25115 | Induct the product rule ~ ... |
ostthlem1 25116 | Lemma for ~ ostth . If tw... |
ostthlem2 25117 | Lemma for ~ ostth . Refin... |
qabsabv 25118 | The regular absolute value... |
padicabv 25119 | The p-adic absolute value ... |
padicabvf 25120 | The p-adic absolute value ... |
padicabvcxp 25121 | All positive powers of the... |
ostth1 25122 | - Lemma for ~ ostth : triv... |
ostth2lem2 25123 | Lemma for ~ ostth2 . (Con... |
ostth2lem3 25124 | Lemma for ~ ostth2 . (Con... |
ostth2lem4 25125 | Lemma for ~ ostth2 . (Con... |
ostth2 25126 | - Lemma for ~ ostth : regu... |
ostth3 25127 | - Lemma for ~ ostth : p-ad... |
ostth 25128 | Ostrowski's theorem, which... |
itvndx 25139 | Index value of the Interva... |
lngndx 25140 | Index value of the "line" ... |
itvid 25141 | Utility theorem: index-ind... |
lngid 25142 | Utility theorem: index-ind... |
trkgstr 25143 | Functionality of a Tarski ... |
trkgbas 25144 | The base set of a Tarski g... |
trkgdist 25145 | The measure of a distance ... |
trkgitv 25146 | The congruence relation in... |
istrkgc 25153 | Property of being a Tarski... |
istrkgb 25154 | Property of being a Tarski... |
istrkgcb 25155 | Property of being a Tarski... |
istrkge 25156 | Property of fulfilling Euc... |
istrkgl 25157 | Building lines from the se... |
istrkgld 25158 | Property of fulfilling the... |
istrkg2ld 25159 | Property of fulfilling the... |
istrkg3ld 25160 | Property of fulfilling the... |
axtgcgrrflx 25161 | Axiom of reflexivity of co... |
axtgcgrid 25162 | Axiom of identity of congr... |
axtgsegcon 25163 | Axiom of segment construct... |
axtg5seg 25164 | Five segments axiom, Axiom... |
axtgbtwnid 25165 | Identity of Betweenness. ... |
axtgpasch 25166 | Axiom of (Inner) Pasch, Ax... |
axtgcont1 25167 | Axiom of Continuity. Axio... |
axtgcont 25168 | Axiom of Continuity. Axio... |
axtglowdim2 25169 | Lower dimension axiom for ... |
axtgupdim2 25170 | Upper dimension axiom for ... |
axtgeucl 25171 | Euclid's Axiom. Axiom A10... |
tgcgrcomimp 25172 | Congruence commutes on the... |
tgcgrcomr 25173 | Congruence commutes on the... |
tgcgrcoml 25174 | Congruence commutes on the... |
tgcgrcomlr 25175 | Congruence commutes on bot... |
tgcgreqb 25176 | Congruence and equality. ... |
tgcgreq 25177 | Congruence and equality. ... |
tgcgrneq 25178 | Congruence and equality. ... |
tgcgrtriv 25179 | Degenerate segments are co... |
tgcgrextend 25180 | Link congruence over a pai... |
tgsegconeq 25181 | Two points that satisfy th... |
tgbtwntriv2 25182 | Betweenness always holds f... |
tgbtwncom 25183 | Betweenness commutes. The... |
tgbtwncomb 25184 | Betweenness commutes, bico... |
tgbtwnne 25185 | Betweenness and inequality... |
tgbtwntriv1 25186 | Betweenness always holds f... |
tgbtwnswapid 25187 | If you can swap the first ... |
tgbtwnintr 25188 | Inner transitivity law for... |
tgbtwnexch3 25189 | Exchange the first endpoin... |
tgbtwnouttr2 25190 | Outer transitivity law for... |
tgbtwnexch2 25191 | Exchange the outer point o... |
tgbtwnouttr 25192 | Outer transitivity law for... |
tgbtwnexch 25193 | Outer transitivity law for... |
tgtrisegint 25194 | A line segment between two... |
tglowdim1 25195 | Lower dimension axiom for ... |
tglowdim1i 25196 | Lower dimension axiom for ... |
tgldimor 25197 | Excluded-middle like state... |
tgldim0eq 25198 | In dimension zero, any two... |
tgldim0itv 25199 | In dimension zero, any two... |
tgldim0cgr 25200 | In dimension zero, any two... |
tgbtwndiff 25201 | There is always a ` c ` di... |
tgdim01 25202 | In geometries of dimension... |
tgifscgr 25203 | Inner five segment congrue... |
tgcgrsub 25204 | Removing identical parts f... |
iscgrg 25207 | The congruence property fo... |
iscgrgd 25208 | The property for two seque... |
iscgrglt 25209 | The property for two seque... |
trgcgrg 25210 | The property for two trian... |
trgcgr 25211 | Triangle congruence. (Con... |
ercgrg 25212 | The shape congruence relat... |
tgcgrxfr 25213 | A line segment can be divi... |
cgr3id 25214 | Reflexivity law for three-... |
cgr3simp1 25215 | Deduce segment congruence ... |
cgr3simp2 25216 | Deduce segment congruence ... |
cgr3simp3 25217 | Deduce segment congruence ... |
cgr3swap12 25218 | Permutation law for three-... |
cgr3swap23 25219 | Permutation law for three-... |
cgr3swap13 25220 | Permutation law for three-... |
cgr3rotr 25221 | Permutation law for three-... |
cgr3rotl 25222 | Permutation law for three-... |
trgcgrcom 25223 | Commutative law for three-... |
cgr3tr 25224 | Transitivity law for three... |
tgbtwnxfr 25225 | A condition for extending ... |
tgcgr4 25226 | Two quadrilaterals to be c... |
isismt 25229 | Property of being an isome... |
ismot 25230 | Property of being an isome... |
motcgr 25231 | Property of a motion: dist... |
idmot 25232 | The identity is a motion. ... |
motf1o 25233 | Motions are bijections. (... |
motcl 25234 | Closure of motions. (Cont... |
motco 25235 | The composition of two mot... |
cnvmot 25236 | The converse of a motion i... |
motplusg 25237 | The operation for motions ... |
motgrp 25238 | The motions of a geometry ... |
motcgrg 25239 | Property of a motion: dist... |
motcgr3 25240 | Property of a motion: dist... |
tglng 25241 | Lines of a Tarski Geometry... |
tglnfn 25242 | Lines as functions. (Cont... |
tglnunirn 25243 | Lines are sets of points. ... |
tglnpt 25244 | Lines are sets of points. ... |
tglngne 25245 | It takes two different poi... |
tglngval 25246 | The line going through poi... |
tglnssp 25247 | Lines are subset of the ge... |
tgellng 25248 | Property of lying on the l... |
tgcolg 25249 | We choose the notation ` (... |
btwncolg1 25250 | Betweenness implies coline... |
btwncolg2 25251 | Betweenness implies coline... |
btwncolg3 25252 | Betweenness implies coline... |
colcom 25253 | Swapping the points defini... |
colrot1 25254 | Rotating the points defini... |
colrot2 25255 | Rotating the points defini... |
ncolcom 25256 | Swapping non-colinear poin... |
ncolrot1 25257 | Rotating non-colinear poin... |
ncolrot2 25258 | Rotating non-colinear poin... |
tgdim01ln 25259 | In geometries of dimension... |
ncoltgdim2 25260 | If there are 3 non-colinea... |
lnxfr 25261 | Transfer law for colineari... |
lnext 25262 | Extend a line with a missi... |
tgfscgr 25263 | Congruence law for the gen... |
lncgr 25264 | Congruence rule for lines.... |
lnid 25265 | Identity law for points on... |
tgidinside 25266 | Law for finding a point in... |
tgbtwnconn1lem1 25267 | Lemma for ~ tgbtwnconn1 . ... |
tgbtwnconn1lem2 25268 | Lemma for ~ tgbtwnconn1 . ... |
tgbtwnconn1lem3 25269 | Lemma for ~ tgbtwnconn1 . ... |
tgbtwnconn1 25270 | Connectivity law for betwe... |
tgbtwnconn2 25271 | Another connectivity law f... |
tgbtwnconn3 25272 | Inner connectivity law for... |
tgbtwnconnln3 25273 | Derive colinearity from be... |
tgbtwnconn22 25274 | Double connectivity law fo... |
tgbtwnconnln1 25275 | Derive colinearity from be... |
tgbtwnconnln2 25276 | Derive colinearity from be... |
legval 25279 | Value of the less-than rel... |
legov 25280 | Value of the less-than rel... |
legov2 25281 | An equivalent definition o... |
legid 25282 | Reflexivity of the less-th... |
btwnleg 25283 | Betweenness implies less-t... |
legtrd 25284 | Transitivity of the less-t... |
legtri3 25285 | Equality from the less-tha... |
legtrid 25286 | Trichotomy law for the les... |
leg0 25287 | Degenerated (zero-length) ... |
legeq 25288 | Deduce equality from "less... |
legbtwn 25289 | Deduce betweenness from "l... |
tgcgrsub2 25290 | Removing identical parts f... |
ltgseg 25291 | The set ` E ` denotes the ... |
ltgov 25292 | Strict "shorter than" geom... |
legov3 25293 | An equivalent definition o... |
legso 25294 | The shorter-than relations... |
ishlg 25297 | Rays : Definition 6.1 of ... |
hlcomb 25298 | The half-line relation com... |
hlcomd 25299 | The half-line relation com... |
hlne1 25300 | The half-line relation imp... |
hlne2 25301 | The half-line relation imp... |
hlln 25302 | The half-line relation imp... |
hleqnid 25303 | The endpoint does not belo... |
hlid 25304 | The half-line relation is ... |
hltr 25305 | The half-line relation is ... |
hlbtwn 25306 | Betweenness is a sufficien... |
btwnhl1 25307 | Deduce half-line from betw... |
btwnhl2 25308 | Deduce half-line from betw... |
btwnhl 25309 | Swap betweenness for a hal... |
lnhl 25310 | Either a point ` C ` on th... |
hlcgrex 25311 | Construct a point on a hal... |
hlcgreulem 25312 | Lemma for ~ hlcgreu . (Co... |
hlcgreu 25313 | The point constructed in ~... |
btwnlng1 25314 | Betweenness implies coline... |
btwnlng2 25315 | Betweenness implies coline... |
btwnlng3 25316 | Betweenness implies coline... |
lncom 25317 | Swapping the points defini... |
lnrot1 25318 | Rotating the points defini... |
lnrot2 25319 | Rotating the points defini... |
ncolne1 25320 | Non-colinear points are di... |
ncolne2 25321 | Non-colinear points are di... |
tgisline 25322 | The property of being a pr... |
tglnne 25323 | It takes two different poi... |
tglndim0 25324 | There are no lines in dime... |
tgelrnln 25325 | The property of being a pr... |
tglineeltr 25326 | Transitivity law for lines... |
tglineelsb2 25327 | If ` S ` lies on PQ , then... |
tglinerflx1 25328 | Reflexivity law for line m... |
tglinerflx2 25329 | Reflexivity law for line m... |
tglinecom 25330 | Commutativity law for line... |
tglinethru 25331 | If ` A ` is a line contain... |
tghilberti1 25332 | There is a line through an... |
tghilberti2 25333 | There is at most one line ... |
tglinethrueu 25334 | There is a unique line goi... |
tglnne0 25335 | A line ` A ` has at least ... |
tglnpt2 25336 | Find a second point on a l... |
tglineintmo 25337 | Two distinct lines interse... |
tglineineq 25338 | Two distinct lines interse... |
tglineneq 25339 | Given three non-colinear p... |
tglineinteq 25340 | Two distinct lines interse... |
ncolncol 25341 | Deduce non-colinearity fro... |
coltr 25342 | A transitivity law for col... |
coltr3 25343 | A transitivity law for col... |
colline 25344 | Three points are colinear ... |
tglowdim2l 25345 | Reformulation of the lower... |
tglowdim2ln 25346 | There is always one point ... |
mirreu3 25349 | Existential uniqueness of ... |
mirval 25350 | Value of the point inversi... |
mirfv 25351 | Value of the point inversi... |
mircgr 25352 | Property of the image by t... |
mirbtwn 25353 | Property of the image by t... |
ismir 25354 | Property of the image by t... |
mirf 25355 | Point inversion as functio... |
mircl 25356 | Closure of the point inver... |
mirmir 25357 | The point inversion functi... |
mircom 25358 | Variation on ~ mirmir . (... |
mirreu 25359 | Any point has a unique ant... |
mireq 25360 | Equality deduction for poi... |
mirinv 25361 | The only invariant point o... |
mirne 25362 | Mirror of non-center point... |
mircinv 25363 | The center point is invari... |
mirf1o 25364 | The point inversion functi... |
miriso 25365 | The point inversion functi... |
mirbtwni 25366 | Point inversion preserves ... |
mirbtwnb 25367 | Point inversion preserves ... |
mircgrs 25368 | Point inversion preserves ... |
mirmir2 25369 | Point inversion of a point... |
mirmot 25370 | Point investion is a motio... |
mirln 25371 | If two points are on the s... |
mirln2 25372 | If a point and its mirror ... |
mirconn 25373 | Point inversion of connect... |
mirhl 25374 | If two points ` X ` and ` ... |
mirbtwnhl 25375 | If the center of the point... |
mirhl2 25376 | Deduce half-line relation ... |
mircgrextend 25377 | Link congruence over a pai... |
mirtrcgr 25378 | Point inversion of one poi... |
mirauto 25379 | Point inversion preserves ... |
miduniq 25380 | Unicity of the middle poin... |
miduniq1 25381 | Unicity of the middle poin... |
miduniq2 25382 | If two point inversions co... |
colmid 25383 | Colinearity and equidistan... |
symquadlem 25384 | Lemma of the symetrial qua... |
krippenlem 25385 | Lemma for ~ krippen . We ... |
krippen 25386 | Krippenlemma (German for c... |
midexlem 25387 | Lemma for the existence of... |
israg 25392 | Property for 3 points A, B... |
ragcom 25393 | Commutative rule for right... |
ragcol 25394 | The right angle property i... |
ragmir 25395 | Right angle property is pr... |
mirrag 25396 | Right angle is conserved b... |
ragtrivb 25397 | Trivial right angle. Theo... |
ragflat2 25398 | Deduce equality from two r... |
ragflat 25399 | Deduce equality from two r... |
ragtriva 25400 | Trivial right angle. Theo... |
ragflat3 25401 | Right angle and colinearit... |
ragcgr 25402 | Right angle and colinearit... |
motrag 25403 | Right angles are preserved... |
ragncol 25404 | Right angle implies non-co... |
perpln1 25405 | Derive a line from perpend... |
perpln2 25406 | Derive a line from perpend... |
isperp 25407 | Property for 2 lines A, B ... |
perpcom 25408 | The "perpendicular" relati... |
perpneq 25409 | Two perpendicular lines ar... |
isperp2 25410 | Property for 2 lines A, B,... |
isperp2d 25411 | One direction of ~ isperp2... |
ragperp 25412 | Deduce that two lines are ... |
footex 25413 | Lemma for ~ foot : existen... |
foot 25414 | From a point ` C ` outside... |
footne 25415 | Uniqueness of the foot poi... |
footeq 25416 | Uniqueness of the foot poi... |
hlperpnel 25417 | A point on a half-line whi... |
perprag 25418 | Deduce a right angle from ... |
perpdragALT 25419 | Deduce a right angle from ... |
perpdrag 25420 | Deduce a right angle from ... |
colperp 25421 | Deduce a perpendicularity ... |
colperpexlem1 25422 | Lemma for ~ colperp . Fir... |
colperpexlem2 25423 | Lemma for ~ colperpex . S... |
colperpexlem3 25424 | Lemma for ~ colperpex . C... |
colperpex 25425 | In dimension 2 and above, ... |
mideulem2 25426 | Lemma for ~ opphllem , whi... |
opphllem 25427 | Lemma 8.24 of [Schwabhause... |
mideulem 25428 | Lemma for ~ mideu . We ca... |
midex 25429 | Existence of the midpoint,... |
mideu 25430 | Existence and uniqueness o... |
islnopp 25431 | The property for two point... |
islnoppd 25432 | Deduce that ` A ` and ` B ... |
oppne1 25433 | Points lying on opposite s... |
oppne2 25434 | Points lying on opposite s... |
oppne3 25435 | Points lying on opposite s... |
oppcom 25436 | Commutativity rule for "op... |
opptgdim2 25437 | If two points opposite to ... |
oppnid 25438 | The "opposite to a line" r... |
opphllem1 25439 | Lemma for ~ opphl . (Cont... |
opphllem2 25440 | Lemma for ~ opphl . Lemma... |
opphllem3 25441 | Lemma for ~ opphl : We as... |
opphllem4 25442 | Lemma for ~ opphl . (Cont... |
opphllem5 25443 | Second part of Lemma 9.4 o... |
opphllem6 25444 | First part of Lemma 9.4 of... |
oppperpex 25445 | Restating ~ colperpex usin... |
opphl 25446 | If two points ` A ` and ` ... |
outpasch 25447 | Axiom of Pasch, outer form... |
hlpasch 25448 | An application of the axio... |
ishpg 25451 | Value of the half-plane re... |
hpgbr 25452 | Half-planes : property for... |
hpgne1 25453 | Points on the open half pl... |
hpgne2 25454 | Points on the open half pl... |
lnopp2hpgb 25455 | Theorem 9.8 of [Schwabhaus... |
lnoppnhpg 25456 | If two points lie on the o... |
hpgerlem 25457 | Lemma for the proof that t... |
hpgid 25458 | The half-plane relation is... |
hpgcom 25459 | The half-plane relation co... |
hpgtr 25460 | The half-plane relation is... |
colopp 25461 | Opposite sides of a line f... |
colhp 25462 | Half-plane relation for co... |
hphl 25463 | If two points are on the s... |
midf 25468 | Midpoint as a function. (... |
midcl 25469 | Closure of the midpoint. ... |
ismidb 25470 | Property of the midpoint. ... |
midbtwn 25471 | Betweenness of midpoint. ... |
midcgr 25472 | Congruence of midpoint. (... |
midid 25473 | Midpoint of a null segment... |
midcom 25474 | Commutativity rule for the... |
mirmid 25475 | Point inversion preserves ... |
lmieu 25476 | Uniqueness of the line mir... |
lmif 25477 | Line mirror as a function.... |
lmicl 25478 | Closure of the line mirror... |
islmib 25479 | Property of the line mirro... |
lmicom 25480 | The line mirroring functio... |
lmilmi 25481 | Line mirroring is an invol... |
lmireu 25482 | Any point has a unique ant... |
lmieq 25483 | Equality deduction for lin... |
lmiinv 25484 | The invariants of the line... |
lmicinv 25485 | The mirroring line is an i... |
lmimid 25486 | If we have a right angle, ... |
lmif1o 25487 | The line mirroring functio... |
lmiisolem 25488 | Lemma for ~ lmiiso . (Con... |
lmiiso 25489 | The line mirroring functio... |
lmimot 25490 | Line mirroring is a motion... |
hypcgrlem1 25491 | Lemma for ~ hypcgr , case ... |
hypcgrlem2 25492 | Lemma for ~ hypcgr , case ... |
hypcgr 25493 | If the catheti of two righ... |
lmiopp 25494 | Line mirroring produces po... |
lnperpex 25495 | Existence of a perpendicul... |
trgcopy 25496 | Triangle construction: a c... |
trgcopyeulem 25497 | Lemma for ~ trgcopyeu . (... |
trgcopyeu 25498 | Triangle construction: a c... |
iscgra 25501 | Property for two angles AB... |
iscgra1 25502 | A special version of ~ isc... |
iscgrad 25503 | Sufficient conditions for ... |
cgrane1 25504 | Angles imply inequality. ... |
cgrane2 25505 | Angles imply inequality. ... |
cgrane3 25506 | Angles imply inequality. ... |
cgrane4 25507 | Angles imply inequality. ... |
cgrahl1 25508 | Angle congruence is indepe... |
cgrahl2 25509 | Angle congruence is indepe... |
cgracgr 25510 | First direction of proposi... |
cgraid 25511 | Angle congruence is reflex... |
cgraswap 25512 | Swap rays in a congruence ... |
cgrcgra 25513 | Triangle congruence implie... |
cgracom 25514 | Angle congruence commutes.... |
cgratr 25515 | Angle congruence is transi... |
cgraswaplr 25516 | Swap both side of angle co... |
cgrabtwn 25517 | Angle congruence preserves... |
cgrahl 25518 | Angle congruence preserves... |
cgracol 25519 | Angle congruence preserves... |
cgrancol 25520 | Angle congruence preserves... |
dfcgra2 25521 | This is the full statement... |
sacgr 25522 | Supplementary angles of co... |
oacgr 25523 | Vertical angle theorem. V... |
acopy 25524 | Angle construction. Theor... |
acopyeu 25525 | Angle construction. Theor... |
isinag 25529 | Property for point ` X ` t... |
inagswap 25530 | Swap the order of the half... |
inaghl 25531 | The "point lie in angle" r... |
isleag 25533 | Geometrical "less than" pr... |
cgrg3col4 25534 | Lemma 11.28 of [Schwabhaus... |
tgsas1 25535 | First congruence theorem: ... |
tgsas 25536 | First congruence theorem: ... |
tgsas2 25537 | First congruence theorem: ... |
tgsas3 25538 | First congruence theorem: ... |
tgasa1 25539 | Second congruence theorem:... |
tgasa 25540 | Second congruence theorem:... |
tgsss1 25541 | Third congruence theorem: ... |
tgsss2 25542 | Third congruence theorem: ... |
tgsss3 25543 | Third congruence theorem: ... |
isoas 25544 | Congruence theorem for iso... |
iseqlg 25547 | Property of a triangle bei... |
iseqlgd 25548 | Condition for a triangle t... |
f1otrgds 25549 | Convenient lemma for ~ f1o... |
f1otrgitv 25550 | Convenient lemma for ~ f1o... |
f1otrg 25551 | A bijection between bases ... |
f1otrge 25552 | A bijection between bases ... |
ttgval 25555 | Define a function to augme... |
ttglem 25556 | Lemma for ~ ttgbas and ~ t... |
ttgbas 25557 | The base set of a complex ... |
ttgplusg 25558 | The addition operation of ... |
ttgsub 25559 | The subtraction operation ... |
ttgvsca 25560 | The scalar product of a co... |
ttgds 25561 | The metric of a complex Hi... |
ttgitvval 25562 | Betweenness for a complex ... |
ttgelitv 25563 | Betweenness for a complex ... |
ttgbtwnid 25564 | Any complex module equippe... |
ttgcontlem1 25565 | Lemma for % ttgcont . (Co... |
xmstrkgc 25566 | Any metric space fulfills ... |
cchhllem 25567 | Lemma for chlbas and chlvs... |
elee 25574 | Membership in a Euclidean ... |
mptelee 25575 | A condition for a mapping ... |
eleenn 25576 | If ` A ` is in ` ( EE `` N... |
eleei 25577 | The forward direction of ~... |
eedimeq 25578 | A point belongs to at most... |
brbtwn 25579 | The binary relationship fo... |
brcgr 25580 | The binary relationship fo... |
fveere 25581 | The function value of a po... |
fveecn 25582 | The function value of a po... |
eqeefv 25583 | Two points are equal iff t... |
eqeelen 25584 | Two points are equal iff t... |
brbtwn2 25585 | Alternate characterization... |
colinearalglem1 25586 | Lemma for ~ colinearalg . ... |
colinearalglem2 25587 | Lemma for ~ colinearalg . ... |
colinearalglem3 25588 | Lemma for ~ colinearalg . ... |
colinearalglem4 25589 | Lemma for ~ colinearalg . ... |
colinearalg 25590 | An algebraic characterizat... |
eleesub 25591 | Membership of a subtractio... |
eleesubd 25592 | Membership of a subtractio... |
axdimuniq 25593 | The unique dimension axiom... |
axcgrrflx 25594 | ` A ` is as far from ` B `... |
axcgrtr 25595 | Congruence is transitive. ... |
axcgrid 25596 | If there is no distance be... |
axsegconlem1 25597 | Lemma for ~ axsegcon . Ha... |
axsegconlem2 25598 | Lemma for ~ axsegcon . Sh... |
axsegconlem3 25599 | Lemma for ~ axsegcon . Sh... |
axsegconlem4 25600 | Lemma for ~ axsegcon . Sh... |
axsegconlem5 25601 | Lemma for ~ axsegcon . Sh... |
axsegconlem6 25602 | Lemma for ~ axsegcon . Sh... |
axsegconlem7 25603 | Lemma for ~ axsegcon . Sh... |
axsegconlem8 25604 | Lemma for ~ axsegcon . Sh... |
axsegconlem9 25605 | Lemma for ~ axsegcon . Sh... |
axsegconlem10 25606 | Lemma for ~ axsegcon . Sh... |
axsegcon 25607 | Any segment ` A B ` can be... |
ax5seglem1 25608 | Lemma for ~ ax5seg . Rexp... |
ax5seglem2 25609 | Lemma for ~ ax5seg . Rexp... |
ax5seglem3a 25610 | Lemma for ~ ax5seg . (Con... |
ax5seglem3 25611 | Lemma for ~ ax5seg . Comb... |
ax5seglem4 25612 | Lemma for ~ ax5seg . Give... |
ax5seglem5 25613 | Lemma for ~ ax5seg . If `... |
ax5seglem6 25614 | Lemma for ~ ax5seg . Give... |
ax5seglem7 25615 | Lemma for ~ ax5seg . An a... |
ax5seglem8 25616 | Lemma for ~ ax5seg . Use ... |
ax5seglem9 25617 | Lemma for ~ ax5seg . Take... |
ax5seg 25618 | The five segment axiom. T... |
axbtwnid 25619 | Points are indivisible. T... |
axpaschlem 25620 | Lemma for ~ axpasch . Set... |
axpasch 25621 | The inner Pasch axiom. Ta... |
axlowdimlem1 25622 | Lemma for ~ axlowdim . Es... |
axlowdimlem2 25623 | Lemma for ~ axlowdim . Sh... |
axlowdimlem3 25624 | Lemma for ~ axlowdim . Se... |
axlowdimlem4 25625 | Lemma for ~ axlowdim . Se... |
axlowdimlem5 25626 | Lemma for ~ axlowdim . Sh... |
axlowdimlem6 25627 | Lemma for ~ axlowdim . Sh... |
axlowdimlem7 25628 | Lemma for ~ axlowdim . Se... |
axlowdimlem8 25629 | Lemma for ~ axlowdim . Ca... |
axlowdimlem9 25630 | Lemma for ~ axlowdim . Ca... |
axlowdimlem10 25631 | Lemma for ~ axlowdim . Se... |
axlowdimlem11 25632 | Lemma for ~ axlowdim . Ca... |
axlowdimlem12 25633 | Lemma for ~ axlowdim . Ca... |
axlowdimlem13 25634 | Lemma for ~ axlowdim . Es... |
axlowdimlem14 25635 | Lemma for ~ axlowdim . Ta... |
axlowdimlem15 25636 | Lemma for ~ axlowdim . Se... |
axlowdimlem16 25637 | Lemma for ~ axlowdim . Se... |
axlowdimlem17 25638 | Lemma for ~ axlowdim . Es... |
axlowdim1 25639 | The lower dimension axiom ... |
axlowdim2 25640 | The lower two-dimensional ... |
axlowdim 25641 | The general lower dimensio... |
axeuclidlem 25642 | Lemma for ~ axeuclid . Ha... |
axeuclid 25643 | Euclid's axiom. Take an a... |
axcontlem1 25644 | Lemma for ~ axcont . Chan... |
axcontlem2 25645 | Lemma for ~ axcont . The ... |
axcontlem3 25646 | Lemma for ~ axcont . Give... |
axcontlem4 25647 | Lemma for ~ axcont . Give... |
axcontlem5 25648 | Lemma for ~ axcont . Comp... |
axcontlem6 25649 | Lemma for ~ axcont . Stat... |
axcontlem7 25650 | Lemma for ~ axcont . Give... |
axcontlem8 25651 | Lemma for ~ axcont . A po... |
axcontlem9 25652 | Lemma for ~ axcont . Give... |
axcontlem10 25653 | Lemma for ~ axcont . Give... |
axcontlem11 25654 | Lemma for ~ axcont . Elim... |
axcontlem12 25655 | Lemma for ~ axcont . Elim... |
axcont 25656 | The axiom of continuity. ... |
eengv 25659 | The value of the Euclidean... |
eengstr 25660 | The Euclidean geometry as ... |
eengbas 25661 | The Base of the Euclidean ... |
ebtwntg 25662 | The betweenness relation u... |
ecgrtg 25663 | The congruence relation us... |
elntg 25664 | The line definition in the... |
eengtrkg 25665 | The geometry structure for... |
eengtrkge 25666 | The geometry structure for... |
edgfndxnn 25669 | The index value of the edg... |
edgfndxid 25670 | The value of the edge func... |
baseltedgf 25671 | The index value of the ` B... |
slotsbaseefdif 25672 | The slots ` Base ` and ` .... |
vtxval 25677 | The set of vertices of a g... |
iedgval 25678 | The set of indexed edges o... |
1vgrex 25679 | A graph with at least one ... |
opvtxval 25680 | The set of vertices of a g... |
opvtxfv 25681 | The set of vertices of a g... |
opvtxov 25682 | The set of vertices of a g... |
opiedgval 25683 | The set of indexed edges o... |
opiedgfv 25684 | The set of indexed edges o... |
opiedgov 25685 | The set of indexed edges o... |
opvtxfvi 25686 | The set of vertices of a g... |
opiedgfvi 25687 | The set of indexed edges o... |
funvtxdm2val 25688 | The set of vertices of an ... |
funiedgdm2val 25689 | The set of indexed edges o... |
funvtxval0 25690 | The set of vertices of an ... |
funvtxdmge2val 25691 | The set of vertices of an ... |
funiedgdmge2val 25692 | The set of indexed edges o... |
basvtxval 25693 | The set of vertices of a g... |
edgfiedgval 25694 | The set of indexed edges o... |
funvtxval 25695 | The set of vertices of a g... |
funiedgval 25696 | The set of indexed edges o... |
structvtxvallem 25697 | Lemma for ~ structvtxval a... |
structvtxval 25698 | The set of vertices of an ... |
structiedg0val 25699 | The set of indexed edges o... |
structgrssvtxlem 25700 | Lemma for ~ structgrssvtx ... |
structgrssvtx 25701 | The set of vertices of a g... |
structgrssiedg 25702 | The set of indexed edges o... |
struct2grstr 25703 | A graph represented as an ... |
struct2grvtx 25704 | The set of vertices of a g... |
struct2griedg 25705 | The set of indexed edges o... |
graop 25706 | Any representation of a gr... |
grastruct 25707 | Any representation of a gr... |
gropd 25708 | If any representation of a... |
grstructd 25709 | If any representation of a... |
gropeld 25710 | If any representation of a... |
grstructeld 25711 | If any representation of a... |
snstrvtxval 25712 | The set of vertices of a g... |
snstriedgval 25713 | The set of indexed edges o... |
vtxval0 25714 | Degenerated case 1 for ver... |
iedgval0 25715 | Degenerated case 1 for edg... |
vtxvalsnop 25716 | Degenerated case 2 for ver... |
iedgvalsnop 25717 | Degenerated case 2 for edg... |
vtxval3sn 25718 | Degenerated case 3 for ver... |
iedgval3sn 25719 | Degenerated case 3 for edg... |
vtxvalprc 25720 | Degenerated case 4 for ver... |
iedgvalprc 25721 | Degenerated case 4 for edg... |
isuhgr 25726 | The predicate "is an undir... |
isushgr 25727 | The predicate "is an undir... |
uhgrf 25728 | The edge function of an un... |
ushgrf 25729 | The edge function of an un... |
uhgrss 25730 | An edge is a subset of ver... |
uhgreq12g 25731 | If two sets have the same ... |
uhgrfun 25732 | The edge function of an un... |
uhgrn0 25733 | An edge is a nonempty subs... |
lpvtx 25734 | The endpoints of a loop (w... |
ushgruhgr 25735 | An undirected simple hyper... |
isuhgrop 25736 | The property of being an u... |
uhgr0e 25737 | The empty graph, with vert... |
uhgr0vb 25738 | The null graph, with no ve... |
uhgr0 25739 | The null graph represented... |
uhgrun 25740 | The union ` U ` of two (un... |
uhgrunop 25741 | The union of two (undirect... |
ushgrun 25742 | The union ` U ` of two (un... |
ushgrunop 25743 | The union of two (undirect... |
uhgrstrrepelem 25744 | Lemma for ~ uhgrstrrepe . ... |
uhgrstrrepe 25745 | Replacing (or adding) the ... |
incistruhgr 25746 | An _incident structure_ ` ... |
isupgr 25751 | The property of being an u... |
wrdupgr 25752 | The property of being an u... |
upgrf 25753 | The edge function of an un... |
upgrfn 25754 | The edge function of an un... |
upgrss 25755 | An edge is a subset of ver... |
upgrn0 25756 | An edge is a nonempty subs... |
upgrle 25757 | An edge of an undirected p... |
upgrfi 25758 | An edge is a finite subset... |
upgrex 25759 | An edge is an unordered pa... |
upgrbi 25760 | Show that an unordered pai... |
isumgr 25761 | The property of being an u... |
isumgrs 25762 | The simplified property of... |
wrdumgr 25763 | The property of being an u... |
umgrf 25764 | The edge function of an un... |
umgrfn 25765 | The edge function of an un... |
umgredg2 25766 | An edge of a multigraph ha... |
umgrbi 25767 | Show that an unordered pai... |
upgruhgr 25768 | An undirected pseudograph ... |
umgrupgr 25769 | An undirected multigraph i... |
umgruhgr 25770 | An undirected multigraph i... |
upgrle2 25771 | An edge of an undirected p... |
umgrnloopv 25772 | In a multigraph, there is ... |
umgredgprv 25773 | In a multigraph, an edge i... |
umgrnloop 25774 | In a multigraph, there is ... |
umgrnloop0 25775 | A multigraph has no loops.... |
umgr0e 25776 | The empty graph, with vert... |
upgr0e 25777 | The empty graph, with vert... |
upgr1elem 25778 | Lemma for ~ upgr1e and ~ u... |
upgr1e 25779 | A pseudograph with one edg... |
upgr0eop 25780 | The empty graph, with vert... |
upgr1eop 25781 | A pseudograph with one edg... |
upgr0eopALT 25782 | Alternate proof of ~ upgr0... |
upgr1eopALT 25783 | Alternate proof of ~ upgr1... |
upgrun 25784 | The union ` U ` of two pse... |
upgrunop 25785 | The union of two pseudogra... |
umgrun 25786 | The union ` U ` of two mul... |
umgrunop 25787 | The union of two multigrap... |
umgrislfupgrlem 25788 | Lemma for ~ umgrislfupgr a... |
umgrislfupgr 25789 | A multigraph is a loop-fre... |
lfgredgge2 25790 | An edge of a loop-free gra... |
lfgrnloop 25791 | A loop-free graph has no l... |
edgaval 25794 | The edges of a graph. (Co... |
edgaopval 25795 | The edges of a graph repre... |
edgaov 25796 | The edges of a graph repre... |
edgastruct 25797 | The edges of a graph repre... |
edgiedgb 25798 | A set is an edge iff it is... |
uhgredgiedgb 25799 | In a hypergraph, a set is ... |
edg0iedg0 25800 | There is no edge in a grap... |
uhgriedg0edg0 25801 | A hypergraph has no edges ... |
uhgredgn0 25802 | An edge of a hypergraph is... |
edguhgr 25803 | An edge of a hypergraph is... |
uhgredgrnv 25804 | An edge of a hypergraph co... |
uhgredgss 25805 | The set of edges of a hype... |
upgredgss 25806 | The set of edges of a pseu... |
umgredgss 25807 | The set of edges of a mult... |
edgupgr 25808 | Properties of an edge of a... |
edgumgr 25809 | Properties of an edge of a... |
uhgrvtxedgiedgb 25810 | In a hypergraph, a vertex ... |
upgredg 25811 | For each edge in a pseudog... |
umgredg 25812 | For each edge in a multigr... |
umgrpredgav 25813 | An edge of a multigraph al... |
upgredg2vtx 25814 | For a vertex incident to a... |
upgredgpr 25815 | If a proper pair (of verti... |
umgredgne 25816 | An edge of a multigraph al... |
umgrnloop2 25817 | A multigraph has no loops.... |
umgredgnlp 25818 | An edge of a multigraph is... |
reluhgra 25823 | The class of all undirecte... |
relushgra 25824 | The class of all undirecte... |
uhgrav 25825 | The classes of vertices an... |
uhgraopelvv 25826 | An undirected hypergraph i... |
isuhgra 25827 | The property of being an u... |
uhgraf 25828 | The edge function of an un... |
uhgrafun 25829 | The edge function of an un... |
isushgra 25830 | The property of being an u... |
ushgraf 25831 | The edge function of an un... |
ushgrauhgra 25832 | An undirected simple hyper... |
uhgraop 25833 | The property of being an u... |
uhgrac 25834 | The property of being an u... |
uhgrass 25835 | An edge is a subset of ver... |
uhgraeq12d 25836 | Equality of hypergraphs. ... |
uhgrares 25837 | A subgraph of a hypergraph... |
uhgra0 25838 | The empty graph, with vert... |
uhgra0v 25839 | The null graph, with no ve... |
uhgraun 25840 | The union of two (undirect... |
relumgra 25843 | The class of all undirecte... |
isumgra 25844 | The property of being an u... |
wrdumgra 25845 | The property of being an u... |
umgraf2 25846 | The edge function of an un... |
umgraf 25847 | The edge function of an un... |
umgrass 25848 | An edge is a subset of ver... |
umgran0 25849 | An edge is a nonempty subs... |
umgrale 25850 | An edge has at most two en... |
umgrafi 25851 | An edge is a finite subset... |
umgraex 25852 | An edge is an unordered pa... |
umgrares 25853 | A subgraph of a graph (for... |
umgra0 25854 | The empty graph, with vert... |
umgra1 25855 | The graph with one edge. ... |
umisuhgra 25856 | An undirected multigraph i... |
umgraun 25857 | The union of two (undirect... |
reluslgra 25863 | The class of all undirecte... |
relusgra 25864 | The class of all undirecte... |
uslgrav 25866 | The classes of vertices an... |
usgrav 25867 | The classes of vertices an... |
edgval 25868 | The edges of a graph. (Co... |
edgopval 25869 | The edges of a graph repre... |
edgov 25870 | The edges of a graph, show... |
edguslgra 25871 | The edges of an undirected... |
isuslgra 25872 | The property of being an u... |
isusgra 25873 | The property of being an u... |
uslgraf 25874 | The edge function of an un... |
usgraf 25875 | The edge function of an un... |
isusgra0 25876 | The property of being an u... |
usgraf0 25877 | The edge function of an un... |
usgrafun 25878 | The edge function of an un... |
usgraop 25879 | An undirected simple graph... |
usgrac 25880 | An undirected simple graph... |
edgss 25881 | The set of edges of an und... |
edg 25882 | An edge of an undirected s... |
isausgra 25883 | The property of an unorder... |
ausisusgra 25884 | The equivalence of the def... |
ausisusgraedg 25885 | The equivalence of the def... |
usgraedgop 25886 | An edge of an undirected s... |
usgraf1o 25887 | The edge function of an un... |
uslgraf1oedg 25888 | The edge function of an un... |
usgraf1 25889 | The edge function of an un... |
usgrass 25890 | An edge is a subset of ver... |
usgraeq12d 25891 | Equality of simple graphs ... |
uslisushgra 25892 | An undirected simple graph... |
uslisumgra 25893 | An undirected simple graph... |
usisuslgra 25894 | An undirected simple graph... |
usisumgra 25895 | An undirected simple graph... |
usisuhgra 25896 | An undirected simple graph... |
elusuhgra 25897 | An undirected simple graph... |
usgrares 25898 | A subgraph of a graph (for... |
usgra0 25899 | The empty graph, with vert... |
usgra0v 25900 | The empty graph with no ve... |
uslgra1 25901 | The graph with one edge, a... |
usgra1 25902 | The graph with one edge, a... |
uslgraun 25903 | The union of two simple gr... |
usgraedg2 25904 | The value of the "edge fun... |
usgraedgprv 25905 | In an undirected graph, an... |
usgraedgrnv 25906 | An edge of an undirected s... |
usgranloopv 25907 | In an undirected simple gr... |
usgranloop 25908 | In an undirected simple gr... |
usgranloop0 25909 | A simple undirected graph ... |
usgraedgrn 25910 | An edge of an undirected s... |
usgra2edg 25911 | If a vertex is adjacent to... |
usgra2edg1 25912 | If a vertex is adjacent to... |
usgrarnedg 25913 | For each edge in a simple ... |
edgprvtx 25914 | An edge of an undirected s... |
usgraedg3 25915 | The value of the "edge fun... |
usgraedg4 25916 | The value of the "edge fun... |
usgraedgreu 25917 | The value of the "edge fun... |
usgrarnedg1 25918 | For each edge in a simple ... |
usgra1v 25919 | A class with one (or no) v... |
usgraidx2vlem1 25920 | Lemma 1 for ~ usgraidx2v .... |
usgraidx2vlem2 25921 | Lemma 2 for ~ usgraidx2v .... |
usgraidx2v 25922 | The mapping of indices of ... |
usgraedgleord 25923 | In a graph the number of e... |
usgraex0elv 25924 | Lemma 0 for ~ usgraexmpl .... |
usgraex1elv 25925 | Lemma 1 for ~ usgraexmpl .... |
usgraex2elv 25926 | Lemma 2 for ~ usgraexmpl .... |
usgraex3elv 25927 | Lemma 3 for ~ usgraexmpl .... |
usgraexmpldifpr 25928 | Lemma for ~ usgraexmpl : a... |
usgraexmplef 25929 | Lemma for ~ usgraexmpl . ... |
usgraexmpl 25930 | ` <. V , E >. ` is a graph... |
usgraexmplvtx 25931 | The vertices ` 0 , 1 , 2 ,... |
usgraexmpledg 25932 | The edges ` { 0 , 1 } , { ... |
usgraexmplc 25933 | ` G = <. V , E >. ` is a g... |
usgraexmplcvtx 25934 | The vertices ` 0 , 1 , 2 ,... |
usgraexmplcedg 25935 | The edges ` { 0 , 1 } , { ... |
fiusgraedgfi 25936 | In a finite graph the numb... |
usgrafisindb0 25937 | The size of a finite simpl... |
usgrafisindb1 25938 | The size of a finite simpl... |
usgrares1 25939 | Restricting an undirected ... |
usgrafilem1 25940 | The domain of the edge fun... |
usgrafilem2 25941 | In a graph with a finite n... |
usgrafisinds 25942 | In a graph with a finite n... |
usgrafisbase 25943 | Induction base for ~ usgra... |
usgrafis 25944 | A simple undirected graph ... |
usgrafiedg 25945 | A simple undirected graph ... |
nbgraop 25952 | The set of neighbors of an... |
nbgraopALT 25953 | Alternate proof of ~ nbgra... |
nbgraop1 25954 | The set of neighbors of an... |
nbgrael 25955 | The set of neighbors of an... |
nbgranv0 25956 | There are no neighbors of ... |
nbusgra 25957 | The set of neighbors of a ... |
nbgra0nb 25958 | A vertex which is not endp... |
nbgraeledg 25959 | A class/vertex is a neighb... |
nbgraisvtx 25960 | Every neighbor of a class/... |
nbgra0edg 25961 | In a graph with no edges, ... |
nbgrassvt 25962 | The neighbors of a vertex ... |
nbgranself 25963 | A vertex in a graph (witho... |
nbgrassovt 25964 | The neighbors of a vertex ... |
nbgranself2 25965 | A class is not a neighbor ... |
nbgrassvwo 25966 | The neighbors of a vertex ... |
nbgrassvwo2 25967 | The neighbors of a vertex ... |
nbgrasym 25968 | A vertex in a graph is a n... |
nbgracnvfv 25969 | Applying the edge function... |
nbgraf1olem1 25970 | Lemma 1 for ~ nbgraf1o . ... |
nbgraf1olem2 25971 | Lemma 2 for ~ nbgraf1o . ... |
nbgraf1olem3 25972 | Lemma 3 for ~ nbgraf1o . ... |
nbgraf1olem4 25973 | Lemma 4 for ~ nbgraf1o . ... |
nbgraf1olem5 25974 | Lemma 5 for ~ nbgraf1o . ... |
nbgraf1o0 25975 | The set of neighbors of a ... |
nbgraf1o 25976 | The set of neighbors of a ... |
nbusgrafi 25977 | The class of neighbors of ... |
nbfiusgrafi 25978 | The class of neighbors of ... |
edgusgranbfin 25979 | The number of neighbors of... |
nb3graprlem1 25980 | Lemma 1 for ~ nb3grapr . ... |
nb3graprlem2 25981 | Lemma 2 for ~ nb3grapr . ... |
nb3grapr 25982 | The neighbors of a vertex ... |
nb3grapr2 25983 | The neighbors of a vertex ... |
nb3gra2nb 25984 | If the neighbors of two ve... |
iscusgra 25985 | The property of being a co... |
iscusgra0 25986 | The property of being a co... |
cusisusgra 25987 | A complete (undirected sim... |
cusgrarn 25988 | In a complete simple graph... |
cusgra0v 25989 | A graph with no vertices (... |
cusgra1v 25990 | A graph with one vertex (a... |
cusgra2v 25991 | A graph with two (differen... |
nbcusgra 25992 | In a complete (undirected ... |
cusgra3v 25993 | A graph with three (differ... |
cusgra3vnbpr 25994 | The neighbors of a vertex ... |
cusgraexilem1 25995 | Lemma 1 for ~ cusgraexi . ... |
cusgraexilem2 25996 | Lemma 2 for ~ cusgraexi . ... |
cusgraexi 25997 | For each set the identity ... |
cusgraexg 25998 | For each set there is an e... |
cusgrasizeindb0 25999 | Base case of the induction... |
cusgrasizeindb1 26000 | Base case of the induction... |
cusgrares 26001 | Restricting a complete sim... |
cusgrasizeindslem1 26002 | Lemma 1 for ~ cusgrasizein... |
cusgrasizeindslem2 26003 | Lemma 2 for ~ cusgrasizein... |
cusgrasizeinds 26004 | Part 1 of induction step i... |
cusgrasize2inds 26005 | Induction step in ~ cusgra... |
cusgrasize 26006 | The size of a finite compl... |
cusgrafilem1 26007 | Lemma 1 for ~ cusgrafi . ... |
cusgrafilem2 26008 | Lemma 2 for ~ cusgrafi . ... |
cusgrafilem3 26009 | Lemma 3 for ~ cusgrafi . ... |
cusgrafi 26010 | If the size of a complete ... |
usgrasscusgra 26011 | An undirected simple graph... |
sizeusglecusglem1 26012 | Lemma 1 for ~ sizeusglecus... |
sizeusglecusglem2 26013 | Lemma 2 for ~ sizeusglecus... |
sizeusglecusg 26014 | The size of an undirected ... |
usgramaxsize 26015 | The maximum size of an und... |
isuvtx 26016 | The set of all universal v... |
uvtxel 26017 | A universal vertex, i.e. a... |
uvtxisvtx 26018 | A universal vertex is a ve... |
uvtx0 26019 | There is no universal vert... |
uvtx01vtx 26020 | If a graph/class has no ed... |
uvtxnbgra 26021 | A universal vertex has all... |
uvtxnm1nbgra 26022 | A universal vertex has ` n... |
uvtxnbgravtx 26023 | A universal vertex is neig... |
cusgrauvtxb 26024 | An undirected simple graph... |
uvtxnb 26025 | A vertex in a undirected s... |
relwlk 26046 | The walks (in an undirecte... |
wlks 26047 | The set of walks (in an un... |
iswlk 26048 | Properties of a pair of fu... |
2mwlk 26049 | The two mappings determini... |
wlkres 26050 | Restrictions of walks (i.e... |
wlkbprop 26051 | Basic properties of a walk... |
iswlkg 26052 | Generalisation of ~ iswlk ... |
wlkcomp 26053 | A walk expressed by proper... |
wlkcompim 26054 | Implications for the prope... |
wlkn0 26055 | The set of vertices of a w... |
wlkop 26056 | A walk (in an undirected s... |
wlkcpr 26057 | A walk as class with two c... |
wlkelwrd 26058 | The components of a walk a... |
edgwlk 26059 | The (connected) edges of a... |
wlklenvm1 26060 | The number of edges of a w... |
wlkon 26061 | The set of walks between t... |
iswlkon 26062 | Properties of a pair of fu... |
wlkonprop 26063 | Properties of a walk betwe... |
wlkoniswlk 26064 | A walk between to vertices... |
wlkonwlk 26065 | A walk is a walk between i... |
trls 26066 | The set of trails (in an u... |
istrl 26067 | Properties of a pair of fu... |
istrl2 26068 | Properties of a pair of fu... |
trliswlk 26069 | A trail is a walk (in an u... |
trlon 26070 | The set of trails between ... |
istrlon 26071 | Properties of a pair of fu... |
trlonprop 26072 | Properties of a trail betw... |
trlonistrl 26073 | A trail between to vertice... |
trlonwlkon 26074 | A trail between two vertic... |
0wlk 26075 | A pair of an empty set (of... |
0trl 26076 | A pair of an empty set (of... |
0wlkon 26077 | A walk of length 0 from a ... |
0trlon 26078 | A trail of length 0 from a... |
2trllemF 26079 | Lemma 5 for ~ constr2trl .... |
2trllemA 26080 | Lemma 1 for ~ constr2trl .... |
2trllemB 26081 | Lemma 2 for ~ constr2trl .... |
2trllemH 26082 | Lemma 3 for ~ constr2trl .... |
2trllemE 26083 | Lemma 4 for ~ constr2trl .... |
2wlklemA 26084 | Lemma for ~ constr2wlk . ... |
2wlklemB 26085 | Lemma for ~ constr2wlk . ... |
2wlklemC 26086 | Lemma for ~ constr2wlk . ... |
2trllemD 26087 | Lemma 4 for ~ constr2trl .... |
2trllemG 26088 | Lemma 7 for ~ constr2trl .... |
wlkntrllem1 26089 | Lemma 1 for ~ wlkntrl : F... |
wlkntrllem2 26090 | Lemma 2 for ~ wlkntrl : T... |
wlkntrllem3 26091 | Lemma 3 for ~ wlkntrl : F... |
wlkntrl 26092 | A walk which is not a trai... |
usgrwlknloop 26093 | In an undirected simple gr... |
2wlklem 26094 | Lemma for ~ is2wlk and ~ 2... |
is2wlk 26095 | Properties of a pair of fu... |
pths 26096 | The set of paths (in an un... |
spths 26097 | The set of simple paths (i... |
ispth 26098 | Properties of a pair of fu... |
isspth 26099 | Properties of a pair of fu... |
0pth 26100 | A pair of an empty set (of... |
0spth 26101 | A pair of an empty set (of... |
pthistrl 26102 | A path is a trail (in an u... |
spthispth 26103 | A simple path is a path (i... |
pthdepisspth 26104 | A path with different star... |
pthon 26105 | The set of paths between t... |
ispthon 26106 | Properties of a pair of fu... |
pthonprop 26107 | Properties of a path betwe... |
pthonispth 26108 | A path between two vertice... |
0pthon 26109 | A path of length 0 from a ... |
0pthon1 26110 | A path of length 0 from a ... |
0pthonv 26111 | For each vertex there is a... |
spthon 26112 | The set of simple paths be... |
isspthon 26113 | Properties of a pair of fu... |
isspthonpth 26114 | Properties of a pair of fu... |
spthonprp 26115 | Properties of a simple pat... |
spthonisspth 26116 | A simple path between to v... |
spthonepeq 26117 | The endpoints of a simple ... |
constr1trl 26118 | Construction of a trail fr... |
1pthonlem1 26119 | Lemma 1 for ~ 1pthon . (C... |
1pthonlem2 26120 | Lemma 2 for ~ 1pthon . (C... |
1pthon 26121 | A path of length 1 from on... |
1pthoncl 26122 | A path of length 1 from on... |
1pthon2v 26123 | For each pair of adjacent ... |
constr2spthlem1 26124 | Lemma 1 for ~ constr2spth ... |
2pthlem1 26125 | Lemma 1 for ~ constr2pth .... |
2pthlem2 26126 | Lemma 2 for ~ constr2pth .... |
2wlklem1 26127 | Lemma 1 for ~ constr2wlk .... |
constr2wlk 26128 | Construction of a walk fro... |
constr2trl 26129 | Construction of a trail fr... |
constr2spth 26130 | A simple path of length 2 ... |
constr2pth 26131 | A path of length 2 from on... |
2pthon 26132 | A path of length 2 from on... |
2pthoncl 26133 | A path of length 2 from on... |
2pthon3v 26134 | For a vertex adjacent to t... |
redwlklem 26135 | Lemma for ~ redwlk . (Con... |
redwlk 26136 | A walk ending at the last ... |
wlkdvspthlem 26137 | Lemma for ~ wlkdvspth . (... |
wlkdvspth 26138 | A walk consisting of diffe... |
usgra2adedgspthlem1 26139 | Lemma 1 for ~ usgra2adedgs... |
usgra2adedgspthlem2 26140 | Lemma 2 for ~ usgra2adedgs... |
usgra2adedgspth 26141 | In an undirected simple gr... |
usgra2adedgwlk 26142 | In an undirected simple gr... |
usgra2adedgwlkon 26143 | In an undirected simple gr... |
usgra2adedgwlkonALT 26144 | Alternate proof for ~ usgr... |
usg2wlk 26145 | In an undirected simple gr... |
usg2wlkon 26146 | In an undirected simple gr... |
usgra2wlkspthlem1 26147 | Lemma 1 for ~ usgra2wlkspt... |
usgra2wlkspthlem2 26148 | Lemma 2 for ~ usgra2wlkspt... |
usgra2wlkspth 26149 | In a undirected simple gra... |
crcts 26150 | The set of circuits (in an... |
cycls 26151 | The set of cycles (in an u... |
iscrct 26152 | Properties of a pair of fu... |
iscycl 26153 | Properties of a pair of fu... |
0crct 26154 | A pair of an empty set (of... |
0cycl 26155 | A pair of an empty set (of... |
crctistrl 26156 | A circuit is a trail. (Co... |
cyclispth 26157 | A cycle is a path. (Contr... |
cycliscrct 26158 | A cycle is a circuit. (Co... |
cyclnspth 26159 | A (non trivial) cycle is n... |
cycliswlk 26160 | A cycle is a walk. (Contr... |
cyclispthon 26161 | A cycle is a path starting... |
fargshiftlem 26162 | If a class is a function, ... |
fargshiftfv 26163 | If a class is a function, ... |
fargshiftf 26164 | If a class is a function, ... |
fargshiftf1 26165 | If a function is 1-1, then... |
fargshiftfo 26166 | If a function is onto, the... |
fargshiftfva 26167 | The values of a shifted fu... |
usgrcyclnl1 26168 | In an undirected simple gr... |
usgrcyclnl2 26169 | In an undirected simple gr... |
3cycl3dv 26170 | In a simple graph, the ver... |
nvnencycllem 26171 | Lemma for ~ 3v3e3cycl1 and... |
3v3e3cycl1 26172 | If there is a cycle of len... |
constr3lem1 26173 | Lemma for ~ constr3trl etc... |
constr3lem2 26174 | Lemma for ~ constr3trl etc... |
constr3lem4 26175 | Lemma for ~ constr3trl etc... |
constr3lem5 26176 | Lemma for ~ constr3trl etc... |
constr3lem6 26177 | Lemma for ~ constr3pthlem3... |
constr3trllem1 26178 | Lemma for ~ constr3trl . ... |
constr3trllem2 26179 | Lemma for ~ constr3trl . ... |
constr3trllem3 26180 | Lemma for ~ constr3trl . ... |
constr3trllem4 26181 | Lemma for ~ constr3trl . ... |
constr3trllem5 26182 | Lemma for ~ constr3trl . ... |
constr3pthlem1 26183 | Lemma for ~ constr3pth . ... |
constr3pthlem2 26184 | Lemma for ~ constr3pth . ... |
constr3pthlem3 26185 | Lemma for ~ constr3pth . ... |
constr3cycllem1 26186 | Lemma for ~ constr3cycl . ... |
constr3trl 26187 | Construction of a trail fr... |
constr3pth 26188 | Construction of a path fro... |
constr3cycl 26189 | Construction of a 3-cycle ... |
constr3cyclp 26190 | Construction of a 3-cycle ... |
constr3cyclpe 26191 | If there are three (differ... |
3v3e3cycl2 26192 | If there are three (differ... |
3v3e3cycl 26193 | If and only if there is a ... |
4cycl4v4e 26194 | If there is a cycle of len... |
4cycl4dv 26195 | In a simple graph, the ver... |
4cycl4dv4e 26196 | If there is a cycle of len... |
dfconngra1 26199 | Alternative definition of ... |
isconngra 26200 | The property of being a co... |
isconngra1 26201 | The property of being a co... |
0conngra 26202 | A class/graph without vert... |
1conngra 26203 | A class/graph with (at mos... |
cusconngra 26204 | A complete (undirected sim... |
wwlk 26209 | The set of walks (in an un... |
wwlkn 26210 | The set of walks (in an un... |
iswwlk 26211 | Properties of a word to re... |
iswwlkn 26212 | Properties of a word to re... |
wwlkprop 26213 | Properties of a walk (in a... |
wwlknprop 26214 | Properties of a walk of a ... |
wwlknimp 26215 | Implications for a set bei... |
wwlksswrd 26216 | Walks (represented by word... |
wwlkn0 26217 | A walk of length 0 is repr... |
wlkiswwlk1 26218 | The sequence of vertices i... |
wlkiswwlk2lem1 26219 | Lemma 1 for ~ wlkiswwlk2 .... |
wlkiswwlk2lem2 26220 | Lemma 2 for ~ wlkiswwlk2 .... |
wlkiswwlk2lem3 26221 | Lemma 3 for ~ wlkiswwlk2 .... |
wlkiswwlk2lem4 26222 | Lemma 4 for ~ wlkiswwlk2 .... |
wlkiswwlk2lem5 26223 | Lemma 5 for ~ wlkiswwlk2 .... |
wlkiswwlk2lem6 26224 | Lemma 6 for ~ wlkiswwlk2 .... |
wlkiswwlk2 26225 | A walk as word corresponds... |
wlkiswwlk 26226 | A walk as word corresponds... |
wlklniswwlkn1 26227 | The sequence of vertices i... |
wlklniswwlkn2 26228 | A walk of length n as word... |
wlklniswwlkn 26229 | A walk of length n as word... |
wwlkiswwlkn 26230 | A walk of a fixed length a... |
wwlksswwlkn 26231 | The walks of a fixed lengt... |
wwlknimpb 26232 | Basic implications for a s... |
wwlkn0s 26233 | The set of all walks as wo... |
vfwlkniswwlkn 26234 | If the edge function of a ... |
2wlkeq 26235 | Conditions for two walks (... |
usg2wlkeq 26236 | Conditions for two walks w... |
usg2wlkeq2 26237 | Conditions for which two w... |
wlknwwlknfun 26238 | Lemma 1 for ~ wlknwwlknbij... |
wlknwwlkninj 26239 | Lemma 2 for ~ wlknwwlknbij... |
wlknwwlknsur 26240 | Lemma 3 for ~ wlknwwlknbij... |
wlknwwlknbij 26241 | Lemma 4 for ~ wlknwwlknbij... |
wlknwwlknbij2 26242 | There is a bijection betwe... |
wlknwwlknen 26243 | The set of walks of a fixe... |
wlknwwlkneqs 26244 | The set of walks of a fixe... |
wlkiswwlkfun 26245 | Lemma 1 for ~ wlkiswwlkbij... |
wlkiswwlkinj 26246 | Lemma 2 for ~ wlkiswwlkbij... |
wlkiswwlksur 26247 | Lemma 3 for ~ wlkiswwlkbij... |
wlkiswwlkbij 26248 | Lemma 4 for ~ wlkiswwlkbij... |
wlkiswwlkbij2 26249 | There is a bijection betwe... |
wwlkeq 26250 | Equality of two walks (as ... |
wwlknred 26251 | Reduction of a walk (as wo... |
wwlknext 26252 | Extension of a walk (as wo... |
wwlknextbi 26253 | Extension of a walk (as wo... |
wwlknredwwlkn 26254 | For each walk (as word) of... |
wwlknredwwlkn0 26255 | For each walk (as word) of... |
wwlkextwrd 26256 | Lemma 0 for ~ wwlkextbij .... |
wwlkextfun 26257 | Lemma 1 for ~ wwlkextbij .... |
wwlkextinj 26258 | Lemma 2 for ~ wwlkextbij .... |
wwlkextsur 26259 | Lemma 3 for ~ wwlkextbij .... |
wwlkextbij0 26260 | Lemma 4 for ~ wwlkextbij .... |
wwlkextbij 26261 | There is a bijection betwe... |
wwlkexthasheq 26262 | The number of the extensio... |
wwlkm1edg 26263 | Removing the trailing edge... |
disjxwwlks 26264 | Sets of walks (as words) e... |
wwlknndef 26265 | Conditions for ` WWalksN `... |
wwlknfi 26266 | The number of walks repres... |
wlknfi 26267 | The number of walks of fix... |
wlknwwlknvbij 26268 | There is a bijection betwe... |
wwlkextproplem1 26269 | Lemma 1 for ~ wwlkextprop ... |
wwlkextproplem2 26270 | Lemma 2 for ~ wwlkextprop ... |
wwlkextproplem3 26271 | Lemma 3 for ~ wwlkextprop ... |
wwlkextprop 26272 | Adding additional properti... |
disjxwwlkn 26273 | Sets of walks (as words) e... |
hashwwlkext 26274 | Number of walks (as words)... |
clwlk 26281 | The set of closed walks (i... |
isclwlk0 26282 | Properties of a pair of fu... |
isclwlkg 26283 | Generalisation of ~ isclwl... |
isclwlk 26284 | Properties of a pair of fu... |
clwlkiswlk 26285 | A closed walk is a walk (i... |
clwlkswlks 26286 | Closed walks are walks (in... |
clwlksarewlks 26287 | Closed walks are walks (in... |
wlkv0 26288 | If there is a walk in an e... |
wlk0 26289 | There is no walk in an emp... |
clwlk0 26290 | There is no closed walk in... |
clwlkcomp 26291 | A closed walk expressed by... |
clwlkcompim 26292 | Implications for the prope... |
0clwlk 26293 | A pair of an empty set (of... |
clwwlk 26294 | The set of closed walks (i... |
clwwlkn 26295 | The set of closed walks (i... |
isclwwlk 26296 | Properties of a word to re... |
isclwwlkn 26297 | Properties of a word to re... |
clwwlkprop 26298 | Properties of a closed wal... |
clwwlkgt0 26299 | A closed walk in an undire... |
clwwlknprop 26300 | Properties of a closed wal... |
clwwlknndef 26301 | Conditions for ` ClWWalksN... |
clwwlkn0 26302 | There is no closed walk of... |
clwwlkn2 26303 | In an undirected simple gr... |
clwwlknimp 26304 | Implications for a set bei... |
clwwlksswrd 26305 | Closed walks (represented ... |
clwwlknfi 26306 | If there is only a finite ... |
clwlkisclwwlklem2a1 26307 | Lemma 1 for ~ clwlkisclwwl... |
clwlkisclwwlklem2a2 26308 | Lemma 3 for ~ clwlkisclwwl... |
clwlkisclwwlklem2a3 26309 | Lemma 3 for ~ clwlkisclwwl... |
clwlkisclwwlklem2fv1 26310 | Lemma 4a for ~ clwlkisclww... |
clwlkisclwwlklem2fv2 26311 | Lemma 4b for ~ clwlkisclww... |
clwlkisclwwlklem2a4 26312 | Lemma 4 for ~ clwlkisclwwl... |
clwlkisclwwlklem2a 26313 | Lemma 2 for ~ clwlkisclwwl... |
clwlkisclwwlklem2 26314 | Lemma for ~ clwlkisclwwlk ... |
clwlkisclwwlklem1 26315 | Lemma for ~ clwlkisclwwlk ... |
clwlkisclwwlklem0 26316 | Lemma for ~ clwlkisclwwlk ... |
clwlkisclwwlk 26317 | A closed walk as word corr... |
clwlkisclwwlk2 26318 | A closed walk corresponds ... |
clwwlkisclwwlkn 26319 | A closed walk of a fixed l... |
clwwlkssclwwlkn 26320 | The closed walks of a fixe... |
clwwlkel 26321 | Obtaining a closed walk (a... |
clwwlkf 26322 | Lemma 1 for ~ clwwlkbij : ... |
clwwlkfv 26323 | Lemma 2 for ~ clwwlkbij : ... |
clwwlkf1 26324 | Lemma 3 for ~ clwwlkbij : ... |
clwwlkfo 26325 | Lemma 4 for ~ clwwlkbij : ... |
clwwlkf1o 26326 | Lemma 5 for ~ clwwlkbij : ... |
clwwlkbij 26327 | There is a bijection betwe... |
clwwlknwwlkncl 26328 | Obtaining a closed walk (a... |
clwwlkvbij 26329 | There is a bijection betwe... |
clwwlkext2edg 26330 | If a word concatenated wit... |
wwlkext2clwwlk 26331 | If a word represents a wal... |
wwlksubclwwlk 26332 | Any prefix of a word repre... |
clwwisshclwwlem1 26333 | Lemma 1 for ~ clwwisshclww... |
clwwisshclwwlem 26334 | Lemma for ~ clwwisshclww .... |
clwwisshclww 26335 | Cyclically shifting a clos... |
clwwisshclwwn 26336 | Cyclically shifting a clos... |
clwwnisshclwwn 26337 | Cyclically shifting a clos... |
erclwwlkrel 26338 | ` .~ ` is a relation. (Co... |
erclwwlkeq 26339 | Two classes are equivalent... |
erclwwlkeqlen 26340 | If two classes are equival... |
erclwwlkref 26341 | ` .~ ` is a reflexive rela... |
erclwwlksym 26342 | ` .~ ` is a symmetric rela... |
erclwwlktr 26343 | ` .~ ` is a transitive rel... |
erclwwlk 26344 | ` .~ ` is an equivalence r... |
eleclclwwlknlem1 26345 | Lemma 1 for ~ eleclclwwlkn... |
eleclclwwlknlem2 26346 | Lemma 2 for ~ eleclclwwlkn... |
clwwlknscsh 26347 | The set of cyclical shifts... |
usg2cwwk2dif 26348 | If a word represents a clo... |
usg2cwwkdifex 26349 | If a word represents a clo... |
erclwwlknrel 26350 | ` .~ ` is a relation. (Co... |
erclwwlkneq 26351 | Two classes are equivalent... |
erclwwlkneqlen 26352 | If two classes are equival... |
erclwwlknref 26353 | ` .~ ` is a reflexive rela... |
erclwwlknsym 26354 | ` .~ ` is a symmetric rela... |
erclwwlkntr 26355 | ` .~ ` is a transitive rel... |
erclwwlkn 26356 | ` .~ ` is an equivalence r... |
qerclwwlknfi 26357 | The quotient set of the se... |
hashclwwlkn0 26358 | The number of closed walks... |
eclclwwlkn1 26359 | An equivalence class accor... |
eleclclwwlkn 26360 | A member of an equivalence... |
hashecclwwlkn1 26361 | The size of every equivale... |
usghashecclwwlk 26362 | The size of every equivale... |
hashclwwlkn 26363 | The size of the set of clo... |
clwwlkndivn 26364 | The size of the set of clo... |
wlklenvp1 26365 | The number of vertices of ... |
wlklenvclwlk 26366 | The number of vertices in ... |
clwlkfclwwlk2wrd 26367 | The second component of a ... |
clwlkfclwwlk1hashn 26368 | The size of the first comp... |
clwlkfclwwlk1hash 26369 | The size of the first comp... |
clwlkfclwwlk2sswd 26370 | The size of a subword of t... |
clwlkfclwwlk 26371 | There is a function betwee... |
clwlkfoclwwlk 26372 | There is an onto function ... |
clwlkf1clwwlklem1 26373 | Lemma 1 for ~ clwlkf1clwwl... |
clwlkf1clwwlklem2 26374 | Lemma 2 for ~ clwlkf1clwwl... |
clwlkf1clwwlklem3 26375 | Lemma 3 for ~ clwlkf1clwwl... |
clwlkf1clwwlklem 26376 | Lemma for ~ clwlkf1clwwlk ... |
clwlkf1clwwlk 26377 | There is a one-to-one func... |
clwlkf1oclwwlk 26378 | There is a one-to-one onto... |
clwlksizeeq 26379 | The size of the set of clo... |
clwlkndivn 26380 | The size of the set of clo... |
el2wlkonotlem 26389 | Lemma for ~ el2wlkonot . ... |
is2wlkonot 26390 | The set of walks of length... |
is2spthonot 26391 | The set of simple paths of... |
2wlkonot 26392 | The set of walks of length... |
2spthonot 26393 | The set of simple paths of... |
2wlksot 26394 | The set of walks of length... |
2spthsot 26395 | The set of simple paths of... |
el2wlkonot 26396 | A walk of length 2 between... |
el2spthonot 26397 | A simple path of length 2 ... |
el2spthonot0 26398 | A simple path of length 2 ... |
el2wlkonotot0 26399 | A walk of length 2 between... |
el2wlkonotot 26400 | A walk of length 2 between... |
el2wlkonotot1 26401 | A walk of length 2 between... |
2wlkonot3v 26402 | If an ordered triple repre... |
2spthonot3v 26403 | If an ordered triple repre... |
2wlkonotv 26404 | If an ordered tripple repr... |
el2wlksoton 26405 | A walk of length 2 between... |
el2spthsoton 26406 | A simple path of length 2 ... |
el2wlksot 26407 | A walk of length 2 between... |
el2pthsot 26408 | A simple path of length 2 ... |
el2wlksotot 26409 | A walk of length 2 between... |
usg2wlkonot 26410 | A walk of length 2 between... |
usg2wotspth 26411 | A walk of length 2 between... |
2pthwlkonot 26412 | For two different vertices... |
2wot2wont 26413 | The set of (simple) paths ... |
2spontn0vne 26414 | If the set of simple paths... |
usg2spthonot 26415 | A simple path of length 2 ... |
usg2spthonot0 26416 | A simple path of length 2 ... |
usg2spthonot1 26417 | A simple path of length 2 ... |
2spot2iun2spont 26418 | The set of simple paths of... |
2spotfi 26419 | In a finite graph, the set... |
vdgrfval 26422 | The value of the vertex de... |
vdgrval 26423 | The value of the vertex de... |
vdgrfival 26424 | The value of the vertex de... |
vdgrf 26425 | The vertex degree function... |
vdgrfif 26426 | The vertex degree function... |
vdgr0 26427 | The degree of a vertex in ... |
vdgrun 26428 | The degree of a vertex in ... |
vdgrfiun 26429 | The degree of a vertex in ... |
vdgr1d 26430 | The vertex degree of a one... |
vdgr1b 26431 | The vertex degree of a one... |
vdgr1c 26432 | The vertex degree of a one... |
vdgr1a 26433 | The vertex degree of a one... |
vdusgraval 26434 | The value of the vertex de... |
vdusgra0nedg 26435 | If a vertex in a simple gr... |
vdgrnn0pnf 26436 | The degree of a vertex is ... |
usgfidegfi 26437 | In a finite graph, the deg... |
usgfiregdegfi 26438 | In a finite graph, the deg... |
hashnbgravd 26439 | The size of the set of the... |
hashnbgravdg 26440 | The size of the set of the... |
nbhashnn0 26441 | The number of the neighbor... |
nbhashuvtx1 26442 | If the number of the neigh... |
nbhashuvtx 26443 | If the number of the neigh... |
uvtxhashnb 26444 | A universal vertex has ` n... |
usgravd0nedg 26445 | If a vertex in a simple gr... |
usgravd00 26446 | If every vertex in a simpl... |
usgrauvtxvdbi 26447 | In a finite undirected sim... |
vdiscusgra 26448 | In a finite complete undir... |
isrgra 26453 | The property of being a k-... |
isrusgra 26454 | The property of being a k-... |
rgraprop 26455 | The properties of a k-regu... |
rusgraprop 26456 | The properties of a k-regu... |
rusgrargra 26457 | A k-regular undirected sim... |
rusisusgra 26458 | Any k-regular undirected s... |
isrusgusrg 26459 | A graph is a k-regular und... |
isrusgusrgcl 26460 | A graph represented by a c... |
isrgrac 26461 | The property of being a k-... |
isrusgrac 26462 | The property of being a k-... |
0egra0rgra 26463 | A graph is 0-regular if it... |
0vgrargra 26464 | A graph with no vertices i... |
cusgraisrusgra 26465 | A complete undirected simp... |
0eusgraiff0rgra 26466 | An undirected simple graph... |
cusgraiffrusgra 26467 | A finite undirected simple... |
0eusgraiff0rgracl 26468 | An undirected simple graph... |
rusgraprop2 26469 | The properties of a k-regu... |
rusgraprop3 26470 | The properties of a k-regu... |
rusgraprop4 26471 | The properties of a k-regu... |
rusgrasn 26472 | If a k-regular undirected ... |
rusgranumwwlkl1 26473 | In a k-regular graph, the ... |
rusgranumwlkl1 26474 | In a k-regular graph, ther... |
rusgranumwlklem0 26475 | Lemma 0 for ~ rusgranumwlk... |
rusgranumwlklem1 26476 | Lemma 1 for ~ rusgranumwlk... |
rusgranumwlklem2 26477 | Lemma 2 for ~ rusgranumwlk... |
rusgranumwlklem3 26478 | Lemma 3 for ~ rusgranumwlk... |
rusgranumwlklem4 26479 | Lemma 4 for ~ rusgranumwlk... |
rusgranumwlkb0 26480 | Induction base 0 for ~ rus... |
rusgranumwlkb1 26481 | Induction base 1 for ~ rus... |
rusgra0edg 26482 | Special case for graphs wi... |
rusgranumwlks 26483 | Induction step for ~ rusgr... |
rusgranumwlk 26484 | In a k-regular graph, the ... |
rusgranumwlkg 26485 | In a k-regular graph, the ... |
rusgranumwwlkg 26486 | In a k-regular graph, the ... |
clwlknclwlkdifs 26487 | The set of walks of length... |
clwlknclwlkdifnum 26488 | In a k-regular graph, the ... |
releupa 26491 | The set ` ( V EulPaths E )... |
iseupa 26492 | The property " ` <. F , P ... |
eupagra 26493 | If an eulerian path exists... |
eupai 26494 | Properties of an Eulerian ... |
eupatrl 26495 | An Eulerian path is a trai... |
eupacl 26496 | An Eulerian path has lengt... |
eupaf1o 26497 | The ` F ` function in an E... |
eupafi 26498 | Any graph with an Eulerian... |
eupapf 26499 | The ` P ` function in an E... |
eupaseg 26500 | The ` N ` -th edge in an e... |
eupa0 26501 | There is an Eulerian path ... |
eupares 26502 | The restriction of an Eule... |
eupap1 26503 | Append one path segment to... |
eupath2lem1 26504 | Lemma for ~ eupath2 . (Co... |
eupath2lem2 26505 | Lemma for ~ eupath2 . (Co... |
eupath2lem3 26506 | Lemma for ~ eupath2 . (Co... |
eupath2 26507 | The only vertices of odd d... |
eupath 26508 | A graph with an Eulerian p... |
vdeg0i 26509 | The base case for the indu... |
umgrabi 26510 | Show that an unordered pai... |
vdegp1ai 26511 | The induction step for a v... |
vdegp1bi 26512 | The induction step for a v... |
vdegp1ci 26513 | The induction step for a v... |
konigsberg 26514 | The Konigsberg Bridge prob... |
isfrgra 26517 | The property of being a fr... |
frisusgrapr 26518 | A friendship graph is an u... |
frisusgra 26519 | A friendship graph is an u... |
frgra0v 26520 | Any graph with no vertex i... |
frgra0 26521 | Any empty graph (graph wit... |
frgraunss 26522 | Any two (different) vertic... |
frgraun 26523 | Any two (different) vertic... |
frisusgranb 26524 | In a friendship graph, the... |
frgra1v 26525 | Any graph with only one ve... |
frgra2v 26526 | Any graph with two (differ... |
frgra3vlem1 26527 | Lemma 1 for ~ frgra3v . (... |
frgra3vlem2 26528 | Lemma 2 for ~ frgra3v . (... |
frgra3v 26529 | Any graph with three verti... |
1vwmgra 26530 | Every graph with one verte... |
3vfriswmgralem 26531 | Lemma for ~ 3vfriswmgra . ... |
3vfriswmgra 26532 | Every friendship graph wit... |
1to2vfriswmgra 26533 | Every friendship graph wit... |
1to3vfriswmgra 26534 | Every friendship graph wit... |
1to3vfriendship 26535 | The friendship theorem for... |
2pthfrgrarn 26536 | Between any two (different... |
2pthfrgrarn2 26537 | Between any two (different... |
2pthfrgra 26538 | Between any two (different... |
3cyclfrgrarn1 26539 | Every vertex in a friendsh... |
3cyclfrgrarn 26540 | Every vertex in a friendsh... |
3cyclfrgrarn2 26541 | Every vertex in a friendsh... |
3cyclfrgra 26542 | Every vertex in a friendsh... |
4cycl2v2nb 26543 | In a (maybe degenerated) 4... |
4cycl2vnunb 26544 | In a 4-cycle, two distinct... |
n4cyclfrgra 26545 | There is no 4-cycle in a f... |
4cyclusnfrgra 26546 | A graph with a 4-cycle is ... |
frgranbnb 26547 | If two neighbors of a spec... |
frconngra 26548 | A friendship graph is conn... |
vdfrgra0 26549 | A vertex in a friendship g... |
vdn0frgrav2 26550 | A vertex in a friendship g... |
vdgn0frgrav2 26551 | A vertex in a friendship g... |
vdn1frgrav2 26552 | Any vertex in a friendship... |
vdgn1frgrav2 26553 | Any vertex in a friendship... |
vdgfrgragt2 26554 | Any vertex in a friendship... |
vdgn1frgrav3 26555 | Any vertex in a friendship... |
usgn0fidegnn0 26556 | In a nonempty, finite grap... |
frgrancvvdeqlem1 26557 | Lemma 1 for ~ frgrancvvdeq... |
frgrancvvdeqlem2 26558 | Lemma 2 for ~ frgrancvvdeq... |
frgrancvvdeqlem3 26559 | Lemma 3 for ~ frgrancvvdeq... |
frgrancvvdeqlem4 26560 | Lemma 4 for ~ frgrancvvdeq... |
frgrancvvdeqlem5 26561 | Lemma 5 for ~ frgrancvvdeq... |
frgrancvvdeqlem6 26562 | Lemma 6 for ~ frgrancvvdeq... |
frgrancvvdeqlem7 26563 | Lemma 7 for ~ frgrancvvdeq... |
frgrancvvdeqlemA 26564 | Lemma A for ~ frgrancvvdeq... |
frgrancvvdeqlemB 26565 | Lemma B for ~ frgrancvvdeq... |
frgrancvvdeqlemC 26566 | Lemma C for ~ frgrancvvdeq... |
frgrancvvdeqlem8 26567 | Lemma 8 for ~ frgrancvvdeq... |
frgrancvvdeqlem9 26568 | Lemma 9 for ~ frgrancvvdeq... |
frgrancvvdeq 26569 | In a finite friendship gra... |
frgrancvvdgeq 26570 | In a friendship graph, two... |
frgrawopreglem1 26571 | Lemma 1 for ~ frgrawopreg ... |
frgrawopreglem2 26572 | Lemma 2 for ~ frgrawopreg ... |
frgrawopreglem3 26573 | Lemma 3 for ~ frgrawopreg ... |
frgrawopreglem4 26574 | Lemma 4 for ~ frgrawopreg ... |
frgrawopreglem5 26575 | Lemma 5 for ~ frgrawopreg ... |
frgrawopreg 26576 | In a friendship graph ther... |
frgrawopreg1 26577 | According to statement 5 i... |
frgrawopreg2 26578 | According to statement 5 i... |
frgraregorufr0 26579 | In a friendship graph ther... |
frgraregorufr 26580 | If there is a vertex havin... |
frgraeu 26581 | Any two (different) vertic... |
frg2woteu 26582 | For two different vertices... |
frg2wotn0 26583 | In a friendship graph, the... |
frg2wot1 26584 | In a friendship graph, the... |
frg2spot1 26585 | In a friendship graph, the... |
frg2woteqm 26586 | There is a (simple) path o... |
frg2woteq 26587 | There is a (simple) path o... |
2spotdisj 26588 | All simple paths of length... |
2spotiundisj 26589 | All simple paths of length... |
frghash2spot 26590 | The number of simple paths... |
2spot0 26591 | If there are no vertices, ... |
usg2spot2nb 26592 | The set of paths of length... |
usgreghash2spotv 26593 | According to statement 7 i... |
usgreg2spot 26594 | In a finite k-regular grap... |
2spotmdisj 26595 | The sets of paths of lengt... |
usgreghash2spot 26596 | In a finite k-regular grap... |
frgregordn0 26597 | If a nonempty friendship g... |
frrusgraord 26598 | If a nonempty finite frien... |
frgraregorufrg 26599 | If there is a vertex havin... |
numclwlk3lem3 26600 | Lemma 3 for ~ numclwwlk3 .... |
extwwlkfablem1 26601 | Lemma 1 for ~ extwwlkfab .... |
extwwlkfablem2lem 26602 | Lemma for ~ extwwlkfablem2... |
clwwlkextfrlem1 26603 | Lemma for ~ numclwwlk2lem1... |
numclwwlkfvc 26604 | Value of function ` C ` , ... |
extwwlkfablem2 26605 | Lemma 2 for ~ extwwlkfab .... |
numclwwlkun 26606 | The set of closed walks in... |
numclwwlkdisj 26607 | The sets of closed walks s... |
numclwwlkovf 26608 | Value of operation ` F ` ,... |
numclwwlkffin 26609 | In a finite graph, the val... |
numclwwlkovfel2 26610 | Properties of an element o... |
numclwwlkovf2 26611 | Value of operation ` F ` f... |
numclwwlkovf2num 26612 | In a k regular graph, ther... |
numclwwlkovf2ex 26613 | Extending a closed walk st... |
numclwwlkovg 26614 | Value of operation ` G ` ,... |
numclwwlkovgel 26615 | Properties of an element o... |
numclwwlkovgelim 26616 | Properties of an element o... |
extwwlkfab 26617 | The set of closed walks (h... |
numclwlk1lem2foa 26618 | Going forth and back form ... |
numclwlk1lem2f 26619 | T is a function. (Contrib... |
numclwlk1lem2fv 26620 | Value of the function T. (... |
numclwlk1lem2f1 26621 | T is a 1-1 function. (Con... |
numclwlk1lem2fo 26622 | T is an onto function. (C... |
numclwlk1lem2f1o 26623 | T is a 1-1 onto function. ... |
numclwlk1lem2 26624 | There is a bijection betwe... |
numclwwlk1 26625 | Statement 9 in [Huneke] p.... |
numclwwlkovq 26626 | Value of operation Q, mapp... |
numclwwlkqhash 26627 | In a k-regular graph, the ... |
numclwwlkovh 26628 | Value of operation H, mapp... |
numclwwlk2lem1 26629 | In a friendship graph, for... |
numclwlk2lem2f 26630 | R is a function. (Contrib... |
numclwlk2lem2fv 26631 | Value of the function R. (... |
numclwlk2lem2f1o 26632 | R is a 1-1 onto function. ... |
numclwwlk2lem3 26633 | In a friendship graph, the... |
numclwwlk2 26634 | Statement 10 in [Huneke] p... |
numclwwlk3lem 26635 | Lemma for ~ numclwwlk3 . ... |
numclwwlk3 26636 | Statement 12 in [Huneke] p... |
numclwwlk4 26637 | The total number of closed... |
numclwwlk5lem 26638 | Lemma for ~ numclwwlk5 . ... |
numclwwlk5 26639 | Statement 13 in [Huneke] p... |
numclwwlk6 26640 | For a prime divisor p of k... |
numclwwlk7 26641 | Statement 14 in [Huneke] p... |
numclwwlk8 26642 | The size of the set of clo... |
frgrareggt1 26643 | If a finite friendship gra... |
frgrareg 26644 | If a finite friendship gra... |
frgraregord013 26645 | If a finite friendship gra... |
frgraregord13 26646 | If a nonempty finite frien... |
frgraogt3nreg 26647 | If a finite friendship gra... |
friendshipgt3 26648 | The friendship theorem for... |
friendship 26649 | The friendship theorem: I... |
conventions 26650 |
... |
conventions-label 26651 |
... |
natded 26652 | Here are typical n... |
ex-natded5.2 26653 | Theorem 5.2 of [Clemente] ... |
ex-natded5.2-2 26654 | A more efficient proof of ... |
ex-natded5.2i 26655 | The same as ~ ex-natded5.2... |
ex-natded5.3 26656 | Theorem 5.3 of [Clemente] ... |
ex-natded5.3-2 26657 | A more efficient proof of ... |
ex-natded5.3i 26658 | The same as ~ ex-natded5.3... |
ex-natded5.5 26659 | Theorem 5.5 of [Clemente] ... |
ex-natded5.7 26660 | Theorem 5.7 of [Clemente] ... |
ex-natded5.7-2 26661 | A more efficient proof of ... |
ex-natded5.8 26662 | Theorem 5.8 of [Clemente] ... |
ex-natded5.8-2 26663 | A more efficient proof of ... |
ex-natded5.13 26664 | Theorem 5.13 of [Clemente]... |
ex-natded5.13-2 26665 | A more efficient proof of ... |
ex-natded9.20 26666 | Theorem 9.20 of [Clemente]... |
ex-natded9.20-2 26667 | A more efficient proof of ... |
ex-natded9.26 26668 | Theorem 9.26 of [Clemente]... |
ex-natded9.26-2 26669 | A more efficient proof of ... |
ex-or 26670 | Example for ~ df-or . Exa... |
ex-an 26671 | Example for ~ df-an . Exa... |
ex-dif 26672 | Example for ~ df-dif . Ex... |
ex-un 26673 | Example for ~ df-un . Exa... |
ex-in 26674 | Example for ~ df-in . Exa... |
ex-uni 26675 | Example for ~ df-uni . Ex... |
ex-ss 26676 | Example for ~ df-ss . Exa... |
ex-pss 26677 | Example for ~ df-pss . Ex... |
ex-pw 26678 | Example for ~ df-pw . Exa... |
ex-pr 26679 | Example for ~ df-pr . (Co... |
ex-br 26680 | Example for ~ df-br . Exa... |
ex-opab 26681 | Example for ~ df-opab . E... |
ex-eprel 26682 | Example for ~ df-eprel . ... |
ex-id 26683 | Example for ~ df-id . Exa... |
ex-po 26684 | Example for ~ df-po . Exa... |
ex-xp 26685 | Example for ~ df-xp . Exa... |
ex-cnv 26686 | Example for ~ df-cnv . Ex... |
ex-co 26687 | Example for ~ df-co . Exa... |
ex-dm 26688 | Example for ~ df-dm . Exa... |
ex-rn 26689 | Example for ~ df-rn . Exa... |
ex-res 26690 | Example for ~ df-res . Ex... |
ex-ima 26691 | Example for ~ df-ima . Ex... |
ex-fv 26692 | Example for ~ df-fv . Exa... |
ex-1st 26693 | Example for ~ df-1st . Ex... |
ex-2nd 26694 | Example for ~ df-2nd . Ex... |
1kp2ke3k 26695 | Example for ~ df-dec , 100... |
ex-fl 26696 | Example for ~ df-fl . Exa... |
ex-ceil 26697 | Example for ~ df-ceil . (... |
ex-mod 26698 | Example for ~ df-mod . (C... |
ex-exp 26699 | Example for ~ df-exp . (C... |
ex-fac 26700 | Example for ~ df-fac . (C... |
ex-bc 26701 | Example for ~ df-bc . (Co... |
ex-hash 26702 | Example for ~ df-hash . (... |
ex-sqrt 26703 | Example for ~ df-sqrt . (... |
ex-abs 26704 | Example for ~ df-abs . (C... |
ex-dvds 26705 | Example for ~ df-dvds : 3 ... |
ex-gcd 26706 | Example for ~ df-gcd . (C... |
ex-lcm 26707 | Example for ~ df-lcm . (C... |
ex-prmo 26708 | Example for ~ df-prmo : ` ... |
aevdemo 26709 | Proof illustrating the com... |
ex-ind-dvds 26710 | Example of a proof by indu... |
avril1 26711 | Poisson d'Avril's Theorem.... |
2bornot2b 26712 | The law of excluded middle... |
helloworld 26713 | The classic "Hello world" ... |
1p1e2apr1 26714 | One plus one equals two. ... |
eqid1 26715 | Law of identity (reflexivi... |
1div0apr 26716 | Division by zero is forbid... |
topnfbey 26717 | Nothing seems to be imposs... |
isplig 26720 | The predicate "is a planar... |
tncp 26721 | In any planar incidence ge... |
lpni 26722 | For any line in a planar i... |
dummylink 26725 | Alias for ~ a1ii that may ... |
id1 26726 | Alias for ~ idALT that may... |
isgrpo 26735 | The predicate "is a group ... |
isgrpoi 26736 | Properties that determine ... |
grpofo 26737 | A group operation maps ont... |
grpocl 26738 | Closure law for a group op... |
grpolidinv 26739 | A group has a left identit... |
grpon0 26740 | The base set of a group is... |
grpoass 26741 | A group operation is assoc... |
grpoidinvlem1 26742 | Lemma for ~ grpoidinv . (... |
grpoidinvlem2 26743 | Lemma for ~ grpoidinv . (... |
grpoidinvlem3 26744 | Lemma for ~ grpoidinv . (... |
grpoidinvlem4 26745 | Lemma for ~ grpoidinv . (... |
grpoidinv 26746 | A group has a left and rig... |
grpoideu 26747 | The left identity element ... |
grporndm 26748 | A group's range in terms o... |
0ngrp 26749 | The empty set is not a gro... |
gidval 26750 | The value of the identity ... |
grpoidval 26751 | Lemma for ~ grpoidcl and o... |
grpoidcl 26752 | The identity element of a ... |
grpoidinv2 26753 | A group's properties using... |
grpolid 26754 | The identity element of a ... |
grporid 26755 | The identity element of a ... |
grporcan 26756 | Right cancellation law for... |
grpoinveu 26757 | The left inverse element o... |
grpoid 26758 | Two ways of saying that an... |
grporn 26759 | The range of a group opera... |
grpoinvfval 26760 | The inverse function of a ... |
grpoinvval 26761 | The inverse of a group ele... |
grpoinvcl 26762 | A group element's inverse ... |
grpoinv 26763 | The properties of a group ... |
grpolinv 26764 | The left inverse of a grou... |
grporinv 26765 | The right inverse of a gro... |
grpoinvid1 26766 | The inverse of a group ele... |
grpoinvid2 26767 | The inverse of a group ele... |
grpolcan 26768 | Left cancellation law for ... |
grpo2inv 26769 | Double inverse law for gro... |
grpoinvf 26770 | Mapping of the inverse fun... |
grpoinvop 26771 | The inverse of the group o... |
grpodivfval 26772 | Group division (or subtrac... |
grpodivval 26773 | Group division (or subtrac... |
grpodivinv 26774 | Group division by an inver... |
grpoinvdiv 26775 | Inverse of a group divisio... |
grpodivf 26776 | Mapping for group division... |
grpodivcl 26777 | Closure of group division ... |
grpodivdiv 26778 | Double group division. (C... |
grpomuldivass 26779 | Associative-type law for m... |
grpodivid 26780 | Division of a group member... |
grponpcan 26781 | Cancellation law for group... |
isablo 26784 | The predicate "is an Abeli... |
ablogrpo 26785 | An Abelian group operation... |
ablocom 26786 | An Abelian group operation... |
ablo32 26787 | Commutative/associative la... |
ablo4 26788 | Commutative/associative la... |
isabloi 26789 | Properties that determine ... |
ablomuldiv 26790 | Law for group multiplicati... |
ablodivdiv 26791 | Law for double group divis... |
ablodivdiv4 26792 | Law for double group divis... |
ablodiv32 26793 | Swap the second and third ... |
ablonnncan 26794 | Cancellation law for group... |
ablonncan 26795 | Cancellation law for group... |
ablonnncan1 26796 | Cancellation law for group... |
vcrel 26799 | The class of all complex v... |
vciOLD 26800 | Obsolete version of ~ cvsi... |
vcsm 26801 | Functionality of th scalar... |
vccl 26802 | Closure of the scalar prod... |
vcidOLD 26803 | Identity element for the s... |
vcdi 26804 | Distributive law for the s... |
vcdir 26805 | Distributive law for the s... |
vcass 26806 | Associative law for the sc... |
vc2OLD 26807 | A vector plus itself is tw... |
vcablo 26808 | Vector addition is an Abel... |
vcgrp 26809 | Vector addition is a group... |
vclcan 26810 | Left cancellation law for ... |
vczcl 26811 | The zero vector is a vecto... |
vc0rid 26812 | The zero vector is a right... |
vc0 26813 | Zero times a vector is the... |
vcz 26814 | Anything times the zero ve... |
vcm 26815 | Minus 1 times a vector is ... |
isvclem 26816 | Lemma for ~ isvcOLD . (Co... |
vcex 26817 | The components of a comple... |
isvcOLD 26818 | The predicate "is a comple... |
isvciOLD 26819 | Properties that determine ... |
cnaddabloOLD 26820 | Obsolete as of 23-Jan-2020... |
cnidOLD 26821 | Obsolete as of 23-Jan-2020... |
cncvcOLD 26822 | Obsolete version of ~ cncv... |
nvss 26832 | Structure of the class of ... |
nvvcop 26833 | A normed complex vector sp... |
nvrel 26841 | The class of all normed co... |
vafval 26842 | Value of the function for ... |
bafval 26843 | Value of the function for ... |
smfval 26844 | Value of the function for ... |
0vfval 26845 | Value of the function for ... |
nmcvfval 26846 | Value of the norm function... |
nvop2 26847 | A normed complex vector sp... |
nvvop 26848 | The vector space component... |
isnvlem 26849 | Lemma for ~ isnv . (Contr... |
nvex 26850 | The components of a normed... |
isnv 26851 | The predicate "is a normed... |
isnvi 26852 | Properties that determine ... |
nvi 26853 | The properties of a normed... |
nvvc 26854 | The vector space component... |
nvablo 26855 | The vector addition operat... |
nvgrp 26856 | The vector addition operat... |
nvgf 26857 | Mapping for the vector add... |
nvsf 26858 | Mapping for the scalar mul... |
nvgcl 26859 | Closure law for the vector... |
nvcom 26860 | The vector addition (group... |
nvass 26861 | The vector addition (group... |
nvadd32 26862 | Commutative/associative la... |
nvrcan 26863 | Right cancellation law for... |
nvadd4 26864 | Rearrangement of 4 terms i... |
nvscl 26865 | Closure law for the scalar... |
nvsid 26866 | Identity element for the s... |
nvsass 26867 | Associative law for the sc... |
nvscom 26868 | Commutative law for the sc... |
nvdi 26869 | Distributive law for the s... |
nvdir 26870 | Distributive law for the s... |
nv2 26871 | A vector plus itself is tw... |
vsfval 26872 | Value of the function for ... |
nvzcl 26873 | Closure law for the zero v... |
nv0rid 26874 | The zero vector is a right... |
nv0lid 26875 | The zero vector is a left ... |
nv0 26876 | Zero times a vector is the... |
nvsz 26877 | Anything times the zero ve... |
nvinv 26878 | Minus 1 times a vector is ... |
nvinvfval 26879 | Function for the negative ... |
nvm 26880 | Vector subtraction in term... |
nvmval 26881 | Value of vector subtractio... |
nvmval2 26882 | Value of vector subtractio... |
nvmfval 26883 | Value of the function for ... |
nvmf 26884 | Mapping for the vector sub... |
nvmcl 26885 | Closure law for the vector... |
nvnnncan1 26886 | Cancellation law for vecto... |
nvmdi 26887 | Distributive law for scala... |
nvnegneg 26888 | Double negative of a vecto... |
nvmul0or 26889 | If a scalar product is zer... |
nvrinv 26890 | A vector minus itself. (C... |
nvlinv 26891 | Minus a vector plus itself... |
nvpncan2 26892 | Cancellation law for vecto... |
nvpncan 26893 | Cancellation law for vecto... |
nvaddsub 26894 | Commutative/associative la... |
nvnpcan 26895 | Cancellation law for a nor... |
nvaddsub4 26896 | Rearrangement of 4 terms i... |
nvmeq0 26897 | The difference between two... |
nvmid 26898 | A vector minus itself is t... |
nvf 26899 | Mapping for the norm funct... |
nvcl 26900 | The norm of a normed compl... |
nvcli 26901 | The norm of a normed compl... |
nvs 26902 | Proportionality property o... |
nvsge0 26903 | The norm of a scalar produ... |
nvm1 26904 | The norm of the negative o... |
nvdif 26905 | The norm of the difference... |
nvpi 26906 | The norm of a vector plus ... |
nvz0 26907 | The norm of a zero vector ... |
nvz 26908 | The norm of a vector is ze... |
nvtri 26909 | Triangle inequality for th... |
nvmtri 26910 | Triangle inequality for th... |
nvabs 26911 | Norm difference property o... |
nvge0 26912 | The norm of a normed compl... |
nvgt0 26913 | A nonzero norm is positive... |
nv1 26914 | From any nonzero vector, c... |
nvop 26915 | A complex inner product sp... |
cnnv 26916 | The set of complex numbers... |
cnnvg 26917 | The vector addition (group... |
cnnvba 26918 | The base set of the normed... |
cnnvs 26919 | The scalar product operati... |
cnnvnm 26920 | The norm operation of the ... |
cnnvm 26921 | The vector subtraction ope... |
elimnv 26922 | Hypothesis elimination lem... |
elimnvu 26923 | Hypothesis elimination lem... |
imsval 26924 | Value of the induced metri... |
imsdval 26925 | Value of the induced metri... |
imsdval2 26926 | Value of the distance func... |
nvnd 26927 | The norm of a normed compl... |
imsdf 26928 | Mapping for the induced me... |
imsmetlem 26929 | Lemma for ~ imsmet . (Con... |
imsmet 26930 | The induced metric of a no... |
imsxmet 26931 | The induced metric of a no... |
cnims 26932 | The metric induced on the ... |
vacn 26933 | Vector addition is jointly... |
nmcvcn 26934 | The norm of a normed compl... |
nmcnc 26935 | The norm of a normed compl... |
smcnlem 26936 | Lemma for ~ smcn . (Contr... |
smcn 26937 | Scalar multiplication is j... |
vmcn 26938 | Vector subtraction is join... |
dipfval 26941 | The inner product function... |
ipval 26942 | Value of the inner product... |
ipval2lem2 26943 | Lemma for ~ ipval3 . (Con... |
ipval2lem3 26944 | Lemma for ~ ipval3 . (Con... |
ipval2lem4 26945 | Lemma for ~ ipval3 . (Con... |
ipval2 26946 | Expansion of the inner pro... |
4ipval2 26947 | Four times the inner produ... |
ipval3 26948 | Expansion of the inner pro... |
ipidsq 26949 | The inner product of a vec... |
ipnm 26950 | Norm expressed in terms of... |
dipcl 26951 | An inner product is a comp... |
ipf 26952 | Mapping for the inner prod... |
dipcj 26953 | The complex conjugate of a... |
ipipcj 26954 | An inner product times its... |
diporthcom 26955 | Orthogonality (meaning inn... |
dip0r 26956 | Inner product with a zero ... |
dip0l 26957 | Inner product with a zero ... |
ipz 26958 | The inner product of a vec... |
dipcn 26959 | Inner product is jointly c... |
sspval 26962 | The set of all subspaces o... |
isssp 26963 | The predicate "is a subspa... |
sspid 26964 | A normed complex vector sp... |
sspnv 26965 | A subspace is a normed com... |
sspba 26966 | The base set of a subspace... |
sspg 26967 | Vector addition on a subsp... |
sspgval 26968 | Vector addition on a subsp... |
ssps 26969 | Scalar multiplication on a... |
sspsval 26970 | Scalar multiplication on a... |
sspmlem 26971 | Lemma for ~ sspm and other... |
sspmval 26972 | Vector addition on a subsp... |
sspm 26973 | Vector subtraction on a su... |
sspz 26974 | The zero vector of a subsp... |
sspn 26975 | The norm on a subspace is ... |
sspnval 26976 | The norm on a subspace in ... |
sspimsval 26977 | The induced metric on a su... |
sspims 26978 | The induced metric on a su... |
lnoval 26991 | The set of linear operator... |
islno 26992 | The predicate "is a linear... |
lnolin 26993 | Basic linearity property o... |
lnof 26994 | A linear operator is a map... |
lno0 26995 | The value of a linear oper... |
lnocoi 26996 | The composition of two lin... |
lnoadd 26997 | Addition property of a lin... |
lnosub 26998 | Subtraction property of a ... |
lnomul 26999 | Scalar multiplication prop... |
nvo00 27000 | Two ways to express a zero... |
nmoofval 27001 | The operator norm function... |
nmooval 27002 | The operator norm function... |
nmosetre 27003 | The set in the supremum of... |
nmosetn0 27004 | The set in the supremum of... |
nmoxr 27005 | The norm of an operator is... |
nmooge0 27006 | The norm of an operator is... |
nmorepnf 27007 | The norm of an operator is... |
nmoreltpnf 27008 | The norm of any operator i... |
nmogtmnf 27009 | The norm of an operator is... |
nmoolb 27010 | A lower bound for an opera... |
nmoubi 27011 | An upper bound for an oper... |
nmoub3i 27012 | An upper bound for an oper... |
nmoub2i 27013 | An upper bound for an oper... |
nmobndi 27014 | Two ways to express that a... |
nmounbi 27015 | Two ways two express that ... |
nmounbseqi 27016 | An unbounded operator dete... |
nmounbseqiALT 27017 | Alternate shorter proof of... |
nmobndseqi 27018 | A bounded sequence determi... |
nmobndseqiALT 27019 | Alternate shorter proof of... |
bloval 27020 | The class of bounded linea... |
isblo 27021 | The predicate "is a bounde... |
isblo2 27022 | The predicate "is a bounde... |
bloln 27023 | A bounded operator is a li... |
blof 27024 | A bounded operator is an o... |
nmblore 27025 | The norm of a bounded oper... |
0ofval 27026 | The zero operator between ... |
0oval 27027 | Value of the zero operator... |
0oo 27028 | The zero operator is an op... |
0lno 27029 | The zero operator is linea... |
nmoo0 27030 | The operator norm of the z... |
0blo 27031 | The zero operator is a bou... |
nmlno0lem 27032 | Lemma for ~ nmlno0i . (Co... |
nmlno0i 27033 | The norm of a linear opera... |
nmlno0 27034 | The norm of a linear opera... |
nmlnoubi 27035 | An upper bound for the ope... |
nmlnogt0 27036 | The norm of a nonzero line... |
lnon0 27037 | The domain of a nonzero li... |
nmblolbii 27038 | A lower bound for the norm... |
nmblolbi 27039 | A lower bound for the norm... |
isblo3i 27040 | The predicate "is a bounde... |
blo3i 27041 | Properties that determine ... |
blometi 27042 | Upper bound for the distan... |
blocnilem 27043 | Lemma for ~ blocni and ~ l... |
blocni 27044 | A linear operator is conti... |
lnocni 27045 | If a linear operator is co... |
blocn 27046 | A linear operator is conti... |
blocn2 27047 | A bounded linear operator ... |
ajfval 27048 | The adjoint function. (Co... |
hmoval 27049 | The set of Hermitian (self... |
ishmo 27050 | The predicate "is a hermit... |
phnv 27053 | Every complex inner produc... |
phrel 27054 | The class of all complex i... |
phnvi 27055 | Every complex inner produc... |
isphg 27056 | The predicate "is a comple... |
phop 27057 | A complex inner product sp... |
cncph 27058 | The set of complex numbers... |
elimph 27059 | Hypothesis elimination lem... |
elimphu 27060 | Hypothesis elimination lem... |
isph 27061 | The predicate "is an inner... |
phpar2 27062 | The parallelogram law for ... |
phpar 27063 | The parallelogram law for ... |
ip0i 27064 | A slight variant of Equati... |
ip1ilem 27065 | Lemma for ~ ip1i . (Contr... |
ip1i 27066 | Equation 6.47 of [Ponnusam... |
ip2i 27067 | Equation 6.48 of [Ponnusam... |
ipdirilem 27068 | Lemma for ~ ipdiri . (Con... |
ipdiri 27069 | Distributive law for inner... |
ipasslem1 27070 | Lemma for ~ ipassi . Show... |
ipasslem2 27071 | Lemma for ~ ipassi . Show... |
ipasslem3 27072 | Lemma for ~ ipassi . Show... |
ipasslem4 27073 | Lemma for ~ ipassi . Show... |
ipasslem5 27074 | Lemma for ~ ipassi . Show... |
ipasslem7 27075 | Lemma for ~ ipassi . Show... |
ipasslem8 27076 | Lemma for ~ ipassi . By ~... |
ipasslem9 27077 | Lemma for ~ ipassi . Conc... |
ipasslem10 27078 | Lemma for ~ ipassi . Show... |
ipasslem11 27079 | Lemma for ~ ipassi . Show... |
ipassi 27080 | Associative law for inner ... |
dipdir 27081 | Distributive law for inner... |
dipdi 27082 | Distributive law for inner... |
ip2dii 27083 | Inner product of two sums.... |
dipass 27084 | Associative law for inner ... |
dipassr 27085 | "Associative" law for seco... |
dipassr2 27086 | "Associative" law for inne... |
dipsubdir 27087 | Distributive law for inner... |
dipsubdi 27088 | Distributive law for inner... |
pythi 27089 | The Pythagorean theorem fo... |
siilem1 27090 | Lemma for ~ sii . (Contri... |
siilem2 27091 | Lemma for ~ sii . (Contri... |
siii 27092 | Inference from ~ sii . (C... |
sii 27093 | Schwarz inequality. Part ... |
sspph 27094 | A subspace of an inner pro... |
ipblnfi 27095 | A function ` F ` generated... |
ip2eqi 27096 | Two vectors are equal iff ... |
phoeqi 27097 | A condition implying that ... |
ajmoi 27098 | Every operator has at most... |
ajfuni 27099 | The adjoint function is a ... |
ajfun 27100 | The adjoint function is a ... |
ajval 27101 | Value of the adjoint funct... |
iscbn 27104 | A complex Banach space is ... |
cbncms 27105 | The induced metric on comp... |
bnnv 27106 | Every complex Banach space... |
bnrel 27107 | The class of all complex B... |
bnsscmcl 27108 | A subspace of a Banach spa... |
cnbn 27109 | The set of complex numbers... |
ubthlem1 27110 | Lemma for ~ ubth . The fu... |
ubthlem2 27111 | Lemma for ~ ubth . Given ... |
ubthlem3 27112 | Lemma for ~ ubth . Prove ... |
ubth 27113 | Uniform Boundedness Theore... |
minvecolem1 27114 | Lemma for ~ minveco . The... |
minvecolem2 27115 | Lemma for ~ minveco . Any... |
minvecolem3 27116 | Lemma for ~ minveco . The... |
minvecolem4a 27117 | Lemma for ~ minveco . ` F ... |
minvecolem4b 27118 | Lemma for ~ minveco . The... |
minvecolem4c 27119 | Lemma for ~ minveco . The... |
minvecolem4 27120 | Lemma for ~ minveco . The... |
minvecolem5 27121 | Lemma for ~ minveco . Dis... |
minvecolem6 27122 | Lemma for ~ minveco . Any... |
minvecolem7 27123 | Lemma for ~ minveco . Sin... |
minveco 27124 | Minimizing vector theorem,... |
ishlo 27127 | The predicate "is a comple... |
hlobn 27128 | Every complex Hilbert spac... |
hlph 27129 | Every complex Hilbert spac... |
hlrel 27130 | The class of all complex H... |
hlnv 27131 | Every complex Hilbert spac... |
hlnvi 27132 | Every complex Hilbert spac... |
hlvc 27133 | Every complex Hilbert spac... |
hlcmet 27134 | The induced metric on a co... |
hlmet 27135 | The induced metric on a co... |
hlpar2 27136 | The parallelogram law sati... |
hlpar 27137 | The parallelogram law sati... |
hlex 27138 | The base set of a Hilbert ... |
hladdf 27139 | Mapping for Hilbert space ... |
hlcom 27140 | Hilbert space vector addit... |
hlass 27141 | Hilbert space vector addit... |
hl0cl 27142 | The Hilbert space zero vec... |
hladdid 27143 | Hilbert space addition wit... |
hlmulf 27144 | Mapping for Hilbert space ... |
hlmulid 27145 | Hilbert space scalar multi... |
hlmulass 27146 | Hilbert space scalar multi... |
hldi 27147 | Hilbert space scalar multi... |
hldir 27148 | Hilbert space scalar multi... |
hlmul0 27149 | Hilbert space scalar multi... |
hlipf 27150 | Mapping for Hilbert space ... |
hlipcj 27151 | Conjugate law for Hilbert ... |
hlipdir 27152 | Distributive law for Hilbe... |
hlipass 27153 | Associative law for Hilber... |
hlipgt0 27154 | The inner product of a Hil... |
hlcompl 27155 | Completeness of a Hilbert ... |
cnchl 27156 | The set of complex numbers... |
ssphl 27157 | A Banach subspace of an in... |
htthlem 27158 | Lemma for ~ htth . The co... |
htth 27159 | Hellinger-Toeplitz Theorem... |
The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
h2hva 27215 | The group (addition) opera... |
h2hsm 27216 | The scalar product operati... |
h2hnm 27217 | The norm function of Hilbe... |
h2hvs 27218 | The vector subtraction ope... |
h2hmetdval 27219 | Value of the distance func... |
h2hcau 27220 | The Cauchy sequences of Hi... |
h2hlm 27221 | The limit sequences of Hil... |
axhilex-zf 27222 | Derive axiom ~ ax-hilex fr... |
axhfvadd-zf 27223 | Derive axiom ~ ax-hfvadd f... |
axhvcom-zf 27224 | Derive axiom ~ ax-hvcom fr... |
axhvass-zf 27225 | Derive axiom ~ ax-hvass fr... |
axhv0cl-zf 27226 | Derive axiom ~ ax-hv0cl fr... |
axhvaddid-zf 27227 | Derive axiom ~ ax-hvaddid ... |
axhfvmul-zf 27228 | Derive axiom ~ ax-hfvmul f... |
axhvmulid-zf 27229 | Derive axiom ~ ax-hvmulid ... |
axhvmulass-zf 27230 | Derive axiom ~ ax-hvmulass... |
axhvdistr1-zf 27231 | Derive axiom ~ ax-hvdistr1... |
axhvdistr2-zf 27232 | Derive axiom ~ ax-hvdistr2... |
axhvmul0-zf 27233 | Derive axiom ~ ax-hvmul0 f... |
axhfi-zf 27234 | Derive axiom ~ ax-hfi from... |
axhis1-zf 27235 | Derive axiom ~ ax-his1 fro... |
axhis2-zf 27236 | Derive axiom ~ ax-his2 fro... |
axhis3-zf 27237 | Derive axiom ~ ax-his3 fro... |
axhis4-zf 27238 | Derive axiom ~ ax-his4 fro... |
axhcompl-zf 27239 | Derive axiom ~ ax-hcompl f... |
hvmulex 27252 | The Hilbert space scalar p... |
hvaddcl 27253 | Closure of vector addition... |
hvmulcl 27254 | Closure of scalar multipli... |
hvmulcli 27255 | Closure inference for scal... |
hvsubf 27256 | Mapping domain and codomai... |
hvsubval 27257 | Value of vector subtractio... |
hvsubcl 27258 | Closure of vector subtract... |
hvaddcli 27259 | Closure of vector addition... |
hvcomi 27260 | Commutation of vector addi... |
hvsubvali 27261 | Value of vector subtractio... |
hvsubcli 27262 | Closure of vector subtract... |
ifhvhv0 27263 | Prove ` if ( A e. ~H , A ,... |
hvaddid2 27264 | Addition with the zero vec... |
hvmul0 27265 | Scalar multiplication with... |
hvmul0or 27266 | If a scalar product is zer... |
hvsubid 27267 | Subtraction of a vector fr... |
hvnegid 27268 | Addition of negative of a ... |
hv2neg 27269 | Two ways to express the ne... |
hvaddid2i 27270 | Addition with the zero vec... |
hvnegidi 27271 | Addition of negative of a ... |
hv2negi 27272 | Two ways to express the ne... |
hvm1neg 27273 | Convert minus one times a ... |
hvaddsubval 27274 | Value of vector addition i... |
hvadd32 27275 | Commutative/associative la... |
hvadd12 27276 | Commutative/associative la... |
hvadd4 27277 | Hilbert vector space addit... |
hvsub4 27278 | Hilbert vector space addit... |
hvaddsub12 27279 | Commutative/associative la... |
hvpncan 27280 | Addition/subtraction cance... |
hvpncan2 27281 | Addition/subtraction cance... |
hvaddsubass 27282 | Associativity of sum and d... |
hvpncan3 27283 | Subtraction and addition o... |
hvmulcom 27284 | Scalar multiplication comm... |
hvsubass 27285 | Hilbert vector space assoc... |
hvsub32 27286 | Hilbert vector space commu... |
hvmulassi 27287 | Scalar multiplication asso... |
hvmulcomi 27288 | Scalar multiplication comm... |
hvmul2negi 27289 | Double negative in scalar ... |
hvsubdistr1 27290 | Scalar multiplication dist... |
hvsubdistr2 27291 | Scalar multiplication dist... |
hvdistr1i 27292 | Scalar multiplication dist... |
hvsubdistr1i 27293 | Scalar multiplication dist... |
hvassi 27294 | Hilbert vector space assoc... |
hvadd32i 27295 | Hilbert vector space commu... |
hvsubassi 27296 | Hilbert vector space assoc... |
hvsub32i 27297 | Hilbert vector space commu... |
hvadd12i 27298 | Hilbert vector space commu... |
hvadd4i 27299 | Hilbert vector space addit... |
hvsubsub4i 27300 | Hilbert vector space addit... |
hvsubsub4 27301 | Hilbert vector space addit... |
hv2times 27302 | Two times a vector. (Cont... |
hvnegdii 27303 | Distribution of negative o... |
hvsubeq0i 27304 | If the difference between ... |
hvsubcan2i 27305 | Vector cancellation law. ... |
hvaddcani 27306 | Cancellation law for vecto... |
hvsubaddi 27307 | Relationship between vecto... |
hvnegdi 27308 | Distribution of negative o... |
hvsubeq0 27309 | If the difference between ... |
hvaddeq0 27310 | If the sum of two vectors ... |
hvaddcan 27311 | Cancellation law for vecto... |
hvaddcan2 27312 | Cancellation law for vecto... |
hvmulcan 27313 | Cancellation law for scala... |
hvmulcan2 27314 | Cancellation law for scala... |
hvsubcan 27315 | Cancellation law for vecto... |
hvsubcan2 27316 | Cancellation law for vecto... |
hvsub0 27317 | Subtraction of a zero vect... |
hvsubadd 27318 | Relationship between vecto... |
hvaddsub4 27319 | Hilbert vector space addit... |
hicl 27321 | Closure of inner product. ... |
hicli 27322 | Closure inference for inne... |
his5 27327 | Associative law for inner ... |
his52 27328 | Associative law for inner ... |
his35 27329 | Move scalar multiplication... |
his35i 27330 | Move scalar multiplication... |
his7 27331 | Distributive law for inner... |
hiassdi 27332 | Distributive/associative l... |
his2sub 27333 | Distributive law for inner... |
his2sub2 27334 | Distributive law for inner... |
hire 27335 | A necessary and sufficient... |
hiidrcl 27336 | Real closure of inner prod... |
hi01 27337 | Inner product with the 0 v... |
hi02 27338 | Inner product with the 0 v... |
hiidge0 27339 | Inner product with self is... |
his6 27340 | Zero inner product with se... |
his1i 27341 | Conjugate law for inner pr... |
abshicom 27342 | Commuted inner products ha... |
hial0 27343 | A vector whose inner produ... |
hial02 27344 | A vector whose inner produ... |
hisubcomi 27345 | Two vector subtractions si... |
hi2eq 27346 | Lemma used to prove equali... |
hial2eq 27347 | Two vectors whose inner pr... |
hial2eq2 27348 | Two vectors whose inner pr... |
orthcom 27349 | Orthogonality commutes. (... |
normlem0 27350 | Lemma used to derive prope... |
normlem1 27351 | Lemma used to derive prope... |
normlem2 27352 | Lemma used to derive prope... |
normlem3 27353 | Lemma used to derive prope... |
normlem4 27354 | Lemma used to derive prope... |
normlem5 27355 | Lemma used to derive prope... |
normlem6 27356 | Lemma used to derive prope... |
normlem7 27357 | Lemma used to derive prope... |
normlem8 27358 | Lemma used to derive prope... |
normlem9 27359 | Lemma used to derive prope... |
normlem7tALT 27360 | Lemma used to derive prope... |
bcseqi 27361 | Equality case of Bunjakova... |
normlem9at 27362 | Lemma used to derive prope... |
dfhnorm2 27363 | Alternate definition of th... |
normf 27364 | The norm function maps fro... |
normval 27365 | The value of the norm of a... |
normcl 27366 | Real closure of the norm o... |
normge0 27367 | The norm of a vector is no... |
normgt0 27368 | The norm of nonzero vector... |
norm0 27369 | The norm of a zero vector.... |
norm-i 27370 | Theorem 3.3(i) of [Beran] ... |
normne0 27371 | A norm is nonzero iff its ... |
normcli 27372 | Real closure of the norm o... |
normsqi 27373 | The square of a norm. (Co... |
norm-i-i 27374 | Theorem 3.3(i) of [Beran] ... |
normsq 27375 | The square of a norm. (Co... |
normsub0i 27376 | Two vectors are equal iff ... |
normsub0 27377 | Two vectors are equal iff ... |
norm-ii-i 27378 | Triangle inequality for no... |
norm-ii 27379 | Triangle inequality for no... |
norm-iii-i 27380 | Theorem 3.3(iii) of [Beran... |
norm-iii 27381 | Theorem 3.3(iii) of [Beran... |
normsubi 27382 | Negative doesn't change th... |
normpythi 27383 | Analogy to Pythagorean the... |
normsub 27384 | Swapping order of subtract... |
normneg 27385 | The norm of a vector equal... |
normpyth 27386 | Analogy to Pythagorean the... |
normpyc 27387 | Corollary to Pythagorean t... |
norm3difi 27388 | Norm of differences around... |
norm3adifii 27389 | Norm of differences around... |
norm3lem 27390 | Lemma involving norm of di... |
norm3dif 27391 | Norm of differences around... |
norm3dif2 27392 | Norm of differences around... |
norm3lemt 27393 | Lemma involving norm of di... |
norm3adifi 27394 | Norm of differences around... |
normpari 27395 | Parallelogram law for norm... |
normpar 27396 | Parallelogram law for norm... |
normpar2i 27397 | Corollary of parallelogram... |
polid2i 27398 | Generalized polarization i... |
polidi 27399 | Polarization identity. Re... |
polid 27400 | Polarization identity. Re... |
hilablo 27401 | Hilbert space vector addit... |
hilid 27402 | The group identity element... |
hilvc 27403 | Hilbert space is a complex... |
hilnormi 27404 | Hilbert space norm in term... |
hilhhi 27405 | Deduce the structure of Hi... |
hhnv 27406 | Hilbert space is a normed ... |
hhva 27407 | The group (addition) opera... |
hhba 27408 | The base set of Hilbert sp... |
hh0v 27409 | The zero vector of Hilbert... |
hhsm 27410 | The scalar product operati... |
hhvs 27411 | The vector subtraction ope... |
hhnm 27412 | The norm function of Hilbe... |
hhims 27413 | The induced metric of Hilb... |
hhims2 27414 | Hilbert space distance met... |
hhmet 27415 | The induced metric of Hilb... |
hhxmet 27416 | The induced metric of Hilb... |
hhmetdval 27417 | Value of the distance func... |
hhip 27418 | The inner product operatio... |
hhph 27419 | The Hilbert space of the H... |
bcsiALT 27420 | Bunjakovaskij-Cauchy-Schwa... |
bcsiHIL 27421 | Bunjakovaskij-Cauchy-Schwa... |
bcs 27422 | Bunjakovaskij-Cauchy-Schwa... |
bcs2 27423 | Corollary of the Bunjakova... |
bcs3 27424 | Corollary of the Bunjakova... |
hcau 27425 | Member of the set of Cauch... |
hcauseq 27426 | A Cauchy sequences on a Hi... |
hcaucvg 27427 | A Cauchy sequence on a Hil... |
seq1hcau 27428 | A sequence on a Hilbert sp... |
hlimi 27429 | Express the predicate: Th... |
hlimseqi 27430 | A sequence with a limit on... |
hlimveci 27431 | Closure of the limit of a ... |
hlimconvi 27432 | Convergence of a sequence ... |
hlim2 27433 | The limit of a sequence on... |
hlimadd 27434 | Limit of the sum of two se... |
hilmet 27435 | The Hilbert space norm det... |
hilxmet 27436 | The Hilbert space norm det... |
hilmetdval 27437 | Value of the distance func... |
hilims 27438 | Hilbert space distance met... |
hhcau 27439 | The Cauchy sequences of Hi... |
hhlm 27440 | The limit sequences of Hil... |
hhcmpl 27441 | Lemma used for derivation ... |
hilcompl 27442 | Lemma used for derivation ... |
hhcms 27444 | The Hilbert space induced ... |
hhhl 27445 | The Hilbert space structur... |
hilcms 27446 | The Hilbert space norm det... |
hilhl 27447 | The Hilbert space of the H... |
issh 27449 | Subspace ` H ` of a Hilber... |
issh2 27450 | Subspace ` H ` of a Hilber... |
shss 27451 | A subspace is a subset of ... |
shel 27452 | A member of a subspace of ... |
shex 27453 | The set of subspaces of a ... |
shssii 27454 | A closed subspace of a Hil... |
sheli 27455 | A member of a subspace of ... |
shelii 27456 | A member of a subspace of ... |
sh0 27457 | The zero vector belongs to... |
shaddcl 27458 | Closure of vector addition... |
shmulcl 27459 | Closure of vector scalar m... |
issh3 27460 | Subspace ` H ` of a Hilber... |
shsubcl 27461 | Closure of vector subtract... |
isch 27463 | Closed subspace ` H ` of a... |
isch2 27464 | Closed subspace ` H ` of a... |
chsh 27465 | A closed subspace is a sub... |
chsssh 27466 | Closed subspaces are subsp... |
chex 27467 | The set of closed subspace... |
chshii 27468 | A closed subspace is a sub... |
ch0 27469 | The zero vector belongs to... |
chss 27470 | A closed subspace of a Hil... |
chel 27471 | A member of a closed subsp... |
chssii 27472 | A closed subspace of a Hil... |
cheli 27473 | A member of a closed subsp... |
chelii 27474 | A member of a closed subsp... |
chlimi 27475 | The limit property of a cl... |
hlim0 27476 | The zero sequence in Hilbe... |
hlimcaui 27477 | If a sequence in Hilbert s... |
hlimf 27478 | Function-like behavior of ... |
hlimuni 27479 | A Hilbert space sequence c... |
hlimreui 27480 | The limit of a Hilbert spa... |
hlimeui 27481 | The limit of a Hilbert spa... |
isch3 27482 | A Hilbert subspace is clos... |
chcompl 27483 | Completeness of a closed s... |
helch 27484 | The unit Hilbert lattice e... |
ifchhv 27485 | Prove ` if ( A e. CH , A ,... |
helsh 27486 | Hilbert space is a subspac... |
shsspwh 27487 | Subspaces are subsets of H... |
chsspwh 27488 | Closed subspaces are subse... |
hsn0elch 27489 | The zero subspace belongs ... |
norm1 27490 | From any nonzero Hilbert s... |
norm1exi 27491 | A normalized vector exists... |
norm1hex 27492 | A normalized vector can ex... |
elch0 27495 | Membership in zero for clo... |
h0elch 27496 | The zero subspace is a clo... |
h0elsh 27497 | The zero subspace is a sub... |
hhssva 27498 | The vector addition operat... |
hhsssm 27499 | The scalar multiplication ... |
hhssnm 27500 | The norm operation on a su... |
issubgoilem 27501 | Lemma for ~ hhssabloilem .... |
hhssabloilem 27502 | Lemma for ~ hhssabloi . F... |
hhssabloi 27503 | Abelian group property of ... |
hhssablo 27504 | Abelian group property of ... |
hhssnv 27505 | Normed complex vector spac... |
hhssnvt 27506 | Normed complex vector spac... |
hhsst 27507 | A member of ` SH ` is a su... |
hhshsslem1 27508 | Lemma for ~ hhsssh . (Con... |
hhshsslem2 27509 | Lemma for ~ hhsssh . (Con... |
hhsssh 27510 | The predicate " ` H ` is a... |
hhsssh2 27511 | The predicate " ` H ` is a... |
hhssba 27512 | The base set of a subspace... |
hhssvs 27513 | The vector subtraction ope... |
hhssvsf 27514 | Mapping of the vector subt... |
hhssph 27515 | Inner product space proper... |
hhssims 27516 | Induced metric of a subspa... |
hhssims2 27517 | Induced metric of a subspa... |
hhssmet 27518 | Induced metric of a subspa... |
hhssmetdval 27519 | Value of the distance func... |
hhsscms 27520 | The induced metric of a cl... |
hhssbn 27521 | Banach space property of a... |
hhsshl 27522 | Hilbert space property of ... |
ocval 27523 | Value of orthogonal comple... |
ocel 27524 | Membership in orthogonal c... |
shocel 27525 | Membership in orthogonal c... |
ocsh 27526 | The orthogonal complement ... |
shocsh 27527 | The orthogonal complement ... |
ocss 27528 | An orthogonal complement i... |
shocss 27529 | An orthogonal complement i... |
occon 27530 | Contraposition law for ort... |
occon2 27531 | Double contraposition for ... |
occon2i 27532 | Double contraposition for ... |
oc0 27533 | The zero vector belongs to... |
ocorth 27534 | Members of a subset and it... |
shocorth 27535 | Members of a subspace and ... |
ococss 27536 | Inclusion in complement of... |
shococss 27537 | Inclusion in complement of... |
shorth 27538 | Members of orthogonal subs... |
ocin 27539 | Intersection of a Hilbert ... |
occon3 27540 | Hilbert lattice contraposi... |
ocnel 27541 | A nonzero vector in the co... |
chocvali 27542 | Value of the orthogonal co... |
shuni 27543 | Two subspaces with trivial... |
chocunii 27544 | Lemma for uniqueness part ... |
pjhthmo 27545 | Projection Theorem, unique... |
occllem 27546 | Lemma for ~ occl . (Contr... |
occl 27547 | Closure of complement of H... |
shoccl 27548 | Closure of complement of H... |
choccl 27549 | Closure of complement of H... |
choccli 27550 | Closure of ` CH ` orthocom... |
shsval 27555 | Value of subspace sum of t... |
shsss 27556 | The subspace sum is a subs... |
shsel 27557 | Membership in the subspace... |
shsel3 27558 | Membership in the subspace... |
shseli 27559 | Membership in subspace sum... |
shscli 27560 | Closure of subspace sum. ... |
shscl 27561 | Closure of subspace sum. ... |
shscom 27562 | Commutative law for subspa... |
shsva 27563 | Vector sum belongs to subs... |
shsel1 27564 | A subspace sum contains a ... |
shsel2 27565 | A subspace sum contains a ... |
shsvs 27566 | Vector subtraction belongs... |
shsub1 27567 | Subspace sum is an upper b... |
shsub2 27568 | Subspace sum is an upper b... |
choc0 27569 | The orthocomplement of the... |
choc1 27570 | The orthocomplement of the... |
chocnul 27571 | Orthogonal complement of t... |
shintcli 27572 | Closure of intersection of... |
shintcl 27573 | The intersection of a none... |
chintcli 27574 | The intersection of a none... |
chintcl 27575 | The intersection (infimum)... |
spanval 27576 | Value of the linear span o... |
hsupval 27577 | Value of supremum of set o... |
chsupval 27578 | The value of the supremum ... |
spancl 27579 | The span of a subset of Hi... |
elspancl 27580 | A member of a span is a ve... |
shsupcl 27581 | Closure of the subspace su... |
hsupcl 27582 | Closure of supremum of set... |
chsupcl 27583 | Closure of supremum of sub... |
hsupss 27584 | Subset relation for suprem... |
chsupss 27585 | Subset relation for suprem... |
hsupunss 27586 | The union of a set of Hilb... |
chsupunss 27587 | The union of a set of clos... |
spanss2 27588 | A subset of Hilbert space ... |
shsupunss 27589 | The union of a set of subs... |
spanid 27590 | A subspace of Hilbert spac... |
spanss 27591 | Ordering relationship for ... |
spanssoc 27592 | The span of a subset of Hi... |
sshjval 27593 | Value of join for subsets ... |
shjval 27594 | Value of join in ` SH ` . ... |
chjval 27595 | Value of join in ` CH ` . ... |
chjvali 27596 | Value of join in ` CH ` . ... |
sshjval3 27597 | Value of join for subsets ... |
sshjcl 27598 | Closure of join for subset... |
shjcl 27599 | Closure of join in ` SH ` ... |
chjcl 27600 | Closure of join in ` CH ` ... |
shjcom 27601 | Commutative law for Hilber... |
shless 27602 | Subset implies subset of s... |
shlej1 27603 | Add disjunct to both sides... |
shlej2 27604 | Add disjunct to both sides... |
shincli 27605 | Closure of intersection of... |
shscomi 27606 | Commutative law for subspa... |
shsvai 27607 | Vector sum belongs to subs... |
shsel1i 27608 | A subspace sum contains a ... |
shsel2i 27609 | A subspace sum contains a ... |
shsvsi 27610 | Vector subtraction belongs... |
shunssi 27611 | Union is smaller than subs... |
shunssji 27612 | Union is smaller than Hilb... |
shsleji 27613 | Subspace sum is smaller th... |
shjcomi 27614 | Commutative law for join i... |
shsub1i 27615 | Subspace sum is an upper b... |
shsub2i 27616 | Subspace sum is an upper b... |
shub1i 27617 | Hilbert lattice join is an... |
shjcli 27618 | Closure of ` CH ` join. (... |
shjshcli 27619 | ` SH ` closure of join. (... |
shlessi 27620 | Subset implies subset of s... |
shlej1i 27621 | Add disjunct to both sides... |
shlej2i 27622 | Add disjunct to both sides... |
shslej 27623 | Subspace sum is smaller th... |
shincl 27624 | Closure of intersection of... |
shub1 27625 | Hilbert lattice join is an... |
shub2 27626 | A subspace is a subset of ... |
shsidmi 27627 | Idempotent law for Hilbert... |
shslubi 27628 | The least upper bound law ... |
shlesb1i 27629 | Hilbert lattice ordering i... |
shsval2i 27630 | An alternate way to expres... |
shsval3i 27631 | An alternate way to expres... |
shmodsi 27632 | The modular law holds for ... |
shmodi 27633 | The modular law is implied... |
pjhthlem1 27634 | Lemma for ~ pjhth . (Cont... |
pjhthlem2 27635 | Lemma for ~ pjhth . (Cont... |
pjhth 27636 | Projection Theorem: Any H... |
pjhtheu 27637 | Projection Theorem: Any H... |
pjhfval 27639 | The value of the projectio... |
pjhval 27640 | Value of a projection. (C... |
pjpreeq 27641 | Equality with a projection... |
pjeq 27642 | Equality with a projection... |
axpjcl 27643 | Closure of a projection in... |
pjhcl 27644 | Closure of a projection in... |
omlsilem 27645 | Lemma for orthomodular law... |
omlsii 27646 | Subspace inference form of... |
omlsi 27647 | Subspace form of orthomodu... |
ococi 27648 | Complement of complement o... |
ococ 27649 | Complement of complement o... |
dfch2 27650 | Alternate definition of th... |
ococin 27651 | The double complement is t... |
hsupval2 27652 | Alternate definition of su... |
chsupval2 27653 | The value of the supremum ... |
sshjval2 27654 | Value of join in the set o... |
chsupid 27655 | A subspace is the supremum... |
chsupsn 27656 | Value of supremum of subse... |
shlub 27657 | Hilbert lattice join is th... |
shlubi 27658 | Hilbert lattice join is th... |
pjhtheu2 27659 | Uniqueness of ` y ` for th... |
pjcli 27660 | Closure of a projection in... |
pjhcli 27661 | Closure of a projection in... |
pjpjpre 27662 | Decomposition of a vector ... |
axpjpj 27663 | Decomposition of a vector ... |
pjclii 27664 | Closure of a projection in... |
pjhclii 27665 | Closure of a projection in... |
pjpj0i 27666 | Decomposition of a vector ... |
pjpji 27667 | Decomposition of a vector ... |
pjpjhth 27668 | Projection Theorem: Any H... |
pjpjhthi 27669 | Projection Theorem: Any H... |
pjop 27670 | Orthocomplement projection... |
pjpo 27671 | Projection in terms of ort... |
pjopi 27672 | Orthocomplement projection... |
pjpoi 27673 | Projection in terms of ort... |
pjoc1i 27674 | Projection of a vector in ... |
pjchi 27675 | Projection of a vector in ... |
pjoccl 27676 | The part of a vector that ... |
pjoc1 27677 | Projection of a vector in ... |
pjomli 27678 | Subspace form of orthomodu... |
pjoml 27679 | Subspace form of orthomodu... |
pjococi 27680 | Proof of orthocomplement t... |
pjoc2i 27681 | Projection of a vector in ... |
pjoc2 27682 | Projection of a vector in ... |
sh0le 27683 | The zero subspace is the s... |
ch0le 27684 | The zero subspace is the s... |
shle0 27685 | No subspace is smaller tha... |
chle0 27686 | No Hilbert lattice element... |
chnlen0 27687 | A Hilbert lattice element ... |
ch0pss 27688 | The zero subspace is a pro... |
orthin 27689 | The intersection of orthog... |
ssjo 27690 | The lattice join of a subs... |
shne0i 27691 | A nonzero subspace has a n... |
shs0i 27692 | Hilbert subspace sum with ... |
shs00i 27693 | Two subspaces are zero iff... |
ch0lei 27694 | The closed subspace zero i... |
chle0i 27695 | No Hilbert closed subspace... |
chne0i 27696 | A nonzero closed subspace ... |
chocini 27697 | Intersection of a closed s... |
chj0i 27698 | Join with lattice zero in ... |
chm1i 27699 | Meet with lattice one in `... |
chjcli 27700 | Closure of ` CH ` join. (... |
chsleji 27701 | Subspace sum is smaller th... |
chseli 27702 | Membership in subspace sum... |
chincli 27703 | Closure of Hilbert lattice... |
chsscon3i 27704 | Hilbert lattice contraposi... |
chsscon1i 27705 | Hilbert lattice contraposi... |
chsscon2i 27706 | Hilbert lattice contraposi... |
chcon2i 27707 | Hilbert lattice contraposi... |
chcon1i 27708 | Hilbert lattice contraposi... |
chcon3i 27709 | Hilbert lattice contraposi... |
chunssji 27710 | Union is smaller than ` CH... |
chjcomi 27711 | Commutative law for join i... |
chub1i 27712 | ` CH ` join is an upper bo... |
chub2i 27713 | ` CH ` join is an upper bo... |
chlubi 27714 | Hilbert lattice join is th... |
chlubii 27715 | Hilbert lattice join is th... |
chlej1i 27716 | Add join to both sides of ... |
chlej2i 27717 | Add join to both sides of ... |
chlej12i 27718 | Add join to both sides of ... |
chlejb1i 27719 | Hilbert lattice ordering i... |
chdmm1i 27720 | De Morgan's law for meet i... |
chdmm2i 27721 | De Morgan's law for meet i... |
chdmm3i 27722 | De Morgan's law for meet i... |
chdmm4i 27723 | De Morgan's law for meet i... |
chdmj1i 27724 | De Morgan's law for join i... |
chdmj2i 27725 | De Morgan's law for join i... |
chdmj3i 27726 | De Morgan's law for join i... |
chdmj4i 27727 | De Morgan's law for join i... |
chnlei 27728 | Equivalent expressions for... |
chjassi 27729 | Associative law for Hilber... |
chj00i 27730 | Two Hilbert lattice elemen... |
chjoi 27731 | The join of a closed subsp... |
chj1i 27732 | Join with Hilbert lattice ... |
chm0i 27733 | Meet with Hilbert lattice ... |
chm0 27734 | Meet with Hilbert lattice ... |
shjshsi 27735 | Hilbert lattice join equal... |
shjshseli 27736 | A closed subspace sum equa... |
chne0 27737 | A nonzero closed subspace ... |
chocin 27738 | Intersection of a closed s... |
chssoc 27739 | A closed subspace less tha... |
chj0 27740 | Join with Hilbert lattice ... |
chslej 27741 | Subspace sum is smaller th... |
chincl 27742 | Closure of Hilbert lattice... |
chsscon3 27743 | Hilbert lattice contraposi... |
chsscon1 27744 | Hilbert lattice contraposi... |
chsscon2 27745 | Hilbert lattice contraposi... |
chpsscon3 27746 | Hilbert lattice contraposi... |
chpsscon1 27747 | Hilbert lattice contraposi... |
chpsscon2 27748 | Hilbert lattice contraposi... |
chjcom 27749 | Commutative law for Hilber... |
chub1 27750 | Hilbert lattice join is gr... |
chub2 27751 | Hilbert lattice join is gr... |
chlub 27752 | Hilbert lattice join is th... |
chlej1 27753 | Add join to both sides of ... |
chlej2 27754 | Add join to both sides of ... |
chlejb1 27755 | Hilbert lattice ordering i... |
chlejb2 27756 | Hilbert lattice ordering i... |
chnle 27757 | Equivalent expressions for... |
chjo 27758 | The join of a closed subsp... |
chabs1 27759 | Hilbert lattice absorption... |
chabs2 27760 | Hilbert lattice absorption... |
chabs1i 27761 | Hilbert lattice absorption... |
chabs2i 27762 | Hilbert lattice absorption... |
chjidm 27763 | Idempotent law for Hilbert... |
chjidmi 27764 | Idempotent law for Hilbert... |
chj12i 27765 | A rearrangement of Hilbert... |
chj4i 27766 | Rearrangement of the join ... |
chjjdiri 27767 | Hilbert lattice join distr... |
chdmm1 27768 | De Morgan's law for meet i... |
chdmm2 27769 | De Morgan's law for meet i... |
chdmm3 27770 | De Morgan's law for meet i... |
chdmm4 27771 | De Morgan's law for meet i... |
chdmj1 27772 | De Morgan's law for join i... |
chdmj2 27773 | De Morgan's law for join i... |
chdmj3 27774 | De Morgan's law for join i... |
chdmj4 27775 | De Morgan's law for join i... |
chjass 27776 | Associative law for Hilber... |
chj12 27777 | A rearrangement of Hilbert... |
chj4 27778 | Rearrangement of the join ... |
ledii 27779 | An ortholattice is distrib... |
lediri 27780 | An ortholattice is distrib... |
lejdii 27781 | An ortholattice is distrib... |
lejdiri 27782 | An ortholattice is distrib... |
ledi 27783 | An ortholattice is distrib... |
spansn0 27784 | The span of the singleton ... |
span0 27785 | The span of the empty set ... |
elspani 27786 | Membership in the span of ... |
spanuni 27787 | The span of a union is the... |
spanun 27788 | The span of a union is the... |
sshhococi 27789 | The join of two Hilbert sp... |
hne0 27790 | Hilbert space has a nonzer... |
chsup0 27791 | The supremum of the empty ... |
h1deoi 27792 | Membership in orthocomplem... |
h1dei 27793 | Membership in 1-dimensiona... |
h1did 27794 | A generating vector belong... |
h1dn0 27795 | A nonzero vector generates... |
h1de2i 27796 | Membership in 1-dimensiona... |
h1de2bi 27797 | Membership in 1-dimensiona... |
h1de2ctlem 27798 | Lemma for ~ h1de2ci . (Co... |
h1de2ci 27799 | Membership in 1-dimensiona... |
spansni 27800 | The span of a singleton in... |
elspansni 27801 | Membership in the span of ... |
spansn 27802 | The span of a singleton in... |
spansnch 27803 | The span of a Hilbert spac... |
spansnsh 27804 | The span of a Hilbert spac... |
spansnchi 27805 | The span of a singleton in... |
spansnid 27806 | A vector belongs to the sp... |
spansnmul 27807 | A scalar product with a ve... |
elspansncl 27808 | A member of a span of a si... |
elspansn 27809 | Membership in the span of ... |
elspansn2 27810 | Membership in the span of ... |
spansncol 27811 | The singletons of collinea... |
spansneleqi 27812 | Membership relation implie... |
spansneleq 27813 | Membership relation that i... |
spansnss 27814 | The span of the singleton ... |
elspansn3 27815 | A member of the span of th... |
elspansn4 27816 | A span membership conditio... |
elspansn5 27817 | A vector belonging to both... |
spansnss2 27818 | The span of the singleton ... |
normcan 27819 | Cancellation-type law that... |
pjspansn 27820 | A projection on the span o... |
spansnpji 27821 | A subset of Hilbert space ... |
spanunsni 27822 | The span of the union of a... |
spanpr 27823 | The span of a pair of vect... |
h1datomi 27824 | A 1-dimensional subspace i... |
h1datom 27825 | A 1-dimensional subspace i... |
cmbr 27827 | Binary relation expressing... |
pjoml2i 27828 | Variation of orthomodular ... |
pjoml3i 27829 | Variation of orthomodular ... |
pjoml4i 27830 | Variation of orthomodular ... |
pjoml5i 27831 | The orthomodular law. Rem... |
pjoml6i 27832 | An equivalent of the ortho... |
cmbri 27833 | Binary relation expressing... |
cmcmlem 27834 | Commutation is symmetric. ... |
cmcmi 27835 | Commutation is symmetric. ... |
cmcm2i 27836 | Commutation with orthocomp... |
cmcm3i 27837 | Commutation with orthocomp... |
cmcm4i 27838 | Commutation with orthocomp... |
cmbr2i 27839 | Alternate definition of th... |
cmcmii 27840 | Commutation is symmetric. ... |
cmcm2ii 27841 | Commutation with orthocomp... |
cmcm3ii 27842 | Commutation with orthocomp... |
cmbr3i 27843 | Alternate definition for t... |
cmbr4i 27844 | Alternate definition for t... |
lecmi 27845 | Comparable Hilbert lattice... |
lecmii 27846 | Comparable Hilbert lattice... |
cmj1i 27847 | A Hilbert lattice element ... |
cmj2i 27848 | A Hilbert lattice element ... |
cmm1i 27849 | A Hilbert lattice element ... |
cmm2i 27850 | A Hilbert lattice element ... |
cmbr3 27851 | Alternate definition for t... |
cm0 27852 | The zero Hilbert lattice e... |
cmidi 27853 | The commutes relation is r... |
pjoml2 27854 | Variation of orthomodular ... |
pjoml3 27855 | Variation of orthomodular ... |
pjoml5 27856 | The orthomodular law. Rem... |
cmcm 27857 | Commutation is symmetric. ... |
cmcm3 27858 | Commutation with orthocomp... |
cmcm2 27859 | Commutation with orthocomp... |
lecm 27860 | Comparable Hilbert lattice... |
fh1 27861 | Foulis-Holland Theorem. I... |
fh2 27862 | Foulis-Holland Theorem. I... |
cm2j 27863 | A lattice element that com... |
fh1i 27864 | Foulis-Holland Theorem. I... |
fh2i 27865 | Foulis-Holland Theorem. I... |
fh3i 27866 | Variation of the Foulis-Ho... |
fh4i 27867 | Variation of the Foulis-Ho... |
cm2ji 27868 | A lattice element that com... |
cm2mi 27869 | A lattice element that com... |
qlax1i 27870 | One of the equations showi... |
qlax2i 27871 | One of the equations showi... |
qlax3i 27872 | One of the equations showi... |
qlax4i 27873 | One of the equations showi... |
qlax5i 27874 | One of the equations showi... |
qlaxr1i 27875 | One of the conditions show... |
qlaxr2i 27876 | One of the conditions show... |
qlaxr4i 27877 | One of the conditions show... |
qlaxr5i 27878 | One of the conditions show... |
qlaxr3i 27879 | A variation of the orthomo... |
chscllem1 27880 | Lemma for ~ chscl . (Cont... |
chscllem2 27881 | Lemma for ~ chscl . (Cont... |
chscllem3 27882 | Lemma for ~ chscl . (Cont... |
chscllem4 27883 | Lemma for ~ chscl . (Cont... |
chscl 27884 | The subspace sum of two cl... |
osumi 27885 | If two closed subspaces of... |
osumcori 27886 | Corollary of ~ osumi . (C... |
osumcor2i 27887 | Corollary of ~ osumi , sho... |
osum 27888 | If two closed subspaces of... |
spansnji 27889 | The subspace sum of a clos... |
spansnj 27890 | The subspace sum of a clos... |
spansnscl 27891 | The subspace sum of a clos... |
sumspansn 27892 | The sum of two vectors bel... |
spansnm0i 27893 | The meet of different one-... |
nonbooli 27894 | A Hilbert lattice with two... |
spansncvi 27895 | Hilbert space has the cove... |
spansncv 27896 | Hilbert space has the cove... |
5oalem1 27897 | Lemma for orthoarguesian l... |
5oalem2 27898 | Lemma for orthoarguesian l... |
5oalem3 27899 | Lemma for orthoarguesian l... |
5oalem4 27900 | Lemma for orthoarguesian l... |
5oalem5 27901 | Lemma for orthoarguesian l... |
5oalem6 27902 | Lemma for orthoarguesian l... |
5oalem7 27903 | Lemma for orthoarguesian l... |
5oai 27904 | Orthoarguesian law 5OA. Th... |
3oalem1 27905 | Lemma for 3OA (weak) ortho... |
3oalem2 27906 | Lemma for 3OA (weak) ortho... |
3oalem3 27907 | Lemma for 3OA (weak) ortho... |
3oalem4 27908 | Lemma for 3OA (weak) ortho... |
3oalem5 27909 | Lemma for 3OA (weak) ortho... |
3oalem6 27910 | Lemma for 3OA (weak) ortho... |
3oai 27911 | 3OA (weak) orthoarguesian ... |
pjorthi 27912 | Projection components on o... |
pjch1 27913 | Property of identity proje... |
pjo 27914 | The orthogonal projection.... |
pjcompi 27915 | Component of a projection.... |
pjidmi 27916 | A projection is idempotent... |
pjadjii 27917 | A projection is self-adjoi... |
pjaddii 27918 | Projection of vector sum i... |
pjinormii 27919 | The inner product of a pro... |
pjmulii 27920 | Projection of (scalar) pro... |
pjsubii 27921 | Projection of vector diffe... |
pjsslem 27922 | Lemma for subset relations... |
pjss2i 27923 | Subset relationship for pr... |
pjssmii 27924 | Projection meet property. ... |
pjssge0ii 27925 | Theorem 4.5(iv)->(v) of [B... |
pjdifnormii 27926 | Theorem 4.5(v)<->(vi) of [... |
pjcji 27927 | The projection on a subspa... |
pjadji 27928 | A projection is self-adjoi... |
pjaddi 27929 | Projection of vector sum i... |
pjinormi 27930 | The inner product of a pro... |
pjsubi 27931 | Projection of vector diffe... |
pjmuli 27932 | Projection of scalar produ... |
pjige0i 27933 | The inner product of a pro... |
pjige0 27934 | The inner product of a pro... |
pjcjt2 27935 | The projection on a subspa... |
pj0i 27936 | The projection of the zero... |
pjch 27937 | Projection of a vector in ... |
pjid 27938 | The projection of a vector... |
pjvec 27939 | The set of vectors belongi... |
pjocvec 27940 | The set of vectors belongi... |
pjocini 27941 | Membership of projection i... |
pjini 27942 | Membership of projection i... |
pjjsi 27943 | A sufficient condition for... |
pjfni 27944 | Functionality of a project... |
pjrni 27945 | The range of a projection.... |
pjfoi 27946 | A projection maps onto its... |
pjfi 27947 | The mapping of a projectio... |
pjvi 27948 | The value of a projection ... |
pjhfo 27949 | A projection maps onto its... |
pjrn 27950 | The range of a projection.... |
pjhf 27951 | The mapping of a projectio... |
pjfn 27952 | Functionality of a project... |
pjsumi 27953 | The projection on a subspa... |
pj11i 27954 | One-to-one correspondence ... |
pjdsi 27955 | Vector decomposition into ... |
pjds3i 27956 | Vector decomposition into ... |
pj11 27957 | One-to-one correspondence ... |
pjmfn 27958 | Functionality of the proje... |
pjmf1 27959 | The projector function map... |
pjoi0 27960 | The inner product of proje... |
pjoi0i 27961 | The inner product of proje... |
pjopythi 27962 | Pythagorean theorem for pr... |
pjopyth 27963 | Pythagorean theorem for pr... |
pjnormi 27964 | The norm of the projection... |
pjpythi 27965 | Pythagorean theorem for pr... |
pjneli 27966 | If a vector does not belon... |
pjnorm 27967 | The norm of the projection... |
pjpyth 27968 | Pythagorean theorem for pr... |
pjnel 27969 | If a vector does not belon... |
pjnorm2 27970 | A vector belongs to the su... |
mayete3i 27971 | Mayet's equation E_3. Par... |
mayetes3i 27972 | Mayet's equation E^*_3, de... |
hosmval 27978 | Value of the sum of two Hi... |
hommval 27979 | Value of the scalar produc... |
hodmval 27980 | Value of the difference of... |
hfsmval 27981 | Value of the sum of two Hi... |
hfmmval 27982 | Value of the scalar produc... |
hosval 27983 | Value of the sum of two Hi... |
homval 27984 | Value of the scalar produc... |
hodval 27985 | Value of the difference of... |
hfsval 27986 | Value of the sum of two Hi... |
hfmval 27987 | Value of the scalar produc... |
hoscl 27988 | Closure of the sum of two ... |
homcl 27989 | Closure of the scalar prod... |
hodcl 27990 | Closure of the difference ... |
ho0val 27993 | Value of the zero Hilbert ... |
ho0f 27994 | Functionality of the zero ... |
df0op2 27995 | Alternate definition of Hi... |
dfiop2 27996 | Alternate definition of Hi... |
hoif 27997 | Functionality of the Hilbe... |
hoival 27998 | The value of the Hilbert s... |
hoico1 27999 | Composition with the Hilbe... |
hoico2 28000 | Composition with the Hilbe... |
hoaddcl 28001 | The sum of Hilbert space o... |
homulcl 28002 | The scalar product of a Hi... |
hoeq 28003 | Equality of Hilbert space ... |
hoeqi 28004 | Equality of Hilbert space ... |
hoscli 28005 | Closure of Hilbert space o... |
hodcli 28006 | Closure of Hilbert space o... |
hocoi 28007 | Composition of Hilbert spa... |
hococli 28008 | Closure of composition of ... |
hocofi 28009 | Mapping of composition of ... |
hocofni 28010 | Functionality of compositi... |
hoaddcli 28011 | Mapping of sum of Hilbert ... |
hosubcli 28012 | Mapping of difference of H... |
hoaddfni 28013 | Functionality of sum of Hi... |
hosubfni 28014 | Functionality of differenc... |
hoaddcomi 28015 | Commutativity of sum of Hi... |
hosubcl 28016 | Mapping of difference of H... |
hoaddcom 28017 | Commutativity of sum of Hi... |
hodsi 28018 | Relationship between Hilbe... |
hoaddassi 28019 | Associativity of sum of Hi... |
hoadd12i 28020 | Commutative/associative la... |
hoadd32i 28021 | Commutative/associative la... |
hocadddiri 28022 | Distributive law for Hilbe... |
hocsubdiri 28023 | Distributive law for Hilbe... |
ho2coi 28024 | Double composition of Hilb... |
hoaddass 28025 | Associativity of sum of Hi... |
hoadd32 28026 | Commutative/associative la... |
hoadd4 28027 | Rearrangement of 4 terms i... |
hocsubdir 28028 | Distributive law for Hilbe... |
hoaddid1i 28029 | Sum of a Hilbert space ope... |
hodidi 28030 | Difference of a Hilbert sp... |
ho0coi 28031 | Composition of the zero op... |
hoid1i 28032 | Composition of Hilbert spa... |
hoid1ri 28033 | Composition of Hilbert spa... |
hoaddid1 28034 | Sum of a Hilbert space ope... |
hodid 28035 | Difference of a Hilbert sp... |
hon0 28036 | A Hilbert space operator i... |
hodseqi 28037 | Subtraction and addition o... |
ho0subi 28038 | Subtraction of Hilbert spa... |
honegsubi 28039 | Relationship between Hilbe... |
ho0sub 28040 | Subtraction of Hilbert spa... |
hosubid1 28041 | The zero operator subtract... |
honegsub 28042 | Relationship between Hilbe... |
homulid2 28043 | An operator equals its sca... |
homco1 28044 | Associative law for scalar... |
homulass 28045 | Scalar product associative... |
hoadddi 28046 | Scalar product distributiv... |
hoadddir 28047 | Scalar product reverse dis... |
homul12 28048 | Swap first and second fact... |
honegneg 28049 | Double negative of a Hilbe... |
hosubneg 28050 | Relationship between opera... |
hosubdi 28051 | Scalar product distributiv... |
honegdi 28052 | Distribution of negative o... |
honegsubdi 28053 | Distribution of negative o... |
honegsubdi2 28054 | Distribution of negative o... |
hosubsub2 28055 | Law for double subtraction... |
hosub4 28056 | Rearrangement of 4 terms i... |
hosubadd4 28057 | Rearrangement of 4 terms i... |
hoaddsubass 28058 | Associative-type law for a... |
hoaddsub 28059 | Law for operator addition ... |
hosubsub 28060 | Law for double subtraction... |
hosubsub4 28061 | Law for double subtraction... |
ho2times 28062 | Two times a Hilbert space ... |
hoaddsubassi 28063 | Associativity of sum and d... |
hoaddsubi 28064 | Law for sum and difference... |
hosd1i 28065 | Hilbert space operator sum... |
hosd2i 28066 | Hilbert space operator sum... |
hopncani 28067 | Hilbert space operator can... |
honpcani 28068 | Hilbert space operator can... |
hosubeq0i 28069 | If the difference between ... |
honpncani 28070 | Hilbert space operator can... |
ho01i 28071 | A condition implying that ... |
ho02i 28072 | A condition implying that ... |
hoeq1 28073 | A condition implying that ... |
hoeq2 28074 | A condition implying that ... |
adjmo 28075 | Every Hilbert space operat... |
adjsym 28076 | Symmetry property of an ad... |
eigrei 28077 | A necessary and sufficient... |
eigre 28078 | A necessary and sufficient... |
eigposi 28079 | A sufficient condition (fi... |
eigorthi 28080 | A necessary and sufficient... |
eigorth 28081 | A necessary and sufficient... |
nmopval 28099 | Value of the norm of a Hil... |
elcnop 28100 | Property defining a contin... |
ellnop 28101 | Property defining a linear... |
lnopf 28102 | A linear Hilbert space ope... |
elbdop 28103 | Property defining a bounde... |
bdopln 28104 | A bounded linear Hilbert s... |
bdopf 28105 | A bounded linear Hilbert s... |
nmopsetretALT 28106 | The set in the supremum of... |
nmopsetretHIL 28107 | The set in the supremum of... |
nmopsetn0 28108 | The set in the supremum of... |
nmopxr 28109 | The norm of a Hilbert spac... |
nmoprepnf 28110 | The norm of a Hilbert spac... |
nmopgtmnf 28111 | The norm of a Hilbert spac... |
nmopreltpnf 28112 | The norm of a Hilbert spac... |
nmopre 28113 | The norm of a bounded oper... |
elbdop2 28114 | Property defining a bounde... |
elunop 28115 | Property defining a unitar... |
elhmop 28116 | Property defining a Hermit... |
hmopf 28117 | A Hermitian operator is a ... |
hmopex 28118 | The class of Hermitian ope... |
nmfnval 28119 | Value of the norm of a Hil... |
nmfnsetre 28120 | The set in the supremum of... |
nmfnsetn0 28121 | The set in the supremum of... |
nmfnxr 28122 | The norm of any Hilbert sp... |
nmfnrepnf 28123 | The norm of a Hilbert spac... |
nlfnval 28124 | Value of the null space of... |
elcnfn 28125 | Property defining a contin... |
ellnfn 28126 | Property defining a linear... |
lnfnf 28127 | A linear Hilbert space fun... |
dfadj2 28128 | Alternate definition of th... |
funadj 28129 | Functionality of the adjoi... |
dmadjss 28130 | The domain of the adjoint ... |
dmadjop 28131 | A member of the domain of ... |
adjeu 28132 | Elementhood in the domain ... |
adjval 28133 | Value of the adjoint funct... |
adjval2 28134 | Value of the adjoint funct... |
cnvadj 28135 | The adjoint function equal... |
funcnvadj 28136 | The converse of the adjoin... |
adj1o 28137 | The adjoint function maps ... |
dmadjrn 28138 | The adjoint of an operator... |
eigvecval 28139 | The set of eigenvectors of... |
eigvalfval 28140 | The eigenvalues of eigenve... |
specval 28141 | The value of the spectrum ... |
speccl 28142 | The spectrum of an operato... |
hhlnoi 28143 | The linear operators of Hi... |
hhnmoi 28144 | The norm of an operator in... |
hhbloi 28145 | A bounded linear operator ... |
hh0oi 28146 | The zero operator in Hilbe... |
hhcno 28147 | The continuous operators o... |
hhcnf 28148 | The continuous functionals... |
dmadjrnb 28149 | The adjoint of an operator... |
nmoplb 28150 | A lower bound for an opera... |
nmopub 28151 | An upper bound for an oper... |
nmopub2tALT 28152 | An upper bound for an oper... |
nmopub2tHIL 28153 | An upper bound for an oper... |
nmopge0 28154 | The norm of any Hilbert sp... |
nmopgt0 28155 | A linear Hilbert space ope... |
cnopc 28156 | Basic continuity property ... |
lnopl 28157 | Basic linearity property o... |
unop 28158 | Basic inner product proper... |
unopf1o 28159 | A unitary operator in Hilb... |
unopnorm 28160 | A unitary operator is idem... |
cnvunop 28161 | The inverse (converse) of ... |
unopadj 28162 | The inverse (converse) of ... |
unoplin 28163 | A unitary operator is line... |
counop 28164 | The composition of two uni... |
hmop 28165 | Basic inner product proper... |
hmopre 28166 | The inner product of the v... |
nmfnlb 28167 | A lower bound for a functi... |
nmfnleub 28168 | An upper bound for the nor... |
nmfnleub2 28169 | An upper bound for the nor... |
nmfnge0 28170 | The norm of any Hilbert sp... |
elnlfn 28171 | Membership in the null spa... |
elnlfn2 28172 | Membership in the null spa... |
cnfnc 28173 | Basic continuity property ... |
lnfnl 28174 | Basic linearity property o... |
adjcl 28175 | Closure of the adjoint of ... |
adj1 28176 | Property of an adjoint Hil... |
adj2 28177 | Property of an adjoint Hil... |
adjeq 28178 | A property that determines... |
adjadj 28179 | Double adjoint. Theorem 3... |
adjvalval 28180 | Value of the value of the ... |
unopadj2 28181 | The adjoint of a unitary o... |
hmopadj 28182 | A Hermitian operator is se... |
hmdmadj 28183 | Every Hermitian operator h... |
hmopadj2 28184 | An operator is Hermitian i... |
hmoplin 28185 | A Hermitian operator is li... |
brafval 28186 | The bra of a vector, expre... |
braval 28187 | A bra-ket juxtaposition, e... |
braadd 28188 | Linearity property of bra ... |
bramul 28189 | Linearity property of bra ... |
brafn 28190 | The bra function is a func... |
bralnfn 28191 | The Dirac bra function is ... |
bracl 28192 | Closure of the bra functio... |
bra0 28193 | The Dirac bra of the zero ... |
brafnmul 28194 | Anti-linearity property of... |
kbfval 28195 | The outer product of two v... |
kbop 28196 | The outer product of two v... |
kbval 28197 | The value of the operator ... |
kbmul 28198 | Multiplication property of... |
kbpj 28199 | If a vector ` A ` has norm... |
eleigvec 28200 | Membership in the set of e... |
eleigvec2 28201 | Membership in the set of e... |
eleigveccl 28202 | Closure of an eigenvector ... |
eigvalval 28203 | The eigenvalue of an eigen... |
eigvalcl 28204 | An eigenvalue is a complex... |
eigvec1 28205 | Property of an eigenvector... |
eighmre 28206 | The eigenvalues of a Hermi... |
eighmorth 28207 | Eigenvectors of a Hermitia... |
nmopnegi 28208 | Value of the norm of the n... |
lnop0 28209 | The value of a linear Hilb... |
lnopmul 28210 | Multiplicative property of... |
lnopli 28211 | Basic scalar product prope... |
lnopfi 28212 | A linear Hilbert space ope... |
lnop0i 28213 | The value of a linear Hilb... |
lnopaddi 28214 | Additive property of a lin... |
lnopmuli 28215 | Multiplicative property of... |
lnopaddmuli 28216 | Sum/product property of a ... |
lnopsubi 28217 | Subtraction property for a... |
lnopsubmuli 28218 | Subtraction/product proper... |
lnopmulsubi 28219 | Product/subtraction proper... |
homco2 28220 | Move a scalar product out ... |
idunop 28221 | The identity function (res... |
0cnop 28222 | The identically zero funct... |
0cnfn 28223 | The identically zero funct... |
idcnop 28224 | The identity function (res... |
idhmop 28225 | The Hilbert space identity... |
0hmop 28226 | The identically zero funct... |
0lnop 28227 | The identically zero funct... |
0lnfn 28228 | The identically zero funct... |
nmop0 28229 | The norm of the zero opera... |
nmfn0 28230 | The norm of the identicall... |
hmopbdoptHIL 28231 | A Hermitian operator is a ... |
hoddii 28232 | Distributive law for Hilbe... |
hoddi 28233 | Distributive law for Hilbe... |
nmop0h 28234 | The norm of any operator o... |
idlnop 28235 | The identity function (res... |
0bdop 28236 | The identically zero opera... |
adj0 28237 | Adjoint of the zero operat... |
nmlnop0iALT 28238 | A linear operator with a z... |
nmlnop0iHIL 28239 | A linear operator with a z... |
nmlnopgt0i 28240 | A linear Hilbert space ope... |
nmlnop0 28241 | A linear operator with a z... |
nmlnopne0 28242 | A linear operator with a n... |
lnopmi 28243 | The scalar product of a li... |
lnophsi 28244 | The sum of two linear oper... |
lnophdi 28245 | The difference of two line... |
lnopcoi 28246 | The composition of two lin... |
lnopco0i 28247 | The composition of a linea... |
lnopeq0lem1 28248 | Lemma for ~ lnopeq0i . Ap... |
lnopeq0lem2 28249 | Lemma for ~ lnopeq0i . (C... |
lnopeq0i 28250 | A condition implying that ... |
lnopeqi 28251 | Two linear Hilbert space o... |
lnopeq 28252 | Two linear Hilbert space o... |
lnopunilem1 28253 | Lemma for ~ lnopunii . (C... |
lnopunilem2 28254 | Lemma for ~ lnopunii . (C... |
lnopunii 28255 | If a linear operator (whos... |
elunop2 28256 | An operator is unitary iff... |
nmopun 28257 | Norm of a unitary Hilbert ... |
unopbd 28258 | A unitary operator is a bo... |
lnophmlem1 28259 | Lemma for ~ lnophmi . (Co... |
lnophmlem2 28260 | Lemma for ~ lnophmi . (Co... |
lnophmi 28261 | A linear operator is Hermi... |
lnophm 28262 | A linear operator is Hermi... |
hmops 28263 | The sum of two Hermitian o... |
hmopm 28264 | The scalar product of a He... |
hmopd 28265 | The difference of two Herm... |
hmopco 28266 | The composition of two com... |
nmbdoplbi 28267 | A lower bound for the norm... |
nmbdoplb 28268 | A lower bound for the norm... |
nmcexi 28269 | Lemma for ~ nmcopexi and ~... |
nmcopexi 28270 | The norm of a continuous l... |
nmcoplbi 28271 | A lower bound for the norm... |
nmcopex 28272 | The norm of a continuous l... |
nmcoplb 28273 | A lower bound for the norm... |
nmophmi 28274 | The norm of the scalar pro... |
bdophmi 28275 | The scalar product of a bo... |
lnconi 28276 | Lemma for ~ lnopconi and ~... |
lnopconi 28277 | A condition equivalent to ... |
lnopcon 28278 | A condition equivalent to ... |
lnopcnbd 28279 | A linear operator is conti... |
lncnopbd 28280 | A continuous linear operat... |
lncnbd 28281 | A continuous linear operat... |
lnopcnre 28282 | A linear operator is conti... |
lnfnli 28283 | Basic property of a linear... |
lnfnfi 28284 | A linear Hilbert space fun... |
lnfn0i 28285 | The value of a linear Hilb... |
lnfnaddi 28286 | Additive property of a lin... |
lnfnmuli 28287 | Multiplicative property of... |
lnfnaddmuli 28288 | Sum/product property of a ... |
lnfnsubi 28289 | Subtraction property for a... |
lnfn0 28290 | The value of a linear Hilb... |
lnfnmul 28291 | Multiplicative property of... |
nmbdfnlbi 28292 | A lower bound for the norm... |
nmbdfnlb 28293 | A lower bound for the norm... |
nmcfnexi 28294 | The norm of a continuous l... |
nmcfnlbi 28295 | A lower bound for the norm... |
nmcfnex 28296 | The norm of a continuous l... |
nmcfnlb 28297 | A lower bound of the norm ... |
lnfnconi 28298 | A condition equivalent to ... |
lnfncon 28299 | A condition equivalent to ... |
lnfncnbd 28300 | A linear functional is con... |
imaelshi 28301 | The image of a subspace un... |
rnelshi 28302 | The range of a linear oper... |
nlelshi 28303 | The null space of a linear... |
nlelchi 28304 | The null space of a contin... |
riesz3i 28305 | A continuous linear functi... |
riesz4i 28306 | A continuous linear functi... |
riesz4 28307 | A continuous linear functi... |
riesz1 28308 | Part 1 of the Riesz repres... |
riesz2 28309 | Part 2 of the Riesz repres... |
cnlnadjlem1 28310 | Lemma for ~ cnlnadji (Theo... |
cnlnadjlem2 28311 | Lemma for ~ cnlnadji . ` G... |
cnlnadjlem3 28312 | Lemma for ~ cnlnadji . By... |
cnlnadjlem4 28313 | Lemma for ~ cnlnadji . Th... |
cnlnadjlem5 28314 | Lemma for ~ cnlnadji . ` F... |
cnlnadjlem6 28315 | Lemma for ~ cnlnadji . ` F... |
cnlnadjlem7 28316 | Lemma for ~ cnlnadji . He... |
cnlnadjlem8 28317 | Lemma for ~ cnlnadji . ` F... |
cnlnadjlem9 28318 | Lemma for ~ cnlnadji . ` F... |
cnlnadji 28319 | Every continuous linear op... |
cnlnadjeui 28320 | Every continuous linear op... |
cnlnadjeu 28321 | Every continuous linear op... |
cnlnadj 28322 | Every continuous linear op... |
cnlnssadj 28323 | Every continuous linear Hi... |
bdopssadj 28324 | Every bounded linear Hilbe... |
bdopadj 28325 | Every bounded linear Hilbe... |
adjbdln 28326 | The adjoint of a bounded l... |
adjbdlnb 28327 | An operator is bounded and... |
adjbd1o 28328 | The mapping of adjoints of... |
adjlnop 28329 | The adjoint of an operator... |
adjsslnop 28330 | Every operator with an adj... |
nmopadjlei 28331 | Property of the norm of an... |
nmopadjlem 28332 | Lemma for ~ nmopadji . (C... |
nmopadji 28333 | Property of the norm of an... |
adjeq0 28334 | An operator is zero iff it... |
adjmul 28335 | The adjoint of the scalar ... |
adjadd 28336 | The adjoint of the sum of ... |
nmoptrii 28337 | Triangle inequality for th... |
nmopcoi 28338 | Upper bound for the norm o... |
bdophsi 28339 | The sum of two bounded lin... |
bdophdi 28340 | The difference between two... |
bdopcoi 28341 | The composition of two bou... |
nmoptri2i 28342 | Triangle-type inequality f... |
adjcoi 28343 | The adjoint of a compositi... |
nmopcoadji 28344 | The norm of an operator co... |
nmopcoadj2i 28345 | The norm of an operator co... |
nmopcoadj0i 28346 | An operator composed with ... |
unierri 28347 | If we approximate a chain ... |
branmfn 28348 | The norm of the bra functi... |
brabn 28349 | The bra of a vector is a b... |
rnbra 28350 | The set of bras equals the... |
bra11 28351 | The bra function maps vect... |
bracnln 28352 | A bra is a continuous line... |
cnvbraval 28353 | Value of the converse of t... |
cnvbracl 28354 | Closure of the converse of... |
cnvbrabra 28355 | The converse bra of the br... |
bracnvbra 28356 | The bra of the converse br... |
bracnlnval 28357 | The vector that a continuo... |
cnvbramul 28358 | Multiplication property of... |
kbass1 28359 | Dirac bra-ket associative ... |
kbass2 28360 | Dirac bra-ket associative ... |
kbass3 28361 | Dirac bra-ket associative ... |
kbass4 28362 | Dirac bra-ket associative ... |
kbass5 28363 | Dirac bra-ket associative ... |
kbass6 28364 | Dirac bra-ket associative ... |
leopg 28365 | Ordering relation for posi... |
leop 28366 | Ordering relation for oper... |
leop2 28367 | Ordering relation for oper... |
leop3 28368 | Operator ordering in terms... |
leoppos 28369 | Binary relation defining a... |
leoprf2 28370 | The ordering relation for ... |
leoprf 28371 | The ordering relation for ... |
leopsq 28372 | The square of a Hermitian ... |
0leop 28373 | The zero operator is a pos... |
idleop 28374 | The identity operator is a... |
leopadd 28375 | The sum of two positive op... |
leopmuli 28376 | The scalar product of a no... |
leopmul 28377 | The scalar product of a po... |
leopmul2i 28378 | Scalar product applied to ... |
leoptri 28379 | The positive operator orde... |
leoptr 28380 | The positive operator orde... |
leopnmid 28381 | A bounded Hermitian operat... |
nmopleid 28382 | A nonzero, bounded Hermiti... |
opsqrlem1 28383 | Lemma for opsqri . (Contr... |
opsqrlem2 28384 | Lemma for opsqri . ` F `` ... |
opsqrlem3 28385 | Lemma for opsqri . (Contr... |
opsqrlem4 28386 | Lemma for opsqri . (Contr... |
opsqrlem5 28387 | Lemma for opsqri . (Contr... |
opsqrlem6 28388 | Lemma for opsqri . (Contr... |
pjhmopi 28389 | A projector is a Hermitian... |
pjlnopi 28390 | A projector is a linear op... |
pjnmopi 28391 | The operator norm of a pro... |
pjbdlni 28392 | A projector is a bounded l... |
pjhmop 28393 | A projection is a Hermitia... |
hmopidmchi 28394 | An idempotent Hermitian op... |
hmopidmpji 28395 | An idempotent Hermitian op... |
hmopidmch 28396 | An idempotent Hermitian op... |
hmopidmpj 28397 | An idempotent Hermitian op... |
pjsdii 28398 | Distributive law for Hilbe... |
pjddii 28399 | Distributive law for Hilbe... |
pjsdi2i 28400 | Chained distributive law f... |
pjcoi 28401 | Composition of projections... |
pjcocli 28402 | Closure of composition of ... |
pjcohcli 28403 | Closure of composition of ... |
pjadjcoi 28404 | Adjoint of composition of ... |
pjcofni 28405 | Functionality of compositi... |
pjss1coi 28406 | Subset relationship for pr... |
pjss2coi 28407 | Subset relationship for pr... |
pjssmi 28408 | Projection meet property. ... |
pjssge0i 28409 | Theorem 4.5(iv)->(v) of [B... |
pjdifnormi 28410 | Theorem 4.5(v)<->(vi) of [... |
pjnormssi 28411 | Theorem 4.5(i)<->(vi) of [... |
pjorthcoi 28412 | Composition of projections... |
pjscji 28413 | The projection of orthogon... |
pjssumi 28414 | The projection on a subspa... |
pjssposi 28415 | Projector ordering can be ... |
pjordi 28416 | The definition of projecto... |
pjssdif2i 28417 | The projection subspace of... |
pjssdif1i 28418 | A necessary and sufficient... |
pjimai 28419 | The image of a projection.... |
pjidmcoi 28420 | A projection is idempotent... |
pjoccoi 28421 | Composition of projections... |
pjtoi 28422 | Subspace sum of projection... |
pjoci 28423 | Projection of orthocomplem... |
pjidmco 28424 | A projection operator is i... |
dfpjop 28425 | Definition of projection o... |
pjhmopidm 28426 | Two ways to express the se... |
elpjidm 28427 | A projection operator is i... |
elpjhmop 28428 | A projection operator is H... |
0leopj 28429 | A projector is a positive ... |
pjadj2 28430 | A projector is self-adjoin... |
pjadj3 28431 | A projector is self-adjoin... |
elpjch 28432 | Reconstruction of the subs... |
elpjrn 28433 | Reconstruction of the subs... |
pjinvari 28434 | A closed subspace ` H ` wi... |
pjin1i 28435 | Lemma for Theorem 1.22 of ... |
pjin2i 28436 | Lemma for Theorem 1.22 of ... |
pjin3i 28437 | Lemma for Theorem 1.22 of ... |
pjclem1 28438 | Lemma for projection commu... |
pjclem2 28439 | Lemma for projection commu... |
pjclem3 28440 | Lemma for projection commu... |
pjclem4a 28441 | Lemma for projection commu... |
pjclem4 28442 | Lemma for projection commu... |
pjci 28443 | Two subspaces commute iff ... |
pjcmul1i 28444 | A necessary and sufficient... |
pjcmul2i 28445 | The projection subspace of... |
pjcohocli 28446 | Closure of composition of ... |
pjadj2coi 28447 | Adjoint of double composit... |
pj2cocli 28448 | Closure of double composit... |
pj3lem1 28449 | Lemma for projection tripl... |
pj3si 28450 | Stronger projection triple... |
pj3i 28451 | Projection triplet theorem... |
pj3cor1i 28452 | Projection triplet corolla... |
pjs14i 28453 | Theorem S-14 of Watanabe, ... |
isst 28456 | Property of a state. (Con... |
ishst 28457 | Property of a complex Hilb... |
sticl 28458 | ` [ 0 , 1 ] ` closure of t... |
stcl 28459 | Real closure of the value ... |
hstcl 28460 | Closure of the value of a ... |
hst1a 28461 | Unit value of a Hilbert-sp... |
hstel2 28462 | Properties of a Hilbert-sp... |
hstorth 28463 | Orthogonality property of ... |
hstosum 28464 | Orthogonal sum property of... |
hstoc 28465 | Sum of a Hilbert-space-val... |
hstnmoc 28466 | Sum of norms of a Hilbert-... |
stge0 28467 | The value of a state is no... |
stle1 28468 | The value of a state is le... |
hstle1 28469 | The norm of the value of a... |
hst1h 28470 | The norm of a Hilbert-spac... |
hst0h 28471 | The norm of a Hilbert-spac... |
hstpyth 28472 | Pythagorean property of a ... |
hstle 28473 | Ordering property of a Hil... |
hstles 28474 | Ordering property of a Hil... |
hstoh 28475 | A Hilbert-space-valued sta... |
hst0 28476 | A Hilbert-space-valued sta... |
sthil 28477 | The value of a state at th... |
stj 28478 | The value of a state on a ... |
sto1i 28479 | The state of a subspace pl... |
sto2i 28480 | The state of the orthocomp... |
stge1i 28481 | If a state is greater than... |
stle0i 28482 | If a state is less than or... |
stlei 28483 | Ordering law for states. ... |
stlesi 28484 | Ordering law for states. ... |
stji1i 28485 | Join of components of Sasa... |
stm1i 28486 | State of component of unit... |
stm1ri 28487 | State of component of unit... |
stm1addi 28488 | Sum of states whose meet i... |
staddi 28489 | If the sum of 2 states is ... |
stm1add3i 28490 | Sum of states whose meet i... |
stadd3i 28491 | If the sum of 3 states is ... |
st0 28492 | The state of the zero subs... |
strlem1 28493 | Lemma for strong state the... |
strlem2 28494 | Lemma for strong state the... |
strlem3a 28495 | Lemma for strong state the... |
strlem3 28496 | Lemma for strong state the... |
strlem4 28497 | Lemma for strong state the... |
strlem5 28498 | Lemma for strong state the... |
strlem6 28499 | Lemma for strong state the... |
stri 28500 | Strong state theorem. The... |
strb 28501 | Strong state theorem (bidi... |
hstrlem2 28502 | Lemma for strong set of CH... |
hstrlem3a 28503 | Lemma for strong set of CH... |
hstrlem3 28504 | Lemma for strong set of CH... |
hstrlem4 28505 | Lemma for strong set of CH... |
hstrlem5 28506 | Lemma for strong set of CH... |
hstrlem6 28507 | Lemma for strong set of CH... |
hstri 28508 | Hilbert space admits a str... |
hstrbi 28509 | Strong CH-state theorem (b... |
largei 28510 | A Hilbert lattice admits a... |
jplem1 28511 | Lemma for Jauch-Piron theo... |
jplem2 28512 | Lemma for Jauch-Piron theo... |
jpi 28513 | The function ` S ` , that ... |
golem1 28514 | Lemma for Godowski's equat... |
golem2 28515 | Lemma for Godowski's equat... |
goeqi 28516 | Godowski's equation, shown... |
stcltr1i 28517 | Property of a strong class... |
stcltr2i 28518 | Property of a strong class... |
stcltrlem1 28519 | Lemma for strong classical... |
stcltrlem2 28520 | Lemma for strong classical... |
stcltrthi 28521 | Theorem for classically st... |
cvbr 28525 | Binary relation expressing... |
cvbr2 28526 | Binary relation expressing... |
cvcon3 28527 | Contraposition law for the... |
cvpss 28528 | The covers relation implie... |
cvnbtwn 28529 | The covers relation implie... |
cvnbtwn2 28530 | The covers relation implie... |
cvnbtwn3 28531 | The covers relation implie... |
cvnbtwn4 28532 | The covers relation implie... |
cvnsym 28533 | The covers relation is not... |
cvnref 28534 | The covers relation is not... |
cvntr 28535 | The covers relation is not... |
spansncv2 28536 | Hilbert space has the cove... |
mdbr 28537 | Binary relation expressing... |
mdi 28538 | Consequence of the modular... |
mdbr2 28539 | Binary relation expressing... |
mdbr3 28540 | Binary relation expressing... |
mdbr4 28541 | Binary relation expressing... |
dmdbr 28542 | Binary relation expressing... |
dmdmd 28543 | The dual modular pair prop... |
mddmd 28544 | The modular pair property ... |
dmdi 28545 | Consequence of the dual mo... |
dmdbr2 28546 | Binary relation expressing... |
dmdi2 28547 | Consequence of the dual mo... |
dmdbr3 28548 | Binary relation expressing... |
dmdbr4 28549 | Binary relation expressing... |
dmdi4 28550 | Consequence of the dual mo... |
dmdbr5 28551 | Binary relation expressing... |
mddmd2 28552 | Relationship between modul... |
mdsl0 28553 | A sublattice condition tha... |
ssmd1 28554 | Ordering implies the modul... |
ssmd2 28555 | Ordering implies the modul... |
ssdmd1 28556 | Ordering implies the dual ... |
ssdmd2 28557 | Ordering implies the dual ... |
dmdsl3 28558 | Sublattice mapping for a d... |
mdsl3 28559 | Sublattice mapping for a m... |
mdslle1i 28560 | Order preservation of the ... |
mdslle2i 28561 | Order preservation of the ... |
mdslj1i 28562 | Join preservation of the o... |
mdslj2i 28563 | Meet preservation of the r... |
mdsl1i 28564 | If the modular pair proper... |
mdsl2i 28565 | If the modular pair proper... |
mdsl2bi 28566 | If the modular pair proper... |
cvmdi 28567 | The covering property impl... |
mdslmd1lem1 28568 | Lemma for ~ mdslmd1i . (C... |
mdslmd1lem2 28569 | Lemma for ~ mdslmd1i . (C... |
mdslmd1lem3 28570 | Lemma for ~ mdslmd1i . (C... |
mdslmd1lem4 28571 | Lemma for ~ mdslmd1i . (C... |
mdslmd1i 28572 | Preservation of the modula... |
mdslmd2i 28573 | Preservation of the modula... |
mdsldmd1i 28574 | Preservation of the dual m... |
mdslmd3i 28575 | Modular pair conditions th... |
mdslmd4i 28576 | Modular pair condition tha... |
csmdsymi 28577 | Cross-symmetry implies M-s... |
mdexchi 28578 | An exchange lemma for modu... |
cvmd 28579 | The covering property impl... |
cvdmd 28580 | The covering property impl... |
ela 28582 | Atoms in a Hilbert lattice... |
elat2 28583 | Expanded membership relati... |
elatcv0 28584 | A Hilbert lattice element ... |
atcv0 28585 | An atom covers the zero su... |
atssch 28586 | Atoms are a subset of the ... |
atelch 28587 | An atom is a Hilbert latti... |
atne0 28588 | An atom is not the Hilbert... |
atss 28589 | A lattice element smaller ... |
atsseq 28590 | Two atoms in a subset rela... |
atcveq0 28591 | A Hilbert lattice element ... |
h1da 28592 | A 1-dimensional subspace i... |
spansna 28593 | The span of the singleton ... |
sh1dle 28594 | A 1-dimensional subspace i... |
ch1dle 28595 | A 1-dimensional subspace i... |
atom1d 28596 | The 1-dimensional subspace... |
superpos 28597 | Superposition Principle. ... |
chcv1 28598 | The Hilbert lattice has th... |
chcv2 28599 | The Hilbert lattice has th... |
chjatom 28600 | The join of a closed subsp... |
shatomici 28601 | The lattice of Hilbert sub... |
hatomici 28602 | The Hilbert lattice is ato... |
hatomic 28603 | A Hilbert lattice is atomi... |
shatomistici 28604 | The lattice of Hilbert sub... |
hatomistici 28605 | ` CH ` is atomistic, i.e. ... |
chpssati 28606 | Two Hilbert lattice elemen... |
chrelati 28607 | The Hilbert lattice is rel... |
chrelat2i 28608 | A consequence of relative ... |
cvati 28609 | If a Hilbert lattice eleme... |
cvbr4i 28610 | An alternate way to expres... |
cvexchlem 28611 | Lemma for ~ cvexchi . (Co... |
cvexchi 28612 | The Hilbert lattice satisf... |
chrelat2 28613 | A consequence of relative ... |
chrelat3 28614 | A consequence of relative ... |
chrelat3i 28615 | A consequence of the relat... |
chrelat4i 28616 | A consequence of relative ... |
cvexch 28617 | The Hilbert lattice satisf... |
cvp 28618 | The Hilbert lattice satisf... |
atnssm0 28619 | The meet of a Hilbert latt... |
atnemeq0 28620 | The meet of distinct atoms... |
atssma 28621 | The meet with an atom's su... |
atcv0eq 28622 | Two atoms covering the zer... |
atcv1 28623 | Two atoms covering the zer... |
atexch 28624 | The Hilbert lattice satisf... |
atomli 28625 | An assertion holding in at... |
atoml2i 28626 | An assertion holding in at... |
atordi 28627 | An ordering law for a Hilb... |
atcvatlem 28628 | Lemma for ~ atcvati . (Co... |
atcvati 28629 | A nonzero Hilbert lattice ... |
atcvat2i 28630 | A Hilbert lattice element ... |
atord 28631 | An ordering law for a Hilb... |
atcvat2 28632 | A Hilbert lattice element ... |
chirredlem1 28633 | Lemma for ~ chirredi . (C... |
chirredlem2 28634 | Lemma for ~ chirredi . (C... |
chirredlem3 28635 | Lemma for ~ chirredi . (C... |
chirredlem4 28636 | Lemma for ~ chirredi . (C... |
chirredi 28637 | The Hilbert lattice is irr... |
chirred 28638 | The Hilbert lattice is irr... |
atcvat3i 28639 | A condition implying that ... |
atcvat4i 28640 | A condition implying exist... |
atdmd 28641 | Two Hilbert lattice elemen... |
atmd 28642 | Two Hilbert lattice elemen... |
atmd2 28643 | Two Hilbert lattice elemen... |
atabsi 28644 | Absorption of an incompara... |
atabs2i 28645 | Absorption of an incompara... |
mdsymlem1 28646 | Lemma for ~ mdsymi . (Con... |
mdsymlem2 28647 | Lemma for ~ mdsymi . (Con... |
mdsymlem3 28648 | Lemma for ~ mdsymi . (Con... |
mdsymlem4 28649 | Lemma for ~ mdsymi . This... |
mdsymlem5 28650 | Lemma for ~ mdsymi . (Con... |
mdsymlem6 28651 | Lemma for ~ mdsymi . This... |
mdsymlem7 28652 | Lemma for ~ mdsymi . Lemm... |
mdsymlem8 28653 | Lemma for ~ mdsymi . Lemm... |
mdsymi 28654 | M-symmetry of the Hilbert ... |
mdsym 28655 | M-symmetry of the Hilbert ... |
dmdsym 28656 | Dual M-symmetry of the Hil... |
atdmd2 28657 | Two Hilbert lattice elemen... |
sumdmdii 28658 | If the subspace sum of two... |
cmmdi 28659 | Commuting subspaces form a... |
cmdmdi 28660 | Commuting subspaces form a... |
sumdmdlem 28661 | Lemma for ~ sumdmdi . The... |
sumdmdlem2 28662 | Lemma for ~ sumdmdi . (Co... |
sumdmdi 28663 | The subspace sum of two Hi... |
dmdbr4ati 28664 | Dual modular pair property... |
dmdbr5ati 28665 | Dual modular pair property... |
dmdbr6ati 28666 | Dual modular pair property... |
dmdbr7ati 28667 | Dual modular pair property... |
mdoc1i 28668 | Orthocomplements form a mo... |
mdoc2i 28669 | Orthocomplements form a mo... |
dmdoc1i 28670 | Orthocomplements form a du... |
dmdoc2i 28671 | Orthocomplements form a du... |
mdcompli 28672 | A condition equivalent to ... |
dmdcompli 28673 | A condition equivalent to ... |
mddmdin0i 28674 | If dual modular implies mo... |
cdjreui 28675 | A member of the sum of dis... |
cdj1i 28676 | Two ways to express " ` A ... |
cdj3lem1 28677 | A property of " ` A ` and ... |
cdj3lem2 28678 | Lemma for ~ cdj3i . Value... |
cdj3lem2a 28679 | Lemma for ~ cdj3i . Closu... |
cdj3lem2b 28680 | Lemma for ~ cdj3i . The f... |
cdj3lem3 28681 | Lemma for ~ cdj3i . Value... |
cdj3lem3a 28682 | Lemma for ~ cdj3i . Closu... |
cdj3lem3b 28683 | Lemma for ~ cdj3i . The s... |
cdj3i 28684 | Two ways to express " ` A ... |
The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
mathbox 28685 | (_This theorem is a dummy ... |
foo3 28686 | A theorem about the univer... |
xfree 28687 | A partial converse to ~ 19... |
xfree2 28688 | A partial converse to ~ 19... |
addltmulALT 28689 | A proof readability experi... |
bian1d 28690 | Adding a superfluous conju... |
or3di 28691 | Distributive law for disju... |
or3dir 28692 | Distributive law for disju... |
3o1cs 28693 | Deduction eliminating disj... |
3o2cs 28694 | Deduction eliminating disj... |
3o3cs 28695 | Deduction eliminating disj... |
spc2ed 28696 | Existential specialization... |
spc2d 28697 | Specialization with 2 quan... |
eqvincg 28698 | A variable introduction la... |
vtocl2d 28699 | Implicit substitution of t... |
ralcom4f 28700 | Commutation of restricted ... |
rexcom4f 28701 | Commutation of restricted ... |
19.9d2rf 28702 | A deduction version of one... |
19.9d2r 28703 | A deduction version of one... |
clelsb3f 28704 | Substitution applied to an... |
sbceqbidf 28705 | Equality theorem for class... |
sbcies 28706 | A special version of class... |
moel 28707 | "At most one" element in a... |
mo5f 28708 | Alternate definition of "a... |
nmo 28709 | Negation of "at most one".... |
moimd 28710 | "At most one" is preserved... |
rmoeqALT 28711 | Equality's restricted exis... |
2reuswap2 28712 | A condition allowing swap ... |
reuxfr3d 28713 | Transfer existential uniqu... |
reuxfr4d 28714 | Transfer existential uniqu... |
rexunirn 28715 | Restricted existential qua... |
rmoxfrdOLD 28716 | Transfer "at most one" res... |
rmoxfrd 28717 | Transfer "at most one" res... |
ssrmo 28718 | "At most one" existential ... |
rmo3f 28719 | Restricted "at most one" u... |
rmo4fOLD 28720 | Restricted "at most one" u... |
rmo4f 28721 | Restricted "at most one" u... |
rabrab 28722 | Abstract builder restricte... |
rabtru 28723 | Abtract builder using the ... |
rabid2f 28724 | An "identity" law for rest... |
rabexgfGS 28725 | Separation Scheme in terms... |
rabsnel 28726 | Truth implied by equality ... |
foresf1o 28727 | From a surjective function... |
rabfodom 28728 | Domination relation for re... |
abrexdomjm 28729 | An indexed set is dominate... |
abrexdom2jm 28730 | An indexed set is dominate... |
abrexexd 28731 | Existence of a class abstr... |
elabreximd 28732 | Class substitution in an i... |
elabreximdv 28733 | Class substitution in an i... |
abrexss 28734 | A necessary condition for ... |
eqri 28735 | Infer equality of classes ... |
rabss3d 28736 | Subclass law for restricte... |
inin 28737 | Intersection with an inter... |
inindif 28738 | See ~ inundif . (Contribu... |
difeq 28739 | Rewriting an equation with... |
indifundif 28740 | A remarkable equation with... |
elpwincl1 28741 | Closure of intersection wi... |
elpwdifcl 28742 | Closure of class differenc... |
elpwiuncl 28743 | Closure of indexed union w... |
elpreq 28744 | Equality wihin a pair. (C... |
ifeqeqx 28745 | An equality theorem tailor... |
elimifd 28746 | Elimination of a condition... |
elim2if 28747 | Elimination of two conditi... |
elim2ifim 28748 | Elimination of two conditi... |
uniinn0 28749 | Sufficient and necessary c... |
uniin1 28750 | Union of intersection. Ge... |
uniin2 28751 | Union of intersection. Ge... |
difuncomp 28752 | Express a class difference... |
pwuniss 28753 | Condition for a class unio... |
elpwunicl 28754 | Closure of a set union wit... |
cbviunf 28755 | Rule used to change the bo... |
iuneq12daf 28756 | Equality deduction for ind... |
iunin1f 28757 | Indexed union of intersect... |
iunxsngf 28758 | A singleton index picks ou... |
ssiun3 28759 | Subset equivalence for an ... |
ssiun2sf 28760 | Subset relationship for an... |
iuninc 28761 | The union of an increasing... |
iundifdifd 28762 | The intersection of a set ... |
iundifdif 28763 | The intersection of a set ... |
iunrdx 28764 | Re-index an indexed union.... |
iunpreima 28765 | Preimage of an indexed uni... |
disjnf 28766 | In case ` x ` is not free ... |
cbvdisjf 28767 | Change bound variables in ... |
disjss1f 28768 | A subset of a disjoint col... |
disjeq1f 28769 | Equality theorem for disjo... |
disjdifprg 28770 | A trivial partition into a... |
disjdifprg2 28771 | A trivial partition of a s... |
disji2f 28772 | Property of a disjoint col... |
disjif 28773 | Property of a disjoint col... |
disjorf 28774 | Two ways to say that a col... |
disjorsf 28775 | Two ways to say that a col... |
disjif2 28776 | Property of a disjoint col... |
disjabrex 28777 | Rewriting a disjoint colle... |
disjabrexf 28778 | Rewriting a disjoint colle... |
disjpreima 28779 | A preimage of a disjoint s... |
disjrnmpt 28780 | Rewriting a disjoint colle... |
disjin 28781 | If a collection is disjoin... |
disjin2 28782 | If a collection is disjoin... |
disjxpin 28783 | Derive a disjunction over ... |
iundisjf 28784 | Rewrite a countable union ... |
iundisj2f 28785 | A disjoint union is disjoi... |
disjrdx 28786 | Re-index a disjunct collec... |
disjex 28787 | Two ways to say that two c... |
disjexc 28788 | A variant of ~ disjex , ap... |
disjunsn 28789 | Append an element to a dis... |
disjun0 28790 | Adding the empty element p... |
disjiunel 28791 | A set of elements B of a d... |
disjuniel 28792 | A set of elements B of a d... |
xpdisjres 28793 | Restriction of a constant ... |
opeldifid 28794 | Ordered pair elementhood o... |
difres 28795 | Case when class difference... |
imadifxp 28796 | Image of the difference wi... |
relfi 28797 | A relation (set) is finite... |
fcoinver 28798 | Build an equivalence relat... |
fcoinvbr 28799 | Binary relation for the eq... |
brabgaf 28800 | The law of concretion for ... |
brelg 28801 | Two things in a binary rel... |
br8d 28802 | Substitution for an eight-... |
opabdm 28803 | Domain of an ordered-pair ... |
opabrn 28804 | Range of an ordered-pair c... |
ssrelf 28805 | A subclass relationship de... |
eqrelrd2 28806 | A version of ~ eqrelrdv2 w... |
erbr3b 28807 | Biconditional for equivale... |
iunsnima 28808 | Image of a singleton by an... |
mptexgf 28809 | If the domain of a functio... |
ac6sf2 28810 | Alternate version of ~ ac6... |
idssxp 28811 | A diagonal set as a subset... |
fnresin 28812 | Restriction of a function ... |
f1o3d 28813 | Describe an implicit one-t... |
rinvf1o 28814 | Sufficient conditions for ... |
fresf1o 28815 | Conditions for a restricti... |
f1mptrn 28816 | Express injection for a ma... |
dfimafnf 28817 | Alternate definition of th... |
funimass4f 28818 | Membership relation for th... |
suppss2f 28819 | Show that the support of a... |
fovcld 28820 | Closure law for an operati... |
ofrn 28821 | The range of the function ... |
ofrn2 28822 | The range of the function ... |
off2 28823 | The function operation pro... |
ofresid 28824 | Applying an operation rest... |
fimarab 28825 | Expressing the image of a ... |
unipreima 28826 | Preimage of a class union.... |
sspreima 28827 | The preimage of a subset i... |
opfv 28828 | Value of a function produc... |
xppreima 28829 | The preimage of a Cartesia... |
xppreima2 28830 | The preimage of a Cartesia... |
elunirn2 28831 | Condition for the membersh... |
abfmpunirn 28832 | Membership in a union of a... |
rabfmpunirn 28833 | Membership in a union of a... |
abfmpeld 28834 | Membership in an element o... |
abfmpel 28835 | Membership in an element o... |
fmptdF 28836 | Domain and co-domain of th... |
mpteq12df 28837 | An equality theorem for th... |
resmptf 28838 | Restriction of the mapping... |
fmptcof2 28839 | Composition of two functio... |
fcomptf 28840 | Express composition of two... |
acunirnmpt 28841 | Axiom of choice for the un... |
acunirnmpt2 28842 | Axiom of choice for the un... |
acunirnmpt2f 28843 | Axiom of choice for the un... |
aciunf1lem 28844 | Choice in an index union. ... |
aciunf1 28845 | Choice in an index union. ... |
cofmpt 28846 | Express composition of a m... |
ofoprabco 28847 | Function operation as a co... |
ofpreima 28848 | Express the preimage of a ... |
ofpreima2 28849 | Express the preimage of a ... |
funcnvmptOLD 28850 | Condition for a function i... |
funcnvmpt 28851 | Condition for a function i... |
funcnv5mpt 28852 | Two ways to say that a fun... |
funcnv4mpt 28853 | Two ways to say that a fun... |
fgreu 28854 | Exactly one point of a fun... |
fcnvgreu 28855 | If the converse of a relat... |
rnmpt2ss 28856 | The range of an operation ... |
mptssALT 28857 | Deduce subset relation of ... |
partfun 28858 | Rewrite a function defined... |
dfcnv2 28859 | Alternative definition of ... |
mpt2mptxf 28860 | Express a two-argument fun... |
gtiso 28861 | Two ways to write a strict... |
isoun 28862 | Infer an isomorphism from ... |
disjdsct 28863 | A disjoint collection is d... |
df1stres 28864 | Definition for a restricti... |
df2ndres 28865 | Definition for a restricti... |
1stpreimas 28866 | The preimage of a singleto... |
1stpreima 28867 | The preimage by ` 1st ` is... |
2ndpreima 28868 | The preimage by ` 2nd ` is... |
curry2ima 28869 | The image of a curried fun... |
supssd 28870 | Inequality deduction for s... |
infssd 28871 | Inequality deduction for i... |
imafi2 28872 | The image by a finite set ... |
unifi3 28873 | If a union is finite, then... |
snct 28874 | A singleton is countable. ... |
prct 28875 | An unordered pair is count... |
fnct 28876 | If the domain of a functio... |
dmct 28877 | The domain of a countable ... |
cnvct 28878 | If a set is countable, so ... |
rnct 28879 | The range of a countable s... |
mptct 28880 | A countable mapping set is... |
mpt2cti 28881 | An operation is countable ... |
abrexct 28882 | An image set of a countabl... |
mptctf 28883 | A countable mapping set is... |
abrexctf 28884 | An image set of a countabl... |
padct 28885 | Index a countable set with... |
cnvoprab 28886 | The converse of a class ab... |
f1od2 28887 | Describe an implicit one-t... |
fcobij 28888 | Composing functions with a... |
fcobijfs 28889 | Composing finitely support... |
suppss3 28890 | Deduce a function's suppor... |
ffs2 28891 | Rewrite a function's suppo... |
ffsrn 28892 | The range of a finitely su... |
resf1o 28893 | Restriction of functions t... |
maprnin 28894 | Restricting the range of t... |
fpwrelmapffslem 28895 | Lemma for ~ fpwrelmapffs .... |
fpwrelmap 28896 | Define a canonical mapping... |
fpwrelmapffs 28897 | Define a canonical mapping... |
addeq0 28898 | Two complex which add up t... |
subeqxfrd 28899 | Transfer two terms of a su... |
znsqcld 28900 | Squaring of nonzero relati... |
nn0sqeq1 28901 | Integer square one. (Cont... |
1neg1t1neg1 28902 | An integer unit times itse... |
lt2addrd 28903 | If the right-hand side of ... |
xgepnf 28904 | An extended real which is ... |
xlemnf 28905 | An extended real which is ... |
xrlelttric 28906 | Trichotomy law for extende... |
xaddeq0 28907 | Two extended reals which a... |
infxrmnf 28908 | The infinimum of a set of ... |
xrinfm 28909 | The extended real numbers ... |
le2halvesd 28910 | A sum is less than the who... |
xraddge02 28911 | A number is less than or e... |
xrge0addge 28912 | A number is less than or e... |
xlt2addrd 28913 | If the right-hand side of ... |
xrsupssd 28914 | Inequality deduction for s... |
xrge0infss 28915 | Any subset of nonnegative ... |
xrge0infssd 28916 | Inequality deduction for i... |
xrge0addcld 28917 | Nonnegative extended reals... |
xrge0subcld 28918 | Condition for closure of n... |
infxrge0lb 28919 | A member of a set of nonne... |
infxrge0glb 28920 | The infimum of a set of no... |
infxrge0gelb 28921 | The infimum of a set of no... |
dfrp2 28922 | Alternate definition of th... |
xrofsup 28923 | The supremum is preserved ... |
supxrnemnf 28924 | The supremum of a nonempty... |
xrhaus 28925 | The topology of the extend... |
joiniooico 28926 | Disjoint joining an open i... |
ubico 28927 | A right-open interval does... |
xeqlelt 28928 | Equality in terms of 'less... |
eliccelico 28929 | Relate elementhood to a cl... |
elicoelioo 28930 | Relate elementhood to a cl... |
iocinioc2 28931 | Intersection between two o... |
xrdifh 28932 | Class difference of a half... |
iocinif 28933 | Relate intersection of two... |
difioo 28934 | The difference between two... |
difico 28935 | The difference between two... |
nndiffz1 28936 | Upper set of the positive ... |
ssnnssfz 28937 | For any finite subset of `... |
fzspl 28938 | Split the last element of ... |
fzdif2 28939 | Split the last element of ... |
fzsplit3 28940 | Split a finite interval of... |
bcm1n 28941 | The proportion of one bino... |
iundisjfi 28942 | Rewrite a countable union ... |
iundisj2fi 28943 | A disjoint union is disjoi... |
iundisjcnt 28944 | Rewrite a countable union ... |
iundisj2cnt 28945 | A countable disjoint union... |
f1ocnt 28946 | Given a countable set ` A ... |
fz1nnct 28947 | NN and integer ranges star... |
fz1nntr 28948 | NN and integer ranges star... |
hashunif 28949 | The cardinality of a disjo... |
numdenneg 28950 | Numerator and denominator ... |
divnumden2 28951 | Calculate the reduced form... |
nnindf 28952 | Principle of Mathematical ... |
nnindd 28953 | Principle of Mathematical ... |
nn0min 28954 | Extracting the minimum pos... |
ltesubnnd 28955 | Subtracting an integer num... |
xdivval 28958 | Value of division: the (un... |
xrecex 28959 | Existence of reciprocal of... |
xmulcand 28960 | Cancellation law for exten... |
xreceu 28961 | Existential uniqueness of ... |
xdivcld 28962 | Closure law for the extend... |
xdivcl 28963 | Closure law for the extend... |
xdivmul 28964 | Relationship between divis... |
rexdiv 28965 | The extended real division... |
xdivrec 28966 | Relationship between divis... |
xdivid 28967 | A number divided by itself... |
xdiv0 28968 | Division into zero is zero... |
xdiv0rp 28969 | Division into zero is zero... |
eliccioo 28970 | Membership in a closed int... |
elxrge02 28971 | Elementhood in the set of ... |
xdivpnfrp 28972 | Plus infinity divided by a... |
rpxdivcld 28973 | Closure law for extended d... |
xrpxdivcld 28974 | Closure law for extended d... |
bhmafibid1 28975 | The Brahmagupta-Fibonacci ... |
bhmafibid2 28976 | The Brahmagupta-Fibonacci ... |
2sqn0 28977 | If the sum of two squares ... |
2sqcoprm 28978 | If the sum of two squares ... |
2sqmod 28979 | Given two decompositions o... |
2sqmo 28980 | There exists at most one d... |
ressplusf 28981 | The group operation functi... |
ressnm 28982 | The norm in a restricted s... |
abvpropd2 28983 | Weaker version of ~ abvpro... |
oppgle 28984 | less-than relation of an o... |
oppglt 28985 | less-than relation of an o... |
ressprs 28986 | The restriction of a preor... |
oduprs 28987 | Being a preset is a self-d... |
posrasymb 28988 | A poset ordering is asymet... |
tospos 28989 | A Toset is a Poset. (Cont... |
resspos 28990 | The restriction of a Poset... |
resstos 28991 | The restriction of a Toset... |
tleile 28992 | In a Toset, two elements m... |
tltnle 28993 | In a Toset, less-than is e... |
odutos 28994 | Being a toset is a self-du... |
tlt2 28995 | In a Toset, two elements m... |
tlt3 28996 | In a Toset, two elements m... |
trleile 28997 | In a Toset, two elements m... |
toslublem 28998 | Lemma for ~ toslub and ~ x... |
toslub 28999 | In a toset, the lowest upp... |
tosglblem 29000 | Lemma for ~ tosglb and ~ x... |
tosglb 29001 | Same theorem as ~ toslub ,... |
clatp0cl 29002 | The poset zero of a comple... |
clatp1cl 29003 | The poset one of a complet... |
xrs0 29006 | The zero of the extended r... |
xrslt 29007 | The "strictly less than" r... |
xrsinvgval 29008 | The inversion operation in... |
xrsmulgzz 29009 | The "multiple" function in... |
xrstos 29010 | The extended real numbers ... |
xrsclat 29011 | The extended real numbers ... |
xrsp0 29012 | The poset 0 of the extende... |
xrsp1 29013 | The poset 1 of the extende... |
ressmulgnn 29014 | Values for the group multi... |
ressmulgnn0 29015 | Values for the group multi... |
xrge0base 29016 | The base of the extended n... |
xrge00 29017 | The zero of the extended n... |
xrge0plusg 29018 | The additive law of the ex... |
xrge0le 29019 | The lower-or-equal relatio... |
xrge0mulgnn0 29020 | The group multiple functio... |
xrge0addass 29021 | Associativity of extended ... |
xrge0addgt0 29022 | The sum of nonnegative and... |
xrge0adddir 29023 | Right-distributivity of ex... |
xrge0adddi 29024 | Left-distributivity of ext... |
xrge0npcan 29025 | Extended nonnegative real ... |
fsumrp0cl 29026 | Closure of a finite sum of... |
abliso 29027 | The image of an Abelian gr... |
isomnd 29032 | A (left) ordered monoid is... |
isogrp 29033 | A (left) ordered group is ... |
ogrpgrp 29034 | An left ordered group is a... |
omndmnd 29035 | A left ordered monoid is a... |
omndtos 29036 | A left ordered monoid is a... |
omndadd 29037 | In an ordered monoid, the ... |
omndaddr 29038 | In a right ordered monoid,... |
omndadd2d 29039 | In a commutative left orde... |
omndadd2rd 29040 | In a left- and right- orde... |
submomnd 29041 | A submonoid of an ordered ... |
xrge0omnd 29042 | The nonnegative extended r... |
omndmul2 29043 | In an ordered monoid, the ... |
omndmul3 29044 | In an ordered monoid, the ... |
omndmul 29045 | In a commutative ordered m... |
ogrpinvOLD 29046 | In an ordered group, the o... |
ogrpinv0le 29047 | In an ordered group, the o... |
ogrpsub 29048 | In an ordered group, the o... |
ogrpaddlt 29049 | In an ordered group, stric... |
ogrpaddltbi 29050 | In a right ordered group, ... |
ogrpaddltrd 29051 | In a right ordered group, ... |
ogrpaddltrbid 29052 | In a right ordered group, ... |
ogrpsublt 29053 | In an ordered group, stric... |
ogrpinv0lt 29054 | In an ordered group, the o... |
ogrpinvlt 29055 | In an ordered group, the o... |
sgnsv 29058 | The sign mapping. (Contri... |
sgnsval 29059 | The sign value. (Contribu... |
sgnsf 29060 | The sign function. (Contr... |
inftmrel 29065 | The infinitesimal relation... |
isinftm 29066 | Express ` x ` is infinites... |
isarchi 29067 | Express the predicate " ` ... |
pnfinf 29068 | Plus infinity is an infini... |
xrnarchi 29069 | The completed real line is... |
isarchi2 29070 | Alternative way to express... |
submarchi 29071 | A submonoid is archimedean... |
isarchi3 29072 | This is the usual definiti... |
archirng 29073 | Property of Archimedean or... |
archirngz 29074 | Property of Archimedean le... |
archiexdiv 29075 | In an Archimedean group, g... |
archiabllem1a 29076 | Lemma for ~ archiabl : In... |
archiabllem1b 29077 | Lemma for ~ archiabl . (C... |
archiabllem1 29078 | Archimedean ordered groups... |
archiabllem2a 29079 | Lemma for ~ archiabl , whi... |
archiabllem2c 29080 | Lemma for ~ archiabl . (C... |
archiabllem2b 29081 | Lemma for ~ archiabl . (C... |
archiabllem2 29082 | Archimedean ordered groups... |
archiabl 29083 | Archimedean left- and righ... |
isslmd 29086 | The predicate "is a semimo... |
slmdlema 29087 | Lemma for properties of a ... |
lmodslmd 29088 | Left semimodules generaliz... |
slmdcmn 29089 | A semimodule is a commutat... |
slmdmnd 29090 | A semimodule is a monoid. ... |
slmdsrg 29091 | The scalar component of a ... |
slmdbn0 29092 | The base set of a semimodu... |
slmdacl 29093 | Closure of ring addition f... |
slmdmcl 29094 | Closure of ring multiplica... |
slmdsn0 29095 | The set of scalars in a se... |
slmdvacl 29096 | Closure of vector addition... |
slmdass 29097 | Semiring left module vecto... |
slmdvscl 29098 | Closure of scalar product ... |
slmdvsdi 29099 | Distributive law for scala... |
slmdvsdir 29100 | Distributive law for scala... |
slmdvsass 29101 | Associative law for scalar... |
slmd0cl 29102 | The ring zero in a semimod... |
slmd1cl 29103 | The ring unit in a semirin... |
slmdvs1 29104 | Scalar product with ring u... |
slmd0vcl 29105 | The zero vector is a vecto... |
slmd0vlid 29106 | Left identity law for the ... |
slmd0vrid 29107 | Right identity law for the... |
slmd0vs 29108 | Zero times a vector is the... |
slmdvs0 29109 | Anything times the zero ve... |
gsumle 29110 | A finite sum in an ordered... |
gsummpt2co 29111 | Split a finite sum into a ... |
gsummpt2d 29112 | Express a finite sum over ... |
gsumvsca1 29113 | Scalar product of a finite... |
gsumvsca2 29114 | Scalar product of a finite... |
gsummptres 29115 | Extend a finite group sum ... |
xrge0tsmsd 29116 | Any finite or infinite sum... |
xrge0tsmsbi 29117 | Any limit of a finite or i... |
xrge0tsmseq 29118 | Any limit of a finite or i... |
rngurd 29119 | Deduce the unit of a ring ... |
ress1r 29120 | ` 1r ` is unaffected by re... |
dvrdir 29121 | Distributive law for the d... |
rdivmuldivd 29122 | Multiplication of two rati... |
ringinvval 29123 | The ring inverse expressed... |
dvrcan5 29124 | Cancellation law for commo... |
subrgchr 29125 | If ` A ` is a subring of `... |
isorng 29130 | An ordered ring is a ring ... |
orngring 29131 | An ordered ring is a ring.... |
orngogrp 29132 | An ordered ring is an orde... |
isofld 29133 | An ordered field is a fiel... |
orngmul 29134 | In an ordered ring, the or... |
orngsqr 29135 | In an ordered ring, all sq... |
ornglmulle 29136 | In an ordered ring, multip... |
orngrmulle 29137 | In an ordered ring, multip... |
ornglmullt 29138 | In an ordered ring, multip... |
orngrmullt 29139 | In an ordered ring, multip... |
orngmullt 29140 | In an ordered ring, the st... |
ofldfld 29141 | An ordered field is a fiel... |
ofldtos 29142 | An ordered field is a tota... |
orng0le1 29143 | In an ordered ring, the ri... |
ofldlt1 29144 | In an ordered field, the r... |
ofldchr 29145 | The characteristic of an o... |
suborng 29146 | Every subring of an ordere... |
subofld 29147 | Every subfield of an order... |
isarchiofld 29148 | Axiom of Archimedes : a ch... |
rhmdvdsr 29149 | A ring homomorphism preser... |
rhmopp 29150 | A ring homomorphism is als... |
elrhmunit 29151 | Ring homomorphisms preserv... |
rhmdvd 29152 | A ring homomorphism preser... |
rhmunitinv 29153 | Ring homomorphisms preserv... |
kerunit 29154 | If a unit element lies in ... |
reldmresv 29157 | The scalar restriction is ... |
resvval 29158 | Value of structure restric... |
resvid2 29159 | General behavior of trivia... |
resvval2 29160 | Value of nontrivial struct... |
resvsca 29161 | Base set of a structure re... |
resvlem 29162 | Other elements of a struct... |
resvbas 29163 | ` Base ` is unaffected by ... |
resvplusg 29164 | ` +g ` is unaffected by sc... |
resvvsca 29165 | ` .s ` is unaffected by sc... |
resvmulr 29166 | ` .s ` is unaffected by sc... |
resv0g 29167 | ` 0g ` is unaffected by sc... |
resv1r 29168 | ` 1r ` is unaffected by sc... |
resvcmn 29169 | Scalar restriction preserv... |
gzcrng 29170 | The gaussian integers form... |
reofld 29171 | The real numbers form an o... |
nn0omnd 29172 | The nonnegative integers f... |
rearchi 29173 | The field of the real numb... |
nn0archi 29174 | The monoid of the nonnegat... |
xrge0slmod 29175 | The extended nonnegative r... |
symgfcoeu 29176 | Uniqueness property of per... |
psgndmfi 29177 | For a finite base set, the... |
psgnid 29178 | Permutation sign of the id... |
pmtrprfv2 29179 | In a transposition of two ... |
pmtrto1cl 29180 | Useful lemma for the follo... |
psgnfzto1stlem 29181 | Lemma for ~ psgnfzto1st . ... |
fzto1stfv1 29182 | Value of our permutation `... |
fzto1st1 29183 | Special case where the per... |
fzto1st 29184 | The function moving one el... |
fzto1stinvn 29185 | Value of the inverse of ou... |
psgnfzto1st 29186 | The permutation sign for m... |
smatfval 29189 | Value of the submatrix. (... |
smatrcl 29190 | Closure of the rectangular... |
smatlem 29191 | Lemma for the next theorem... |
smattl 29192 | Entries of a submatrix, to... |
smattr 29193 | Entries of a submatrix, to... |
smatbl 29194 | Entries of a submatrix, bo... |
smatbr 29195 | Entries of a submatrix, bo... |
smatcl 29196 | Closure of the square subm... |
matmpt2 29197 | Write a square matrix as a... |
1smat1 29198 | The submatrix of the ident... |
submat1n 29199 | One case where the submatr... |
submatres 29200 | Special case where the sub... |
submateqlem1 29201 | Lemma for ~ submateq . (C... |
submateqlem2 29202 | Lemma for ~ submateq . (C... |
submateq 29203 | Sufficient condition for t... |
submatminr1 29204 | If we take a submatrix by ... |
lmatval 29207 | Value of the literal matri... |
lmatfval 29208 | Entries of a literal matri... |
lmatfvlem 29209 | Useful lemma to extract li... |
lmatcl 29210 | Closure of the literal mat... |
lmat22lem 29211 | Lemma for ~ lmat22e11 and ... |
lmat22e11 29212 | Entry of a 2x2 literal mat... |
lmat22e12 29213 | Entry of a 2x2 literal mat... |
lmat22e21 29214 | Entry of a 2x2 literal mat... |
lmat22e22 29215 | Entry of a 2x2 literal mat... |
lmat22det 29216 | The determinant of a liter... |
mdetpmtr1 29217 | The determinant of a matri... |
mdetpmtr2 29218 | The determinant of a matri... |
mdetpmtr12 29219 | The determinant of a matri... |
mdetlap1 29220 | A Laplace expansion of the... |
madjusmdetlem1 29221 | Lemma for ~ madjusmdet . ... |
madjusmdetlem2 29222 | Lemma for ~ madjusmdet . ... |
madjusmdetlem3 29223 | Lemma for ~ madjusmdet . ... |
madjusmdetlem4 29224 | Lemma for ~ madjusmdet . ... |
madjusmdet 29225 | Express the cofactor of th... |
mdetlap 29226 | Laplace expansion of the d... |
fvproj 29227 | Value of a function on pai... |
fimaproj 29228 | Image of a cartesian produ... |
txomap 29229 | Given two open maps ` F ` ... |
qtopt1 29230 | If every equivalence class... |
qtophaus 29231 | If an open map's graph in ... |
circtopn 29232 | The topology of the unit c... |
circcn 29233 | The function gluing the re... |
reff 29234 | For any cover refinement, ... |
locfinreflem 29235 | A locally finite refinemen... |
locfinref 29236 | A locally finite refinemen... |
iscref 29239 | The property that every op... |
crefeq 29240 | Equality theorem for the "... |
creftop 29241 | A space where every open c... |
crefi 29242 | The property that every op... |
crefdf 29243 | A formulation of ~ crefi e... |
crefss 29244 | The "every open cover has ... |
cmpcref 29245 | Equivalent definition of c... |
cmpfiref 29246 | Every open cover of a Comp... |
ldlfcntref 29249 | Every open cover of a Lind... |
ispcmp 29252 | The predicate "is a paraco... |
cmppcmp 29253 | Every compact space is par... |
dispcmp 29254 | Every discrete space is pa... |
pcmplfin 29255 | Given a paracompact topolo... |
pcmplfinf 29256 | Given a paracompact topolo... |
metidval 29261 | Value of the metric identi... |
metidss 29262 | As a relation, the metric ... |
metidv 29263 | ` A ` and ` B ` identify b... |
metideq 29264 | Basic property of the metr... |
metider 29265 | The metric identification ... |
pstmval 29266 | Value of the metric induce... |
pstmfval 29267 | Function value of the metr... |
pstmxmet 29268 | The metric induced by a ps... |
hauseqcn 29269 | In a Hausdorff topology, t... |
unitsscn 29270 | The closed unit is a subse... |
elunitrn 29271 | The closed unit is a subse... |
elunitcn 29272 | The closed unit is a subse... |
elunitge0 29273 | An element of the closed u... |
unitssxrge0 29274 | The closed unit is a subse... |
unitdivcld 29275 | Necessary conditions for a... |
iistmd 29276 | The closed unit forms a to... |
unicls 29277 | The union of the closed se... |
tpr2tp 29278 | The usual topology on ` ( ... |
tpr2uni 29279 | The usual topology on ` ( ... |
xpinpreima 29280 | Rewrite the cartesian prod... |
xpinpreima2 29281 | Rewrite the cartesian prod... |
sqsscirc1 29282 | The complex square of side... |
sqsscirc2 29283 | The complex square of side... |
cnre2csqlem 29284 | Lemma for ~ cnre2csqima . ... |
cnre2csqima 29285 | Image of a centered square... |
tpr2rico 29286 | For any point of an open s... |
cnvordtrestixx 29287 | The restriction of the 'gr... |
prsdm 29288 | Domain of the relation of ... |
prsrn 29289 | Range of the relation of a... |
prsss 29290 | Relation of a subpreset. ... |
prsssdm 29291 | Domain of a subpreset rela... |
ordtprsval 29292 | Value of the order topolog... |
ordtprsuni 29293 | Value of the order topolog... |
ordtcnvNEW 29294 | The order dual generates t... |
ordtrestNEW 29295 | The subspace topology of a... |
ordtrest2NEWlem 29296 | Lemma for ~ ordtrest2NEW .... |
ordtrest2NEW 29297 | An interval-closed set ` A... |
ordtconlem1 29298 | Connectedness in the order... |
ordtcon 29299 | Connectedness in the order... |
mndpluscn 29300 | A mapping that is both a h... |
mhmhmeotmd 29301 | Deduce a Topological Monoi... |
rmulccn 29302 | Multiplication by a real c... |
raddcn 29303 | Addition in the real numbe... |
xrmulc1cn 29304 | The operation multiplying ... |
fmcncfil 29305 | The image of a Cauchy filt... |
xrge0hmph 29306 | The extended nonnegative r... |
xrge0iifcnv 29307 | Define a bijection from ` ... |
xrge0iifcv 29308 | The defined function's val... |
xrge0iifiso 29309 | The defined bijection from... |
xrge0iifhmeo 29310 | Expose a homeomorphism fro... |
xrge0iifhom 29311 | The defined function from ... |
xrge0iif1 29312 | Condition for the defined ... |
xrge0iifmhm 29313 | The defined function from ... |
xrge0pluscn 29314 | The addition operation of ... |
xrge0mulc1cn 29315 | The operation multiplying ... |
xrge0tps 29316 | The extended nonnegative r... |
xrge0topn 29317 | The topology of the extend... |
xrge0haus 29318 | The topology of the extend... |
xrge0tmdOLD 29319 | The extended nonnegative r... |
xrge0tmd 29320 | The extended nonnegative r... |
lmlim 29321 | Relate a limit in a given ... |
lmlimxrge0 29322 | Relate a limit in the nonn... |
rge0scvg 29323 | Implication of convergence... |
fsumcvg4 29324 | A serie with finite suppor... |
pnfneige0 29325 | A neighborhood of ` +oo ` ... |
lmxrge0 29326 | Express "sequence ` F ` co... |
lmdvg 29327 | If a monotonic sequence of... |
lmdvglim 29328 | If a monotonic real number... |
pl1cn 29329 | A univariate polynomial is... |
zringnm 29332 | The norm (function) for a ... |
zzsnm 29333 | The norm of the ring of th... |
zlm0 29334 | Zero of a ` ZZ ` -module. ... |
zlm1 29335 | Unit of a ` ZZ ` -module (... |
zlmds 29336 | Distance in a ` ZZ ` -modu... |
zlmtset 29337 | Topology in a ` ZZ ` -modu... |
zlmnm 29338 | Norm of a ` ZZ ` -module (... |
zhmnrg 29339 | The ` ZZ ` -module built f... |
nmmulg 29340 | The norm of a group produc... |
zrhnm 29341 | The norm of the image by `... |
cnzh 29342 | The ` ZZ ` -module of ` CC... |
rezh 29343 | The ` ZZ ` -module of ` RR... |
qqhval 29346 | Value of the canonical hom... |
zrhf1ker 29347 | The kernel of the homomorp... |
zrhchr 29348 | The kernel of the homomorp... |
zrhker 29349 | The kernel of the homomorp... |
zrhunitpreima 29350 | The preimage by ` ZRHom ` ... |
elzrhunit 29351 | Condition for the image by... |
elzdif0 29352 | Lemma for ~ qqhval2 . (Co... |
qqhval2lem 29353 | Lemma for ~ qqhval2 . (Co... |
qqhval2 29354 | Value of the canonical hom... |
qqhvval 29355 | Value of the canonical hom... |
qqh0 29356 | The image of ` 0 ` by the ... |
qqh1 29357 | The image of ` 1 ` by the ... |
qqhf 29358 | ` QQHom ` as a function. ... |
qqhvq 29359 | The image of a quotient by... |
qqhghm 29360 | The ` QQHom ` homomorphism... |
qqhrhm 29361 | The ` QQHom ` homomorphism... |
qqhnm 29362 | The norm of the image by `... |
qqhcn 29363 | The ` QQHom ` homomorphism... |
qqhucn 29364 | The ` QQHom ` homomorphism... |
rrhval 29368 | Value of the canonical hom... |
rrhcn 29369 | If the topology of ` R ` i... |
rrhf 29370 | If the topology of ` R ` i... |
isrrext 29372 | Express the property " ` R... |
rrextnrg 29373 | An extension of ` RR ` is ... |
rrextdrg 29374 | An extension of ` RR ` is ... |
rrextnlm 29375 | The norm of an extension o... |
rrextchr 29376 | The ring characteristic of... |
rrextcusp 29377 | An extension of ` RR ` is ... |
rrexttps 29378 | An extension of ` RR ` is ... |
rrexthaus 29379 | The topology of an extensi... |
rrextust 29380 | The uniformity of an exten... |
rerrext 29381 | The field of the real numb... |
cnrrext 29382 | The field of the complex n... |
qqtopn 29383 | The topology of the field ... |
rrhfe 29384 | If ` R ` is an extension o... |
rrhcne 29385 | If ` R ` is an extension o... |
rrhqima 29386 | The ` RRHom ` homomorphism... |
rrh0 29387 | The image of ` 0 ` by the ... |
xrhval 29390 | The value of the embedding... |
zrhre 29391 | The ` ZRHom ` homomorphism... |
qqhre 29392 | The ` QQHom ` homomorphism... |
rrhre 29393 | The ` RRHom ` homomorphism... |
relmntop 29396 | Manifold is a relation. (... |
ismntoplly 29397 | Property of being a manifo... |
ismntop 29398 | Property of being a manifo... |
nexple 29399 | A lower bound for an expon... |
indv 29402 | Value of the indicator fun... |
indval 29403 | Value of the indicator fun... |
indval2 29404 | Alternate value of the ind... |
indf 29405 | An indicator function as a... |
indfval 29406 | Value of the indicator fun... |
pr01ssre 29407 | The range of the indicator... |
ind1 29408 | Value of the indicator fun... |
ind0 29409 | Value of the indicator fun... |
ind1a 29410 | Value of the indicator fun... |
indpi1 29411 | Preimage of the singleton ... |
indsum 29412 | Finite sum of a product wi... |
indf1o 29413 | The bijection between a po... |
indpreima 29414 | A function with range ` { ... |
indf1ofs 29415 | The bijection between fini... |
esumex 29418 | An extended sum is a set b... |
esumcl 29419 | Closure for extended sum i... |
esumeq12dvaf 29420 | Equality deduction for ext... |
esumeq12dva 29421 | Equality deduction for ext... |
esumeq12d 29422 | Equality deduction for ext... |
esumeq1 29423 | Equality theorem for an ex... |
esumeq1d 29424 | Equality theorem for an ex... |
esumeq2 29425 | Equality theorem for exten... |
esumeq2d 29426 | Equality deduction for ext... |
esumeq2dv 29427 | Equality deduction for ext... |
esumeq2sdv 29428 | Equality deduction for ext... |
nfesum1 29429 | Bound-variable hypothesis ... |
nfesum2 29430 | Bound-variable hypothesis ... |
cbvesum 29431 | Change bound variable in a... |
cbvesumv 29432 | Change bound variable in a... |
esumid 29433 | Identify the extended sum ... |
esumgsum 29434 | A finite extended sum is t... |
esumval 29435 | Develop the value of the e... |
esumel 29436 | The extended sum is a limi... |
esumnul 29437 | Extended sum over the empt... |
esum0 29438 | Extended sum of zero. (Co... |
esumf1o 29439 | Re-index an extended sum u... |
esumc 29440 | Convert from the collectio... |
esumrnmpt 29441 | Rewrite an extended sum in... |
esumsplit 29442 | Split an extended sum into... |
esummono 29443 | Extended sum is monotonic.... |
esumpad 29444 | Extend an extended sum by ... |
esumpad2 29445 | Remove zeroes from an exte... |
esumadd 29446 | Addition of infinite sums.... |
esumle 29447 | If all of the terms of an ... |
gsumesum 29448 | Relate a group sum on ` ( ... |
esumlub 29449 | The extended sum is the lo... |
esumaddf 29450 | Addition of infinite sums.... |
esumlef 29451 | If all of the terms of an ... |
esumcst 29452 | The extended sum of a cons... |
esumsnf 29453 | The extended sum of a sing... |
esumsn 29454 | The extended sum of a sing... |
esumpr 29455 | Extended sum over a pair. ... |
esumpr2 29456 | Extended sum over a pair, ... |
esumrnmpt2 29457 | Rewrite an extended sum in... |
esumfzf 29458 | Formulating a partial exte... |
esumfsup 29459 | Formulating an extended su... |
esumfsupre 29460 | Formulating an extended su... |
esumss 29461 | Change the index set to a ... |
esumpinfval 29462 | The value of the extended ... |
esumpfinvallem 29463 | Lemma for ~ esumpfinval . ... |
esumpfinval 29464 | The value of the extended ... |
esumpfinvalf 29465 | Same as ~ esumpfinval , mi... |
esumpinfsum 29466 | The value of the extended ... |
esumpcvgval 29467 | The value of the extended ... |
esumpmono 29468 | The partial sums in an ext... |
esumcocn 29469 | Lemma for ~ esummulc2 and ... |
esummulc1 29470 | An extended sum multiplied... |
esummulc2 29471 | An extended sum multiplied... |
esumdivc 29472 | An extended sum divided by... |
hashf2 29473 | Lemma for ~ hasheuni . (C... |
hasheuni 29474 | The cardinality of a disjo... |
esumcvg 29475 | The sequence of partial su... |
esumcvg2 29476 | Simpler version of ~ esumc... |
esumcvgsum 29477 | The value of the extended ... |
esumsup 29478 | Express an extended sum as... |
esumgect 29479 | "Send ` n ` to ` +oo ` " i... |
esumcvgre 29480 | All terms of a converging ... |
esum2dlem 29481 | Lemma for ~ esum2d (finite... |
esum2d 29482 | Write a double extended su... |
esumiun 29483 | Sum over a non necessarily... |
ofceq 29486 | Equality theorem for funct... |
ofcfval 29487 | Value of an operation appl... |
ofcval 29488 | Evaluate a function/consta... |
ofcfn 29489 | The function operation pro... |
ofcfeqd2 29490 | Equality theorem for funct... |
ofcfval3 29491 | General value of ` ( F oFC... |
ofcf 29492 | The function/constant oper... |
ofcfval2 29493 | The function operation exp... |
ofcfval4 29494 | The function/constant oper... |
ofcc 29495 | Left operation by a consta... |
ofcof 29496 | Relate function operation ... |
sigaex 29499 | Lemma for ~ issiga and ~ i... |
sigaval 29500 | The set of sigma-algebra w... |
issiga 29501 | An alternative definition ... |
isrnsigaOLD 29502 | The property of being a si... |
isrnsiga 29503 | The property of being a si... |
0elsiga 29504 | A sigma-algebra contains t... |
baselsiga 29505 | A sigma-algebra contains i... |
sigasspw 29506 | A sigma-algebra is a set o... |
sigaclcu 29507 | A sigma-algebra is closed ... |
sigaclcuni 29508 | A sigma-algebra is closed ... |
sigaclfu 29509 | A sigma-algebra is closed ... |
sigaclcu2 29510 | A sigma-algebra is closed ... |
sigaclfu2 29511 | A sigma-algebra is closed ... |
sigaclcu3 29512 | A sigma-algebra is closed ... |
issgon 29513 | Property of being a sigma-... |
sgon 29514 | A sigma-algebra is a sigma... |
elsigass 29515 | An element of a sigma-alge... |
elrnsiga 29516 | Dropping the base informat... |
isrnsigau 29517 | The property of being a si... |
unielsiga 29518 | A sigma-algebra contains i... |
dmvlsiga 29519 | Lebesgue-measurable subset... |
pwsiga 29520 | Any power set forms a sigm... |
prsiga 29521 | The smallest possible sigm... |
sigaclci 29522 | A sigma-algebra is closed ... |
difelsiga 29523 | A sigma-algebra is closed ... |
unelsiga 29524 | A sigma-algebra is closed ... |
inelsiga 29525 | A sigma-algebra is closed ... |
sigainb 29526 | Building a sigma-algebra f... |
insiga 29527 | The intersection of a coll... |
sigagenval 29530 | Value of the generated sig... |
sigagensiga 29531 | A generated sigma-algebra ... |
sgsiga 29532 | A generated sigma-algebra ... |
unisg 29533 | The sigma-algebra generate... |
dmsigagen 29534 | A sigma-algebra can be gen... |
sssigagen 29535 | A set is a subset of the s... |
sssigagen2 29536 | A subset of the generating... |
elsigagen 29537 | Any element of a set is al... |
elsigagen2 29538 | Any countable union of ele... |
sigagenss 29539 | The generated sigma-algebr... |
sigagenss2 29540 | Sufficient condition for i... |
sigagenid 29541 | The sigma-algebra generate... |
ispisys 29542 | The property of being a pi... |
ispisys2 29543 | The property of being a pi... |
inelpisys 29544 | Pi-systems are closed unde... |
sigapisys 29545 | All sigma-algebras are pi-... |
isldsys 29546 | The property of being a la... |
pwldsys 29547 | The power set of the unive... |
unelldsys 29548 | Lambda-systems are closed ... |
sigaldsys 29549 | All sigma-algebras are lam... |
ldsysgenld 29550 | The intersection of all la... |
sigapildsyslem 29551 | Lemma for ~ sigapildsys . ... |
sigapildsys 29552 | Sigma-algebra are exactly ... |
ldgenpisyslem1 29553 | Lemma for ~ ldgenpisys . ... |
ldgenpisyslem2 29554 | Lemma for ~ ldgenpisys . ... |
ldgenpisyslem3 29555 | Lemma for ~ ldgenpisys . ... |
ldgenpisys 29556 | The lambda system ` E ` ge... |
dynkin 29557 | Dynkin's lambda-pi theorem... |
isros 29558 | The property of being a ri... |
rossspw 29559 | A ring of sets is a collec... |
0elros 29560 | A ring of sets contains th... |
unelros 29561 | A ring of sets is closed u... |
difelros 29562 | A ring of sets is closed u... |
inelros 29563 | A ring of sets is closed u... |
fiunelros 29564 | A ring of sets is closed u... |
issros 29565 | The property of being a se... |
srossspw 29566 | A semi-ring of sets is a c... |
0elsros 29567 | A semi-ring of sets contai... |
inelsros 29568 | A semi-ring of sets is clo... |
diffiunisros 29569 | In semiring of sets, compl... |
rossros 29570 | Rings of sets are semi-rin... |
brsiga 29573 | The Borel Algebra on real ... |
brsigarn 29574 | The Borel Algebra is a sig... |
brsigasspwrn 29575 | The Borel Algebra is a set... |
unibrsiga 29576 | The union of the Borel Alg... |
cldssbrsiga 29577 | A Borel Algebra contains a... |
sxval 29580 | Value of the product sigma... |
sxsiga 29581 | A product sigma-algebra is... |
sxsigon 29582 | A product sigma-algebra is... |
sxuni 29583 | The base set of a product ... |
elsx 29584 | The cartesian product of t... |
measbase 29587 | The base set of a measure ... |
measval 29588 | The value of the ` measure... |
ismeas 29589 | The property of being a me... |
isrnmeas 29590 | The property of being a me... |
dmmeas 29591 | The domain of a measure is... |
measbasedom 29592 | The base set of a measure ... |
measfrge0 29593 | A measure is a function ov... |
measfn 29594 | A measure is a function on... |
measvxrge0 29595 | The values of a measure ar... |
measvnul 29596 | The measure of the empty s... |
measge0 29597 | A measure is nonnegative. ... |
measle0 29598 | If the measure of a given ... |
measvun 29599 | The measure of a countable... |
measxun2 29600 | The measure the union of t... |
measun 29601 | The measure the union of t... |
measvunilem 29602 | Lemma for ~ measvuni . (C... |
measvunilem0 29603 | Lemma for ~ measvuni . (C... |
measvuni 29604 | The measure of a countable... |
measssd 29605 | A measure is monotone with... |
measunl 29606 | A measure is sub-additive ... |
measiuns 29607 | The measure of the union o... |
measiun 29608 | A measure is sub-additive.... |
meascnbl 29609 | A measure is continuous fr... |
measinblem 29610 | Lemma for ~ measinb . (Co... |
measinb 29611 | Building a measure restric... |
measres 29612 | Building a measure restric... |
measinb2 29613 | Building a measure restric... |
measdivcstOLD 29614 | Division of a measure by a... |
measdivcst 29615 | Division of a measure by a... |
cntmeas 29616 | The Counting measure is a ... |
pwcntmeas 29617 | The counting measure is a ... |
cntnevol 29618 | Counting and Lebesgue meas... |
voliune 29619 | The Lebesgue measure funct... |
volfiniune 29620 | The Lebesgue measure funct... |
volmeas 29621 | The Lebesgue measure is a ... |
ddeval1 29624 | Value of the delta measure... |
ddeval0 29625 | Value of the delta measure... |
ddemeas 29626 | The Dirac delta measure is... |
relae 29630 | 'almost everywhere' is a r... |
brae 29631 | 'almost everywhere' relati... |
braew 29632 | 'almost everywhere' relati... |
truae 29633 | A truth holds almost every... |
aean 29634 | A conjunction holds almost... |
faeval 29636 | Value of the 'almost every... |
relfae 29637 | The 'almost everywhere' bu... |
brfae 29638 | 'almost everywhere' relati... |
ismbfm 29641 | The predicate " ` F ` is a... |
elunirnmbfm 29642 | The property of being a me... |
mbfmfun 29643 | A measurable function is a... |
mbfmf 29644 | A measurable function as a... |
isanmbfm 29645 | The predicate to be a meas... |
mbfmcnvima 29646 | The preimage by a measurab... |
mbfmbfm 29647 | A measurable function to a... |
mbfmcst 29648 | A constant function is mea... |
1stmbfm 29649 | The first projection map i... |
2ndmbfm 29650 | The second projection map ... |
imambfm 29651 | If the sigma-algebra in th... |
cnmbfm 29652 | A continuous function is m... |
mbfmco 29653 | The composition of two mea... |
mbfmco2 29654 | The pair building of two m... |
mbfmvolf 29655 | Measurable functions with ... |
elmbfmvol2 29656 | Measurable functions with ... |
mbfmcnt 29657 | All functions are measurab... |
br2base 29658 | The base set for the gener... |
dya2ub 29659 | An upper bound for a dyadi... |
sxbrsigalem0 29660 | The closed half-spaces of ... |
sxbrsigalem3 29661 | The sigma-algebra generate... |
dya2iocival 29662 | The function ` I ` returns... |
dya2iocress 29663 | Dyadic intervals are subse... |
dya2iocbrsiga 29664 | Dyadic intervals are Borel... |
dya2icobrsiga 29665 | Dyadic intervals are Borel... |
dya2icoseg 29666 | For any point and any clos... |
dya2icoseg2 29667 | For any point and any open... |
dya2iocrfn 29668 | The function returning dya... |
dya2iocct 29669 | The dyadic rectangle set i... |
dya2iocnrect 29670 | For any point of an open r... |
dya2iocnei 29671 | For any point of an open s... |
dya2iocuni 29672 | Every open set of ` ( RR X... |
dya2iocucvr 29673 | The dyadic rectangular set... |
sxbrsigalem1 29674 | The Borel algebra on ` ( R... |
sxbrsigalem2 29675 | The sigma-algebra generate... |
sxbrsigalem4 29676 | The Borel algebra on ` ( R... |
sxbrsigalem5 29677 | First direction for ~ sxbr... |
sxbrsigalem6 29678 | First direction for ~ sxbr... |
sxbrsiga 29679 | The product sigma-algebra ... |
omsval 29682 | Value of the function mapp... |
omsfval 29683 | Value of the outer measure... |
omscl 29684 | A closure lemma for the co... |
omsf 29685 | A constructed outer measur... |
oms0 29686 | A constructed outer measur... |
omsmon 29687 | A constructed outer measur... |
omssubaddlem 29688 | For any small margin ` E `... |
omssubadd 29689 | A constructed outer measur... |
carsgval 29692 | Value of the Caratheodory ... |
carsgcl 29693 | Closure of the Caratheodor... |
elcarsg 29694 | Property of being a Catath... |
baselcarsg 29695 | The universe set, ` O ` , ... |
0elcarsg 29696 | The empty set is Caratheod... |
carsguni 29697 | The union of all Caratheod... |
elcarsgss 29698 | Caratheodory measurable se... |
difelcarsg 29699 | The Caratheodory measurabl... |
inelcarsg 29700 | The Caratheodory measurabl... |
unelcarsg 29701 | The Caratheodory-measurabl... |
difelcarsg2 29702 | The Caratheodory-measurabl... |
carsgmon 29703 | Utility lemma: Apply mono... |
carsgsigalem 29704 | Lemma for the following th... |
fiunelcarsg 29705 | The Caratheodory measurabl... |
carsgclctunlem1 29706 | Lemma for ~ carsgclctun . ... |
carsggect 29707 | The outer measure is count... |
carsgclctunlem2 29708 | Lemma for ~ carsgclctun . ... |
carsgclctunlem3 29709 | Lemma for ~ carsgclctun . ... |
carsgclctun 29710 | The Caratheodory measurabl... |
carsgsiga 29711 | The Caratheodory measurabl... |
omsmeas 29712 | The restriction of a const... |
pmeasmono 29713 | This theorem's hypotheses ... |
pmeasadd 29714 | A premeasure on a ring of ... |
itgeq12dv 29715 | Equality theorem for an in... |
sitgval 29721 | Value of the simple functi... |
issibf 29722 | The predicate " ` F ` is a... |
sibf0 29723 | The constant zero function... |
sibfmbl 29724 | A simple function is measu... |
sibff 29725 | A simple function is a fun... |
sibfrn 29726 | A simple function has fini... |
sibfima 29727 | Any preimage of a singleto... |
sibfinima 29728 | The measure of the interse... |
sibfof 29729 | Applying function operatio... |
sitgfval 29730 | Value of the Bochner integ... |
sitgclg 29731 | Closure of the Bochner int... |
sitgclbn 29732 | Closure of the Bochner int... |
sitgclcn 29733 | Closure of the Bochner int... |
sitgclre 29734 | Closure of the Bochner int... |
sitg0 29735 | The integral of the consta... |
sitgf 29736 | The integral for simple fu... |
sitgaddlemb 29737 | Lemma for * sitgadd . (Co... |
sitmval 29738 | Value of the simple functi... |
sitmfval 29739 | Value of the integral dist... |
sitmcl 29740 | Closure of the integral di... |
sitmf 29741 | The integral metric as a f... |
oddpwdc 29743 | Lemma for ~ eulerpart . T... |
oddpwdcv 29744 | Lemma for ~ eulerpart : va... |
eulerpartlemsv1 29745 | Lemma for ~ eulerpart . V... |
eulerpartlemelr 29746 | Lemma for ~ eulerpart . (... |
eulerpartlemsv2 29747 | Lemma for ~ eulerpart . V... |
eulerpartlemsf 29748 | Lemma for ~ eulerpart . (... |
eulerpartlems 29749 | Lemma for ~ eulerpart . (... |
eulerpartlemsv3 29750 | Lemma for ~ eulerpart . V... |
eulerpartlemgc 29751 | Lemma for ~ eulerpart . (... |
eulerpartleme 29752 | Lemma for ~ eulerpart . (... |
eulerpartlemv 29753 | Lemma for ~ eulerpart . (... |
eulerpartlemo 29754 | Lemma for ~ eulerpart : ` ... |
eulerpartlemd 29755 | Lemma for ~ eulerpart : ` ... |
eulerpartlem1 29756 | Lemma for ~ eulerpart . (... |
eulerpartlemb 29757 | Lemma for ~ eulerpart . T... |
eulerpartlemt0 29758 | Lemma for ~ eulerpart . (... |
eulerpartlemf 29759 | Lemma for ~ eulerpart : O... |
eulerpartlemt 29760 | Lemma for ~ eulerpart . (... |
eulerpartgbij 29761 | Lemma for ~ eulerpart : T... |
eulerpartlemgv 29762 | Lemma for ~ eulerpart : va... |
eulerpartlemr 29763 | Lemma for ~ eulerpart . (... |
eulerpartlemmf 29764 | Lemma for ~ eulerpart . (... |
eulerpartlemgvv 29765 | Lemma for ~ eulerpart : va... |
eulerpartlemgu 29766 | Lemma for ~ eulerpart : R... |
eulerpartlemgh 29767 | Lemma for ~ eulerpart : T... |
eulerpartlemgf 29768 | Lemma for ~ eulerpart : I... |
eulerpartlemgs2 29769 | Lemma for ~ eulerpart : T... |
eulerpartlemn 29770 | Lemma for ~ eulerpart . (... |
eulerpart 29771 | Euler's theorem on partiti... |
subiwrd 29774 | Lemma for ~ sseqp1 . (Con... |
subiwrdlen 29775 | Length of a subword of an ... |
iwrdsplit 29776 | Lemma for ~ sseqp1 . (Con... |
sseqval 29777 | Value of the strong sequen... |
sseqfv1 29778 | Value of the strong sequen... |
sseqfn 29779 | A strong recursive sequenc... |
sseqmw 29780 | Lemma for ~ sseqf amd ~ ss... |
sseqf 29781 | A strong recursive sequenc... |
sseqfres 29782 | The first elements in the ... |
sseqfv2 29783 | Value of the strong sequen... |
sseqp1 29784 | Value of the strong sequen... |
fiblem 29787 | Lemma for ~ fib0 , ~ fib1 ... |
fib0 29788 | Value of the Fibonacci seq... |
fib1 29789 | Value of the Fibonacci seq... |
fibp1 29790 | Value of the Fibonacci seq... |
fib2 29791 | Value of the Fibonacci seq... |
fib3 29792 | Value of the Fibonacci seq... |
fib4 29793 | Value of the Fibonacci seq... |
fib5 29794 | Value of the Fibonacci seq... |
fib6 29795 | Value of the Fibonacci seq... |
elprob 29798 | The property of being a pr... |
domprobmeas 29799 | A probability measure is a... |
domprobsiga 29800 | The domain of a probabilit... |
probtot 29801 | The probability of the uni... |
prob01 29802 | A probability is an elemen... |
probnul 29803 | The probability of the emp... |
unveldomd 29804 | The universe is an element... |
unveldom 29805 | The universe is an element... |
nuleldmp 29806 | The empty set is an elemen... |
probcun 29807 | The probability of the uni... |
probun 29808 | The probability of the uni... |
probdif 29809 | The probability of the dif... |
probinc 29810 | A probability law is incre... |
probdsb 29811 | The probability of the com... |
probmeasd 29812 | A probability measure is a... |
probvalrnd 29813 | The value of a probability... |
probtotrnd 29814 | The probability of the uni... |
totprobd 29815 | Law of total probability, ... |
totprob 29816 | Law of total probability. ... |
probfinmeasbOLD 29817 | Build a probability measur... |
probfinmeasb 29818 | Build a probability measur... |
probmeasb 29819 | Build a probability from a... |
cndprobval 29822 | The value of the condition... |
cndprobin 29823 | An identity linking condit... |
cndprob01 29824 | The conditional probabilit... |
cndprobtot 29825 | The conditional probabilit... |
cndprobnul 29826 | The conditional probabilit... |
cndprobprob 29827 | The conditional probabilit... |
bayesth 29828 | Bayes Theorem. (Contribut... |
rrvmbfm 29831 | A real-valued random varia... |
isrrvv 29832 | Elementhood to the set of ... |
rrvvf 29833 | A real-valued random varia... |
rrvfn 29834 | A real-valued random varia... |
rrvdm 29835 | The domain of a random var... |
rrvrnss 29836 | The range of a random vari... |
rrvf2 29837 | A real-valued random varia... |
rrvdmss 29838 | The domain of a random var... |
rrvfinvima 29839 | For a real-value random va... |
0rrv 29840 | The constant function equa... |
rrvadd 29841 | The sum of two random vari... |
rrvmulc 29842 | A random variable multipli... |
rrvsum 29843 | An indexed sum of random v... |
orvcval 29846 | Value of the preimage mapp... |
orvcval2 29847 | Another way to express the... |
elorvc 29848 | Elementhood of a preimage.... |
orvcval4 29849 | The value of the preimage ... |
orvcoel 29850 | If the relation produces o... |
orvccel 29851 | If the relation produces c... |
elorrvc 29852 | Elementhood of a preimage ... |
orrvcval4 29853 | The value of the preimage ... |
orrvcoel 29854 | If the relation produces o... |
orrvccel 29855 | If the relation produces c... |
orvcgteel 29856 | Preimage maps produced by ... |
orvcelval 29857 | Preimage maps produced by ... |
orvcelel 29858 | Preimage maps produced by ... |
dstrvval 29859 | The value of the distribut... |
dstrvprob 29860 | The distribution of a rand... |
orvclteel 29861 | Preimage maps produced by ... |
dstfrvel 29862 | Elementhood of preimage ma... |
dstfrvunirn 29863 | The limit of all preimage ... |
orvclteinc 29864 | Preimage maps produced by ... |
dstfrvinc 29865 | A cumulative distribution ... |
dstfrvclim1 29866 | The limit of the cumulativ... |
coinfliplem 29867 | Division in the extended r... |
coinflipprob 29868 | The ` P ` we defined for c... |
coinflipspace 29869 | The space of our coin-flip... |
coinflipuniv 29870 | The universe of our coin-f... |
coinfliprv 29871 | The ` X ` we defined for c... |
coinflippv 29872 | The probability of heads i... |
coinflippvt 29873 | The probability of tails i... |
ballotlemoex 29874 | ` O ` is a set. (Contribu... |
ballotlem1 29875 | The size of the universe i... |
ballotlemelo 29876 | Elementhood in ` O ` . (C... |
ballotlem2 29877 | The probability that the f... |
ballotlemfval 29878 | The value of F. (Contribut... |
ballotlemfelz 29879 | ` ( F `` C ) ` has values ... |
ballotlemfp1 29880 | If the ` J ` th ballot is ... |
ballotlemfc0 29881 | ` F ` takes value 0 betwee... |
ballotlemfcc 29882 | ` F ` takes value 0 betwee... |
ballotlemfmpn 29883 | ` ( F `` C ) ` finishes co... |
ballotlemfval0 29884 | ` ( F `` C ) ` always star... |
ballotleme 29885 | Elements of ` E ` . (Cont... |
ballotlemodife 29886 | Elements of ` ( O \ E ) ` ... |
ballotlem4 29887 | If the first pick is a vot... |
ballotlem5 29888 | If A is not ahead througho... |
ballotlemi 29889 | Value of ` I ` for a given... |
ballotlemiex 29890 | Properties of ` ( I `` C )... |
ballotlemi1 29891 | The first tie cannot be re... |
ballotlemii 29892 | The first tie cannot be re... |
ballotlemsup 29893 | The set of zeroes of ` F `... |
ballotlemimin 29894 | ` ( I `` C ) ` is the firs... |
ballotlemic 29895 | If the first vote is for B... |
ballotlem1c 29896 | If the first vote is for A... |
ballotlemsval 29897 | Value of ` S ` . (Contrib... |
ballotlemsv 29898 | Value of ` S ` evaluated a... |
ballotlemsgt1 29899 | ` S ` maps values less tha... |
ballotlemsdom 29900 | Domain of ` S ` for a give... |
ballotlemsel1i 29901 | The range ` ( 1 ... ( I ``... |
ballotlemsf1o 29902 | The defined ` S ` is a bij... |
ballotlemsi 29903 | The image by ` S ` of the ... |
ballotlemsima 29904 | The image by ` S ` of an i... |
ballotlemieq 29905 | If two countings share the... |
ballotlemrval 29906 | Value of ` R ` . (Contrib... |
ballotlemscr 29907 | The image of ` ( R `` C ) ... |
ballotlemrv 29908 | Value of ` R ` evaluated a... |
ballotlemrv1 29909 | Value of ` R ` before the ... |
ballotlemrv2 29910 | Value of ` R ` after the t... |
ballotlemro 29911 | Range of ` R ` is included... |
ballotlemgval 29912 | Expand the value of ` .^ `... |
ballotlemgun 29913 | A property of the defined ... |
ballotlemfg 29914 | Express the value of ` ( F... |
ballotlemfrc 29915 | Express the value of ` ( F... |
ballotlemfrci 29916 | Reverse counting preserves... |
ballotlemfrceq 29917 | Value of ` F ` for a rever... |
ballotlemfrcn0 29918 | Value of ` F ` for a rever... |
ballotlemrc 29919 | Range of ` R ` . (Contrib... |
ballotlemirc 29920 | Applying ` R ` does not ch... |
ballotlemrinv0 29921 | Lemma for ~ ballotlemrinv ... |
ballotlemrinv 29922 | ` R ` is its own inverse :... |
ballotlem1ri 29923 | When the vote on the first... |
ballotlem7 29924 | ` R ` is a bijection betwe... |
ballotlem8 29925 | There are as many counting... |
ballotth 29926 | Bertrand's ballot problem ... |
sgncl 29927 | Closure of the signum. (C... |
sgnclre 29928 | Closure of the signum. (C... |
sgnneg 29929 | Negation of the signum. (... |
sgn3da 29930 | A conditional containing a... |
sgnmul 29931 | Signum of a product. (Con... |
sgnmulrp2 29932 | Multiplication by a positi... |
sgnsub 29933 | Subtraction of a number of... |
sgnnbi 29934 | Negative signum. (Contrib... |
sgnpbi 29935 | Positive signum. (Contrib... |
sgn0bi 29936 | Zero signum. (Contributed... |
sgnsgn 29937 | Signum is idempotent. (Co... |
sgnmulsgn 29938 | If two real numbers are of... |
sgnmulsgp 29939 | If two real numbers are of... |
fzssfzo 29940 | Condition for an integer i... |
gsumncl 29941 | Closure of a group sum in ... |
gsumnunsn 29942 | Closure of a group sum in ... |
wrdres 29943 | Condition for the restrict... |
wrdsplex 29944 | Existence of a split of a ... |
ccatmulgnn0dir 29945 | Concatenation of words fol... |
ofcccat 29946 | Letterwise operations on w... |
ofcs1 29947 | Letterwise operations on a... |
ofcs2 29948 | Letterwise operations on a... |
plymul02 29949 | Product of a polynomial wi... |
plymulx0 29950 | Coefficients of a polynomi... |
plymulx 29951 | Coefficients of a polynomi... |
plyrecld 29952 | Closure of a polynomial wi... |
signsplypnf 29953 | The quotient of a polynomi... |
signsply0 29954 | Lemma for the rule of sign... |
signspval 29955 | The value of the skipping ... |
signsw0glem 29956 | Neutral element property o... |
signswbase 29957 | The base of ` W ` is the t... |
signswplusg 29958 | The operation of ` W ` . ... |
signsw0g 29959 | The neutral element of ` W... |
signswmnd 29960 | ` W ` is a monoid structur... |
signswrid 29961 | The zero-skipping operatio... |
signswlid 29962 | The zero-skipping operatio... |
signswn0 29963 | The zero-skipping operatio... |
signswch 29964 | The zero-skipping operatio... |
signslema 29965 | Computational part of sign... |
signstfv 29966 | Value of the zero-skipping... |
signstfval 29967 | Value of the zero-skipping... |
signstcl 29968 | Closure of the zero skippi... |
signstf 29969 | The zero skipping sign wor... |
signstlen 29970 | Length of the zero skippin... |
signstf0 29971 | Sign of a single letter wo... |
signstfvn 29972 | Zero-skipping sign in a wo... |
signsvtn0 29973 | If the last letter is non ... |
signstfvp 29974 | Zero-skipping sign in a wo... |
signstfvneq0 29975 | In case the first letter i... |
signstfvcl 29976 | Closure of the zero skippi... |
signstfvc 29977 | Zero-skipping sign in a wo... |
signstres 29978 | Restriction of a zero skip... |
signstfveq0a 29979 | Lemma for ~ signstfveq0 . ... |
signstfveq0 29980 | In case the last letter is... |
signsvvfval 29981 | The value of ` V ` , which... |
signsvvf 29982 | ` V ` is a function. (Con... |
signsvf0 29983 | There is no change of sign... |
signsvf1 29984 | In a single-letter word, w... |
signsvfn 29985 | Number of changes in a wor... |
signsvtp 29986 | Adding a letter of the sam... |
signsvtn 29987 | Adding a letter of a diffe... |
signsvfpn 29988 | Adding a letter of the sam... |
signsvfnn 29989 | Adding a letter of a diffe... |
signlem0 29990 | Adding a zero as the highe... |
signshf 29991 | ` H ` , corresponding to t... |
signshwrd 29992 | ` H ` , corresponding to t... |
signshlen 29993 | Length of ` H ` , correspo... |
signshnz 29994 | ` H ` is not the empty wor... |
istrkg2d 29997 | Property of fulfilling dim... |
axtglowdim2OLD 29998 | Lower dimension axiom for ... |
axtgupdim2OLD 29999 | Upper dimension axiom for ... |
afsval 30002 | Value of the AFS relation ... |
brafs 30003 | Binary relationship form o... |
tg5segofs 30004 | Rephrase ~ axtg5seg using ... |
bnj170 30017 | ` /\ ` -manipulation. (Co... |
bnj240 30018 | ` /\ ` -manipulation. (Co... |
bnj248 30019 | ` /\ ` -manipulation. (Co... |
bnj250 30020 | ` /\ ` -manipulation. (Co... |
bnj251 30021 | ` /\ ` -manipulation. (Co... |
bnj252 30022 | ` /\ ` -manipulation. (Co... |
bnj253 30023 | ` /\ ` -manipulation. (Co... |
bnj255 30024 | ` /\ ` -manipulation. (Co... |
bnj256 30025 | ` /\ ` -manipulation. (Co... |
bnj257 30026 | ` /\ ` -manipulation. (Co... |
bnj258 30027 | ` /\ ` -manipulation. (Co... |
bnj268 30028 | ` /\ ` -manipulation. (Co... |
bnj290 30029 | ` /\ ` -manipulation. (Co... |
bnj291 30030 | ` /\ ` -manipulation. (Co... |
bnj312 30031 | ` /\ ` -manipulation. (Co... |
bnj334 30032 | ` /\ ` -manipulation. (Co... |
bnj345 30033 | ` /\ ` -manipulation. (Co... |
bnj422 30034 | ` /\ ` -manipulation. (Co... |
bnj432 30035 | ` /\ ` -manipulation. (Co... |
bnj446 30036 | ` /\ ` -manipulation. (Co... |
bnj21 30037 | First-order logic and set ... |
bnj23 30038 | First-order logic and set ... |
bnj31 30039 | First-order logic and set ... |
bnj62 30040 | First-order logic and set ... |
bnj89 30041 | First-order logic and set ... |
bnj90 30042 | First-order logic and set ... |
bnj101 30043 | First-order logic and set ... |
bnj105 30044 | First-order logic and set ... |
bnj115 30045 | First-order logic and set ... |
bnj132 30046 | First-order logic and set ... |
bnj133 30047 | First-order logic and set ... |
bnj142OLD 30048 | First-order logic and set ... |
bnj145OLD 30049 | First-order logic and set ... |
bnj156 30050 | First-order logic and set ... |
bnj158 30051 | First-order logic and set ... |
bnj168 30052 | First-order logic and set ... |
bnj206 30053 | First-order logic and set ... |
bnj216 30054 | First-order logic and set ... |
bnj219 30055 | First-order logic and set ... |
bnj226 30056 | First-order logic and set ... |
bnj228 30057 | First-order logic and set ... |
bnj519 30058 | First-order logic and set ... |
bnj521 30059 | First-order logic and set ... |
bnj524 30060 | First-order logic and set ... |
bnj525 30061 | First-order logic and set ... |
bnj534 30062 | First-order logic and set ... |
bnj538 30063 | First-order logic and set ... |
bnj538OLD 30064 | First-order logic and set ... |
bnj529 30065 | First-order logic and set ... |
bnj551 30066 | First-order logic and set ... |
bnj563 30067 | First-order logic and set ... |
bnj564 30068 | First-order logic and set ... |
bnj593 30069 | First-order logic and set ... |
bnj596 30070 | First-order logic and set ... |
bnj610 30071 | Pass from equality ( ` x =... |
bnj642 30072 | ` /\ ` -manipulation. (Co... |
bnj643 30073 | ` /\ ` -manipulation. (Co... |
bnj645 30074 | ` /\ ` -manipulation. (Co... |
bnj658 30075 | ` /\ ` -manipulation. (Co... |
bnj667 30076 | ` /\ ` -manipulation. (Co... |
bnj705 30077 | ` /\ ` -manipulation. (Co... |
bnj706 30078 | ` /\ ` -manipulation. (Co... |
bnj707 30079 | ` /\ ` -manipulation. (Co... |
bnj708 30080 | ` /\ ` -manipulation. (Co... |
bnj721 30081 | ` /\ ` -manipulation. (Co... |
bnj832 30082 | ` /\ ` -manipulation. (Co... |
bnj835 30083 | ` /\ ` -manipulation. (Co... |
bnj836 30084 | ` /\ ` -manipulation. (Co... |
bnj837 30085 | ` /\ ` -manipulation. (Co... |
bnj769 30086 | ` /\ ` -manipulation. (Co... |
bnj770 30087 | ` /\ ` -manipulation. (Co... |
bnj771 30088 | ` /\ ` -manipulation. (Co... |
bnj887 30089 | ` /\ ` -manipulation. (Co... |
bnj918 30090 | First-order logic and set ... |
bnj919 30091 | First-order logic and set ... |
bnj923 30092 | First-order logic and set ... |
bnj927 30093 | First-order logic and set ... |
bnj930 30094 | First-order logic and set ... |
bnj931 30095 | First-order logic and set ... |
bnj937 30096 | First-order logic and set ... |
bnj941 30097 | First-order logic and set ... |
bnj945 30098 | Technical lemma for ~ bnj6... |
bnj946 30099 | First-order logic and set ... |
bnj951 30100 | ` /\ ` -manipulation. (Co... |
bnj956 30101 | First-order logic and set ... |
bnj976 30102 | First-order logic and set ... |
bnj982 30103 | First-order logic and set ... |
bnj1019 30104 | First-order logic and set ... |
bnj1023 30105 | First-order logic and set ... |
bnj1095 30106 | First-order logic and set ... |
bnj1096 30107 | First-order logic and set ... |
bnj1098 30108 | First-order logic and set ... |
bnj1101 30109 | First-order logic and set ... |
bnj1113 30110 | First-order logic and set ... |
bnj1109 30111 | First-order logic and set ... |
bnj1131 30112 | First-order logic and set ... |
bnj1138 30113 | First-order logic and set ... |
bnj1142 30114 | First-order logic and set ... |
bnj1143 30115 | First-order logic and set ... |
bnj1146 30116 | First-order logic and set ... |
bnj1149 30117 | First-order logic and set ... |
bnj1185 30118 | First-order logic and set ... |
bnj1196 30119 | First-order logic and set ... |
bnj1198 30120 | First-order logic and set ... |
bnj1209 30121 | First-order logic and set ... |
bnj1211 30122 | First-order logic and set ... |
bnj1213 30123 | First-order logic and set ... |
bnj1212 30124 | First-order logic and set ... |
bnj1219 30125 | First-order logic and set ... |
bnj1224 30126 | First-order logic and set ... |
bnj1230 30127 | First-order logic and set ... |
bnj1232 30128 | First-order logic and set ... |
bnj1235 30129 | First-order logic and set ... |
bnj1239 30130 | First-order logic and set ... |
bnj1238 30131 | First-order logic and set ... |
bnj1241 30132 | First-order logic and set ... |
bnj1247 30133 | First-order logic and set ... |
bnj1254 30134 | First-order logic and set ... |
bnj1262 30135 | First-order logic and set ... |
bnj1266 30136 | First-order logic and set ... |
bnj1265 30137 | First-order logic and set ... |
bnj1275 30138 | First-order logic and set ... |
bnj1276 30139 | First-order logic and set ... |
bnj1292 30140 | First-order logic and set ... |
bnj1293 30141 | First-order logic and set ... |
bnj1294 30142 | First-order logic and set ... |
bnj1299 30143 | First-order logic and set ... |
bnj1304 30144 | First-order logic and set ... |
bnj1316 30145 | First-order logic and set ... |
bnj1317 30146 | First-order logic and set ... |
bnj1322 30147 | First-order logic and set ... |
bnj1340 30148 | First-order logic and set ... |
bnj1345 30149 | First-order logic and set ... |
bnj1350 30150 | First-order logic and set ... |
bnj1351 30151 | First-order logic and set ... |
bnj1352 30152 | First-order logic and set ... |
bnj1361 30153 | First-order logic and set ... |
bnj1366 30154 | First-order logic and set ... |
bnj1379 30155 | First-order logic and set ... |
bnj1383 30156 | First-order logic and set ... |
bnj1385 30157 | First-order logic and set ... |
bnj1386 30158 | First-order logic and set ... |
bnj1397 30159 | First-order logic and set ... |
bnj1400 30160 | First-order logic and set ... |
bnj1405 30161 | First-order logic and set ... |
bnj1422 30162 | First-order logic and set ... |
bnj1424 30163 | First-order logic and set ... |
bnj1436 30164 | First-order logic and set ... |
bnj1441 30165 | First-order logic and set ... |
bnj1454 30166 | First-order logic and set ... |
bnj1459 30167 | First-order logic and set ... |
bnj1464 30168 | Conversion of implicit sub... |
bnj1465 30169 | First-order logic and set ... |
bnj1468 30170 | Conversion of implicit sub... |
bnj1476 30171 | First-order logic and set ... |
bnj1502 30172 | First-order logic and set ... |
bnj1503 30173 | First-order logic and set ... |
bnj1517 30174 | First-order logic and set ... |
bnj1521 30175 | First-order logic and set ... |
bnj1533 30176 | First-order logic and set ... |
bnj1534 30177 | First-order logic and set ... |
bnj1536 30178 | First-order logic and set ... |
bnj1538 30179 | First-order logic and set ... |
bnj1541 30180 | First-order logic and set ... |
bnj1542 30181 | First-order logic and set ... |
bnj110 30182 | Well-founded induction res... |
bnj157 30183 | Well-founded induction res... |
bnj66 30184 | Technical lemma for ~ bnj6... |
bnj91 30185 | First-order logic and set ... |
bnj92 30186 | First-order logic and set ... |
bnj93 30187 | Technical lemma for ~ bnj9... |
bnj95 30188 | Technical lemma for ~ bnj1... |
bnj96 30189 | Technical lemma for ~ bnj1... |
bnj97 30190 | Technical lemma for ~ bnj1... |
bnj98 30191 | Technical lemma for ~ bnj1... |
bnj106 30192 | First-order logic and set ... |
bnj118 30193 | First-order logic and set ... |
bnj121 30194 | First-order logic and set ... |
bnj124 30195 | Technical lemma for ~ bnj1... |
bnj125 30196 | Technical lemma for ~ bnj1... |
bnj126 30197 | Technical lemma for ~ bnj1... |
bnj130 30198 | Technical lemma for ~ bnj1... |
bnj149 30199 | Technical lemma for ~ bnj1... |
bnj150 30200 | Technical lemma for ~ bnj1... |
bnj151 30201 | Technical lemma for ~ bnj1... |
bnj154 30202 | Technical lemma for ~ bnj1... |
bnj155 30203 | Technical lemma for ~ bnj1... |
bnj153 30204 | Technical lemma for ~ bnj8... |
bnj207 30205 | Technical lemma for ~ bnj8... |
bnj213 30206 | First-order logic and set ... |
bnj222 30207 | Technical lemma for ~ bnj2... |
bnj229 30208 | Technical lemma for ~ bnj5... |
bnj517 30209 | Technical lemma for ~ bnj5... |
bnj518 30210 | Technical lemma for ~ bnj8... |
bnj523 30211 | Technical lemma for ~ bnj8... |
bnj526 30212 | Technical lemma for ~ bnj8... |
bnj528 30213 | Technical lemma for ~ bnj8... |
bnj535 30214 | Technical lemma for ~ bnj8... |
bnj539 30215 | Technical lemma for ~ bnj8... |
bnj540 30216 | Technical lemma for ~ bnj8... |
bnj543 30217 | Technical lemma for ~ bnj8... |
bnj544 30218 | Technical lemma for ~ bnj8... |
bnj545 30219 | Technical lemma for ~ bnj8... |
bnj546 30220 | Technical lemma for ~ bnj8... |
bnj548 30221 | Technical lemma for ~ bnj8... |
bnj553 30222 | Technical lemma for ~ bnj8... |
bnj554 30223 | Technical lemma for ~ bnj8... |
bnj556 30224 | Technical lemma for ~ bnj8... |
bnj557 30225 | Technical lemma for ~ bnj8... |
bnj558 30226 | Technical lemma for ~ bnj8... |
bnj561 30227 | Technical lemma for ~ bnj8... |
bnj562 30228 | Technical lemma for ~ bnj8... |
bnj570 30229 | Technical lemma for ~ bnj8... |
bnj571 30230 | Technical lemma for ~ bnj8... |
bnj605 30231 | Technical lemma. This lem... |
bnj581 30232 | Technical lemma for ~ bnj5... |
bnj589 30233 | Technical lemma for ~ bnj8... |
bnj590 30234 | Technical lemma for ~ bnj8... |
bnj591 30235 | Technical lemma for ~ bnj8... |
bnj594 30236 | Technical lemma for ~ bnj8... |
bnj580 30237 | Technical lemma for ~ bnj5... |
bnj579 30238 | Technical lemma for ~ bnj8... |
bnj602 30239 | Equality theorem for the `... |
bnj607 30240 | Technical lemma for ~ bnj8... |
bnj609 30241 | Technical lemma for ~ bnj8... |
bnj611 30242 | Technical lemma for ~ bnj8... |
bnj600 30243 | Technical lemma for ~ bnj8... |
bnj601 30244 | Technical lemma for ~ bnj8... |
bnj852 30245 | Technical lemma for ~ bnj6... |
bnj864 30246 | Technical lemma for ~ bnj6... |
bnj865 30247 | Technical lemma for ~ bnj6... |
bnj873 30248 | Technical lemma for ~ bnj6... |
bnj849 30249 | Technical lemma for ~ bnj6... |
bnj882 30250 | Definition (using hypothes... |
bnj18eq1 30251 | Equality theorem for trans... |
bnj893 30252 | Property of ` _trCl ` . U... |
bnj900 30253 | Technical lemma for ~ bnj6... |
bnj906 30254 | Property of ` _trCl ` . (... |
bnj908 30255 | Technical lemma for ~ bnj6... |
bnj911 30256 | Technical lemma for ~ bnj6... |
bnj916 30257 | Technical lemma for ~ bnj6... |
bnj917 30258 | Technical lemma for ~ bnj6... |
bnj934 30259 | Technical lemma for ~ bnj6... |
bnj929 30260 | Technical lemma for ~ bnj6... |
bnj938 30261 | Technical lemma for ~ bnj6... |
bnj944 30262 | Technical lemma for ~ bnj6... |
bnj953 30263 | Technical lemma for ~ bnj6... |
bnj958 30264 | Technical lemma for ~ bnj6... |
bnj1000 30265 | Technical lemma for ~ bnj8... |
bnj965 30266 | Technical lemma for ~ bnj8... |
bnj964 30267 | Technical lemma for ~ bnj6... |
bnj966 30268 | Technical lemma for ~ bnj6... |
bnj967 30269 | Technical lemma for ~ bnj6... |
bnj969 30270 | Technical lemma for ~ bnj6... |
bnj970 30271 | Technical lemma for ~ bnj6... |
bnj910 30272 | Technical lemma for ~ bnj6... |
bnj978 30273 | Technical lemma for ~ bnj6... |
bnj981 30274 | Technical lemma for ~ bnj6... |
bnj983 30275 | Technical lemma for ~ bnj6... |
bnj984 30276 | Technical lemma for ~ bnj6... |
bnj985 30277 | Technical lemma for ~ bnj6... |
bnj986 30278 | Technical lemma for ~ bnj6... |
bnj996 30279 | Technical lemma for ~ bnj6... |
bnj998 30280 | Technical lemma for ~ bnj6... |
bnj999 30281 | Technical lemma for ~ bnj6... |
bnj1001 30282 | Technical lemma for ~ bnj6... |
bnj1006 30283 | Technical lemma for ~ bnj6... |
bnj1014 30284 | Technical lemma for ~ bnj6... |
bnj1015 30285 | Technical lemma for ~ bnj6... |
bnj1018 30286 | Technical lemma for ~ bnj6... |
bnj1020 30287 | Technical lemma for ~ bnj6... |
bnj1021 30288 | Technical lemma for ~ bnj6... |
bnj907 30289 | Technical lemma for ~ bnj6... |
bnj1029 30290 | Property of ` _trCl ` . (... |
bnj1033 30291 | Technical lemma for ~ bnj6... |
bnj1034 30292 | Technical lemma for ~ bnj6... |
bnj1039 30293 | Technical lemma for ~ bnj6... |
bnj1040 30294 | Technical lemma for ~ bnj6... |
bnj1047 30295 | Technical lemma for ~ bnj6... |
bnj1049 30296 | Technical lemma for ~ bnj6... |
bnj1052 30297 | Technical lemma for ~ bnj6... |
bnj1053 30298 | Technical lemma for ~ bnj6... |
bnj1071 30299 | Technical lemma for ~ bnj6... |
bnj1083 30300 | Technical lemma for ~ bnj6... |
bnj1090 30301 | Technical lemma for ~ bnj6... |
bnj1093 30302 | Technical lemma for ~ bnj6... |
bnj1097 30303 | Technical lemma for ~ bnj6... |
bnj1110 30304 | Technical lemma for ~ bnj6... |
bnj1112 30305 | Technical lemma for ~ bnj6... |
bnj1118 30306 | Technical lemma for ~ bnj6... |
bnj1121 30307 | Technical lemma for ~ bnj6... |
bnj1123 30308 | Technical lemma for ~ bnj6... |
bnj1030 30309 | Technical lemma for ~ bnj6... |
bnj1124 30310 | Property of ` _trCl ` . (... |
bnj1133 30311 | Technical lemma for ~ bnj6... |
bnj1128 30312 | Technical lemma for ~ bnj6... |
bnj1127 30313 | Property of ` _trCl ` . (... |
bnj1125 30314 | Property of ` _trCl ` . (... |
bnj1145 30315 | Technical lemma for ~ bnj6... |
bnj1147 30316 | Property of ` _trCl ` . (... |
bnj1137 30317 | Property of ` _trCl ` . (... |
bnj1148 30318 | Property of ` _pred ` . (... |
bnj1136 30319 | Technical lemma for ~ bnj6... |
bnj1152 30320 | Technical lemma for ~ bnj6... |
bnj1154 30321 | Property of ` Fr ` . (Con... |
bnj1171 30322 | Technical lemma for ~ bnj6... |
bnj1172 30323 | Technical lemma for ~ bnj6... |
bnj1173 30324 | Technical lemma for ~ bnj6... |
bnj1174 30325 | Technical lemma for ~ bnj6... |
bnj1175 30326 | Technical lemma for ~ bnj6... |
bnj1176 30327 | Technical lemma for ~ bnj6... |
bnj1177 30328 | Technical lemma for ~ bnj6... |
bnj1186 30329 | Technical lemma for ~ bnj6... |
bnj1190 30330 | Technical lemma for ~ bnj6... |
bnj1189 30331 | Technical lemma for ~ bnj6... |
bnj69 30332 | Existence of a minimal ele... |
bnj1228 30333 | Existence of a minimal ele... |
bnj1204 30334 | Well-founded induction. T... |
bnj1234 30335 | Technical lemma for ~ bnj6... |
bnj1245 30336 | Technical lemma for ~ bnj6... |
bnj1256 30337 | Technical lemma for ~ bnj6... |
bnj1259 30338 | Technical lemma for ~ bnj6... |
bnj1253 30339 | Technical lemma for ~ bnj6... |
bnj1279 30340 | Technical lemma for ~ bnj6... |
bnj1286 30341 | Technical lemma for ~ bnj6... |
bnj1280 30342 | Technical lemma for ~ bnj6... |
bnj1296 30343 | Technical lemma for ~ bnj6... |
bnj1309 30344 | Technical lemma for ~ bnj6... |
bnj1307 30345 | Technical lemma for ~ bnj6... |
bnj1311 30346 | Technical lemma for ~ bnj6... |
bnj1318 30347 | Technical lemma for ~ bnj6... |
bnj1326 30348 | Technical lemma for ~ bnj6... |
bnj1321 30349 | Technical lemma for ~ bnj6... |
bnj1364 30350 | Property of ` _FrSe ` . (... |
bnj1371 30351 | Technical lemma for ~ bnj6... |
bnj1373 30352 | Technical lemma for ~ bnj6... |
bnj1374 30353 | Technical lemma for ~ bnj6... |
bnj1384 30354 | Technical lemma for ~ bnj6... |
bnj1388 30355 | Technical lemma for ~ bnj6... |
bnj1398 30356 | Technical lemma for ~ bnj6... |
bnj1413 30357 | Property of ` _trCl ` . (... |
bnj1408 30358 | Technical lemma for ~ bnj1... |
bnj1414 30359 | Property of ` _trCl ` . (... |
bnj1415 30360 | Technical lemma for ~ bnj6... |
bnj1416 30361 | Technical lemma for ~ bnj6... |
bnj1418 30362 | Property of ` _pred ` . (... |
bnj1417 30363 | Technical lemma for ~ bnj6... |
bnj1421 30364 | Technical lemma for ~ bnj6... |
bnj1444 30365 | Technical lemma for ~ bnj6... |
bnj1445 30366 | Technical lemma for ~ bnj6... |
bnj1446 30367 | Technical lemma for ~ bnj6... |
bnj1447 30368 | Technical lemma for ~ bnj6... |
bnj1448 30369 | Technical lemma for ~ bnj6... |
bnj1449 30370 | Technical lemma for ~ bnj6... |
bnj1442 30371 | Technical lemma for ~ bnj6... |
bnj1450 30372 | Technical lemma for ~ bnj6... |
bnj1423 30373 | Technical lemma for ~ bnj6... |
bnj1452 30374 | Technical lemma for ~ bnj6... |
bnj1466 30375 | Technical lemma for ~ bnj6... |
bnj1467 30376 | Technical lemma for ~ bnj6... |
bnj1463 30377 | Technical lemma for ~ bnj6... |
bnj1489 30378 | Technical lemma for ~ bnj6... |
bnj1491 30379 | Technical lemma for ~ bnj6... |
bnj1312 30380 | Technical lemma for ~ bnj6... |
bnj1493 30381 | Technical lemma for ~ bnj6... |
bnj1497 30382 | Technical lemma for ~ bnj6... |
bnj1498 30383 | Technical lemma for ~ bnj6... |
bnj60 30384 | Well-founded recursion, pa... |
bnj1514 30385 | Technical lemma for ~ bnj1... |
bnj1518 30386 | Technical lemma for ~ bnj1... |
bnj1519 30387 | Technical lemma for ~ bnj1... |
bnj1520 30388 | Technical lemma for ~ bnj1... |
bnj1501 30389 | Technical lemma for ~ bnj1... |
bnj1500 30390 | Well-founded recursion, pa... |
bnj1525 30391 | Technical lemma for ~ bnj1... |
bnj1529 30392 | Technical lemma for ~ bnj1... |
bnj1523 30393 | Technical lemma for ~ bnj1... |
bnj1522 30394 | Well-founded recursion, pa... |
quartfull 30401 | The quartic equation, writ... |
deranglem 30402 | Lemma for derangements. (... |
derangval 30403 | Define the derangement fun... |
derangf 30404 | The derangement number is ... |
derang0 30405 | The derangement number of ... |
derangsn 30406 | The derangement number of ... |
derangenlem 30407 | One half of ~ derangen . ... |
derangen 30408 | The derangement number is ... |
subfacval 30409 | The subfactorial is define... |
derangen2 30410 | Write the derangement numb... |
subfacf 30411 | The subfactorial is a func... |
subfaclefac 30412 | The subfactorial is less t... |
subfac0 30413 | The subfactorial at zero. ... |
subfac1 30414 | The subfactorial at one. ... |
subfacp1lem1 30415 | Lemma for ~ subfacp1 . Th... |
subfacp1lem2a 30416 | Lemma for ~ subfacp1 . Pr... |
subfacp1lem2b 30417 | Lemma for ~ subfacp1 . Pr... |
subfacp1lem3 30418 | Lemma for ~ subfacp1 . In... |
subfacp1lem4 30419 | Lemma for ~ subfacp1 . Th... |
subfacp1lem5 30420 | Lemma for ~ subfacp1 . In... |
subfacp1lem6 30421 | Lemma for ~ subfacp1 . By... |
subfacp1 30422 | A two-term recurrence for ... |
subfacval2 30423 | A closed-form expression f... |
subfaclim 30424 | The subfactorial converges... |
subfacval3 30425 | Another closed form expres... |
derangfmla 30426 | The derangements formula, ... |
erdszelem1 30427 | Lemma for ~ erdsze . (Con... |
erdszelem2 30428 | Lemma for ~ erdsze . (Con... |
erdszelem3 30429 | Lemma for ~ erdsze . (Con... |
erdszelem4 30430 | Lemma for ~ erdsze . (Con... |
erdszelem5 30431 | Lemma for ~ erdsze . (Con... |
erdszelem6 30432 | Lemma for ~ erdsze . (Con... |
erdszelem7 30433 | Lemma for ~ erdsze . (Con... |
erdszelem8 30434 | Lemma for ~ erdsze . (Con... |
erdszelem9 30435 | Lemma for ~ erdsze . (Con... |
erdszelem10 30436 | Lemma for ~ erdsze . (Con... |
erdszelem11 30437 | Lemma for ~ erdsze . (Con... |
erdsze 30438 | The Erdős-Szekeres th... |
erdsze2lem1 30439 | Lemma for ~ erdsze2 . (Co... |
erdsze2lem2 30440 | Lemma for ~ erdsze2 . (Co... |
erdsze2 30441 | Generalize the statement o... |
kur14lem1 30442 | Lemma for ~ kur14 . (Cont... |
kur14lem2 30443 | Lemma for ~ kur14 . Write... |
kur14lem3 30444 | Lemma for ~ kur14 . A clo... |
kur14lem4 30445 | Lemma for ~ kur14 . Compl... |
kur14lem5 30446 | Lemma for ~ kur14 . Closu... |
kur14lem6 30447 | Lemma for ~ kur14 . If ` ... |
kur14lem7 30448 | Lemma for ~ kur14 : main p... |
kur14lem8 30449 | Lemma for ~ kur14 . Show ... |
kur14lem9 30450 | Lemma for ~ kur14 . Since... |
kur14lem10 30451 | Lemma for ~ kur14 . Disch... |
kur14 30452 | Kuratowski's closure-compl... |
ispcon 30459 | The property of being a pa... |
pconcn 30460 | The property of being a pa... |
pcontop 30461 | A simply connected space i... |
isscon 30462 | The property of being a si... |
sconpcon 30463 | A simply connected space i... |
scontop 30464 | A simply connected space i... |
sconpht 30465 | A closed path in a simply ... |
cnpcon 30466 | An image of a path-connect... |
pconcon 30467 | A path-connected space is ... |
txpcon 30468 | The topological product of... |
ptpcon 30469 | The topological product of... |
indispcon 30470 | The indiscrete topology (o... |
conpcon 30471 | A connected and locally pa... |
qtoppcon 30472 | A quotient of a path-conne... |
pconpi1 30473 | All fundamental groups in ... |
sconpht2 30474 | Any two paths in a simply ... |
sconpi1 30475 | A path-connected topologic... |
txsconlem 30476 | Lemma for ~ txscon . (Con... |
txscon 30477 | The topological product of... |
cvxpcon 30478 | A convex subset of the com... |
cvxscon 30479 | A convex subset of the com... |
blscon 30480 | An open ball in the comple... |
cnllyscon 30481 | The topology of the comple... |
rescon 30482 | A subset of ` RR ` is simp... |
iooscon 30483 | An open interval is simply... |
iccscon 30484 | A closed interval is simpl... |
retopscon 30485 | The real numbers are simpl... |
iccllyscon 30486 | A closed interval is local... |
rellyscon 30487 | The real numbers are local... |
iiscon 30488 | The unit interval is simpl... |
iillyscon 30489 | The unit interval is local... |
iinllycon 30490 | The unit interval is local... |
fncvm 30493 | Lemma for covering maps. ... |
cvmscbv 30494 | Change bound variables in ... |
iscvm 30495 | The property of being a co... |
cvmtop1 30496 | Reverse closure for a cove... |
cvmtop2 30497 | Reverse closure for a cove... |
cvmcn 30498 | A covering map is a contin... |
cvmcov 30499 | Property of a covering map... |
cvmsrcl 30500 | Reverse closure for an eve... |
cvmsi 30501 | One direction of ~ cvmsval... |
cvmsval 30502 | Elementhood in the set ` S... |
cvmsss 30503 | An even covering is a subs... |
cvmsn0 30504 | An even covering is nonemp... |
cvmsuni 30505 | An even covering of ` U ` ... |
cvmsdisj 30506 | An even covering of ` U ` ... |
cvmshmeo 30507 | Every element of an even c... |
cvmsf1o 30508 | ` F ` , localized to an el... |
cvmscld 30509 | The sets of an even coveri... |
cvmsss2 30510 | An open subset of an evenl... |
cvmcov2 30511 | The covering map property ... |
cvmseu 30512 | Every element in ` U. T ` ... |
cvmsiota 30513 | Identify the unique elemen... |
cvmopnlem 30514 | Lemma for ~ cvmopn . (Con... |
cvmfolem 30515 | Lemma for ~ cvmfo . (Cont... |
cvmopn 30516 | A covering map is an open ... |
cvmliftmolem1 30517 | Lemma for ~ cvmliftmo . (... |
cvmliftmolem2 30518 | Lemma for ~ cvmliftmo . (... |
cvmliftmoi 30519 | A lift of a continuous fun... |
cvmliftmo 30520 | A lift of a continuous fun... |
cvmliftlem1 30521 | Lemma for ~ cvmlift . In ... |
cvmliftlem2 30522 | Lemma for ~ cvmlift . ` W ... |
cvmliftlem3 30523 | Lemma for ~ cvmlift . Sin... |
cvmliftlem4 30524 | Lemma for ~ cvmlift . The... |
cvmliftlem5 30525 | Lemma for ~ cvmlift . Def... |
cvmliftlem6 30526 | Lemma for ~ cvmlift . Ind... |
cvmliftlem7 30527 | Lemma for ~ cvmlift . Pro... |
cvmliftlem8 30528 | Lemma for ~ cvmlift . The... |
cvmliftlem9 30529 | Lemma for ~ cvmlift . The... |
cvmliftlem10 30530 | Lemma for ~ cvmlift . The... |
cvmliftlem11 30531 | Lemma for ~ cvmlift . (Co... |
cvmliftlem13 30532 | Lemma for ~ cvmlift . The... |
cvmliftlem14 30533 | Lemma for ~ cvmlift . Put... |
cvmliftlem15 30534 | Lemma for ~ cvmlift . Dis... |
cvmlift 30535 | One of the important prope... |
cvmfo 30536 | A covering map is an onto ... |
cvmliftiota 30537 | Write out a function ` H `... |
cvmlift2lem1 30538 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem9a 30539 | Lemma for ~ cvmlift2 and ~... |
cvmlift2lem2 30540 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem3 30541 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem4 30542 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem5 30543 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem6 30544 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem7 30545 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem8 30546 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem9 30547 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem10 30548 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem11 30549 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem12 30550 | Lemma for ~ cvmlift2 . (C... |
cvmlift2lem13 30551 | Lemma for ~ cvmlift2 . (C... |
cvmlift2 30552 | A two-dimensional version ... |
cvmliftphtlem 30553 | Lemma for ~ cvmliftpht . ... |
cvmliftpht 30554 | If ` G ` and ` H ` are pat... |
cvmlift3lem1 30555 | Lemma for ~ cvmlift3 . (C... |
cvmlift3lem2 30556 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem3 30557 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem4 30558 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem5 30559 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem6 30560 | Lemma for ~ cvmlift3 . (C... |
cvmlift3lem7 30561 | Lemma for ~ cvmlift3 . (C... |
cvmlift3lem8 30562 | Lemma for ~ cvmlift2 . (C... |
cvmlift3lem9 30563 | Lemma for ~ cvmlift2 . (C... |
cvmlift3 30564 | A general version of ~ cvm... |
snmlff 30565 | The function ` F ` from ~ ... |
snmlfval 30566 | The function ` F ` from ~ ... |
snmlval 30567 | The property " ` A ` is si... |
snmlflim 30568 | If ` A ` is simply normal,... |
mvtval 30651 | The set of variable typeco... |
mrexval 30652 | The set of "raw expression... |
mexval 30653 | The set of expressions, wh... |
mexval2 30654 | The set of expressions, wh... |
mdvval 30655 | The set of disjoint variab... |
mvrsval 30656 | The set of variables in an... |
mvrsfpw 30657 | The set of variables in an... |
mrsubffval 30658 | The substitution of some v... |
mrsubfval 30659 | The substitution of some v... |
mrsubval 30660 | The substitution of some v... |
mrsubcv 30661 | The value of a substituted... |
mrsubvr 30662 | The value of a substituted... |
mrsubff 30663 | A substitution is a functi... |
mrsubrn 30664 | Although it is defined for... |
mrsubff1 30665 | When restricted to complet... |
mrsubff1o 30666 | When restricted to complet... |
mrsub0 30667 | The value of the substitut... |
mrsubf 30668 | A substitution is a functi... |
mrsubccat 30669 | Substitution distributes o... |
mrsubcn 30670 | A substitution does not ch... |
elmrsubrn 30671 | Characterization of the su... |
mrsubco 30672 | The composition of two sub... |
mrsubvrs 30673 | The set of variables in a ... |
msubffval 30674 | A substitution applied to ... |
msubfval 30675 | A substitution applied to ... |
msubval 30676 | A substitution applied to ... |
msubrsub 30677 | A substitution applied to ... |
msubty 30678 | The type of a substituted ... |
elmsubrn 30679 | Characterization of substi... |
msubrn 30680 | Although it is defined for... |
msubff 30681 | A substitution is a functi... |
msubco 30682 | The composition of two sub... |
msubf 30683 | A substitution is a functi... |
mvhfval 30684 | Value of the function mapp... |
mvhval 30685 | Value of the function mapp... |
mpstval 30686 | A pre-statement is an orde... |
elmpst 30687 | Property of being a pre-st... |
msrfval 30688 | Value of the reduct of a p... |
msrval 30689 | Value of the reduct of a p... |
mpstssv 30690 | A pre-statement is an orde... |
mpst123 30691 | Decompose a pre-statement ... |
mpstrcl 30692 | The elements of a pre-stat... |
msrf 30693 | The reduct of a pre-statem... |
msrrcl 30694 | If ` X ` and ` Y ` have th... |
mstaval 30695 | Value of the set of statem... |
msrid 30696 | The reduct of a statement ... |
msrfo 30697 | The reduct of a pre-statem... |
mstapst 30698 | A statement is a pre-state... |
elmsta 30699 | Property of being a statem... |
ismfs 30700 | A formal system is a tuple... |
mfsdisj 30701 | The constants and variable... |
mtyf2 30702 | The type function maps var... |
mtyf 30703 | The type function maps var... |
mvtss 30704 | The set of variable typeco... |
maxsta 30705 | An axiom is a statement. ... |
mvtinf 30706 | Each variable typecode has... |
msubff1 30707 | When restricted to complet... |
msubff1o 30708 | When restricted to complet... |
mvhf 30709 | The function mapping varia... |
mvhf1 30710 | The function mapping varia... |
msubvrs 30711 | The set of variables in a ... |
mclsrcl 30712 | Reverse closure for the cl... |
mclsssvlem 30713 | Lemma for ~ mclsssv . (Co... |
mclsval 30714 | The function mapping varia... |
mclsssv 30715 | The closure of a set of ex... |
ssmclslem 30716 | Lemma for ~ ssmcls . (Con... |
vhmcls 30717 | All variable hypotheses ar... |
ssmcls 30718 | The original expressions a... |
ss2mcls 30719 | The closure is monotonic u... |
mclsax 30720 | The closure is closed unde... |
mclsind 30721 | Induction theorem for clos... |
mppspstlem 30722 | Lemma for ~ mppspst . (Co... |
mppsval 30723 | Definition of a provable p... |
elmpps 30724 | Definition of a provable p... |
mppspst 30725 | A provable pre-statement i... |
mthmval 30726 | A theorem is a pre-stateme... |
elmthm 30727 | A theorem is a pre-stateme... |
mthmi 30728 | A statement whose reduct i... |
mthmsta 30729 | A theorem is a pre-stateme... |
mppsthm 30730 | A provable pre-statement i... |
mthmblem 30731 | Lemma for ~ mthmb . (Cont... |
mthmb 30732 | If two statements have the... |
mthmpps 30733 | Given a theorem, there is ... |
mclsppslem 30734 | The closure is closed unde... |
mclspps 30735 | The closure is closed unde... |
problem1 30812 | Practice problem 1. Clues... |
problem2 30813 | Practice problem 2. Clues... |
problem2OLD 30814 | Practice problem 2. Clues... |
problem3 30815 | Practice problem 3. Clues... |
problem4 30816 | Practice problem 4. Clues... |
problem5 30817 | Practice problem 5. Clues... |
quad3 30818 | Variant of quadratic equat... |
climuzcnv 30819 | Utility lemma to convert b... |
sinccvglem 30820 | ` ( ( sin `` x ) / x ) ~~>... |
sinccvg 30821 | ` ( ( sin `` x ) / x ) ~~>... |
circum 30822 | The circumference of a cir... |
elfzm12 30823 | Membership in a curtailed ... |
nn0seqcvg 30824 | A strictly-decreasing nonn... |
lediv2aALT 30825 | Division of both sides of ... |
abs2sqlei 30826 | The absolute values of two... |
abs2sqlti 30827 | The absolute values of two... |
abs2sqle 30828 | The absolute values of two... |
abs2sqlt 30829 | The absolute values of two... |
abs2difi 30830 | Difference of absolute val... |
abs2difabsi 30831 | Absolute value of differen... |
axextprim 30832 | ~ ax-ext without distinct ... |
axrepprim 30833 | ~ ax-rep without distinct ... |
axunprim 30834 | ~ ax-un without distinct v... |
axpowprim 30835 | ~ ax-pow without distinct ... |
axregprim 30836 | ~ ax-reg without distinct ... |
axinfprim 30837 | ~ ax-inf without distinct ... |
axacprim 30838 | ~ ax-ac without distinct v... |
untelirr 30839 | We call a class "untanged"... |
untuni 30840 | The union of a class is un... |
untsucf 30841 | If a class is untangled, t... |
unt0 30842 | The null set is untangled.... |
untint 30843 | If there is an untangled e... |
efrunt 30844 | If ` A ` is well-founded b... |
untangtr 30845 | A transitive class is unta... |
3orel1 30846 | Partial elimination of a t... |
3orel2 30847 | Partial elimination of a t... |
3orel3 30848 | Partial elimination of a t... |
3pm3.2ni 30849 | Triple negated disjunction... |
3jaodd 30850 | Double deduction form of ~... |
3orit 30851 | Closed form of ~ 3ori . (... |
biimpexp 30852 | A biconditional in the ant... |
3orel13 30853 | Elimination of two disjunc... |
nepss 30854 | Two classes are inequal if... |
3ccased 30855 | Triple disjunction form of... |
dfso3 30856 | Expansion of the definitio... |
brtpid1 30857 | A binary relationship invo... |
brtpid2 30858 | A binary relationship invo... |
brtpid3 30859 | A binary relationship invo... |
ceqsrexv2 30860 | Alternate elimitation of a... |
iota5f 30861 | A method for computing iot... |
ceqsralv2 30862 | Alternate elimination of a... |
sqdivzi 30863 | Distribution of square ove... |
subdivcomb1 30864 | Bring a term in a subtract... |
subdivcomb2 30865 | Bring a term in a subtract... |
supfz 30866 | The supremum of a finite s... |
inffz 30867 | The infimum of a finite se... |
inffzOLD 30868 | The infimum of a finite se... |
fz0n 30869 | The sequence ` ( 0 ... ( N... |
shftvalg 30870 | Value of a sequence shifte... |
divcnvlin 30871 | Limit of the ratio of two ... |
climlec3 30872 | Comparison of a constant t... |
logi 30873 | Calculate the logarithm of... |
iexpire 30874 | ` _i ` raised to itself is... |
bcneg1 30875 | The binomial coefficent ov... |
bcm1nt 30876 | The proportion of one bion... |
bcprod 30877 | A product identity for bin... |
bccolsum 30878 | A column-sum rule for bino... |
iprodefisumlem 30879 | Lemma for ~ iprodefisum . ... |
iprodefisum 30880 | Applying the exponential f... |
iprodgam 30881 | An infinite product versio... |
faclimlem1 30882 | Lemma for ~ faclim . Clos... |
faclimlem2 30883 | Lemma for ~ faclim . Show... |
faclimlem3 30884 | Lemma for ~ faclim . Alge... |
faclim 30885 | An infinite product expres... |
iprodfac 30886 | An infinite product expres... |
faclim2 30887 | Another factorial limit du... |
pdivsq 30888 | Condition for a prime divi... |
dvdspw 30889 | Exponentiation law for div... |
gcd32 30890 | Swap the second and third ... |
gcdabsorb 30891 | Absorption law for gcd. (... |
brtp 30892 | A condition for a binary r... |
dftr6 30893 | A potential definition of ... |
coep 30894 | Composition with epsilon. ... |
coepr 30895 | Composition with the conve... |
dffr5 30896 | A quantifier free definiti... |
dfso2 30897 | Quantifier free definition... |
dfpo2 30898 | Quantifier free definition... |
br8 30899 | Substitution for an eight-... |
br6 30900 | Substitution for a six-pla... |
br4 30901 | Substitution for a four-pl... |
dfres3 30902 | Alternate definition of re... |
cnvco1 30903 | Another distributive law o... |
cnvco2 30904 | Another distributive law o... |
eldm3 30905 | Quantifier-free definition... |
elrn3 30906 | Quantifier-free definition... |
pocnv 30907 | The converse of a partial ... |
socnv 30908 | The converse of a strict o... |
funpsstri 30909 | A condition for subset tri... |
fundmpss 30910 | If a class ` F ` is a prop... |
fvresval 30911 | The value of a function at... |
funsseq 30912 | Given two functions with e... |
fununiq 30913 | The uniqueness condition o... |
funbreq 30914 | An equality condition for ... |
mpteq12d 30915 | An equality inference for ... |
fprb 30916 | A condition for functionho... |
br1steq 30917 | Uniqueness condition for b... |
br2ndeq 30918 | Uniqueness condition for b... |
br1steqg 30919 | Uniqueness condition for b... |
br2ndeqg 30920 | Uniqueness condition for b... |
dfdm5 30921 | Definition of domain in te... |
dfrn5 30922 | Definition of range in ter... |
opelco3 30923 | Alternate way of saying th... |
elima4 30924 | Quantifier-free expression... |
fv1stcnv 30925 | The value of the converse ... |
fv2ndcnv 30926 | The value of the converse ... |
setinds 30927 | Principle of ` _E ` induct... |
setinds2f 30928 | ` _E ` induction schema, u... |
setinds2 30929 | ` _E ` induction schema, u... |
elpotr 30930 | A class of transitive sets... |
dford5reg 30931 | Given ~ ax-reg , an ordina... |
dfon2lem1 30932 | Lemma for ~ dfon2 . (Cont... |
dfon2lem2 30933 | Lemma for ~ dfon2 . (Cont... |
dfon2lem3 30934 | Lemma for ~ dfon2 . All s... |
dfon2lem4 30935 | Lemma for ~ dfon2 . If tw... |
dfon2lem5 30936 | Lemma for ~ dfon2 . Two s... |
dfon2lem6 30937 | Lemma for ~ dfon2 . A tra... |
dfon2lem7 30938 | Lemma for ~ dfon2 . All e... |
dfon2lem8 30939 | Lemma for ~ dfon2 . The i... |
dfon2lem9 30940 | Lemma for ~ dfon2 . A cla... |
dfon2 30941 | ` On ` consists of all set... |
domep 30942 | The domain of the epsilon ... |
rdgprc0 30943 | The value of the recursive... |
rdgprc 30944 | The value of the recursive... |
dfrdg2 30945 | Alternate definition of th... |
dfrdg3 30946 | Generalization of ~ dfrdg2... |
axextdfeq 30947 | A version of ~ ax-ext for ... |
ax8dfeq 30948 | A version of ~ ax-8 for us... |
axextdist 30949 | ~ ax-ext with distinctors ... |
axext4dist 30950 | ~ axext4 with distinctors ... |
19.12b 30951 | Version of ~ 19.12vv with ... |
exnel 30952 | There is always a set not ... |
distel 30953 | Distinctors in terms of me... |
axextndbi 30954 | ~ axextnd as a bicondition... |
hbntg 30955 | A more general form of ~ h... |
hbimtg 30956 | A more general and closed ... |
hbaltg 30957 | A more general and closed ... |
hbng 30958 | A more general form of ~ h... |
hbimg 30959 | A more general form of ~ h... |
tfisg 30960 | A closed form of ~ tfis . ... |
dftrpred2 30963 | A definition of the transi... |
trpredeq1 30964 | Equality theorem for trans... |
trpredeq2 30965 | Equality theorem for trans... |
trpredeq3 30966 | Equality theorem for trans... |
trpredeq1d 30967 | Equality deduction for tra... |
trpredeq2d 30968 | Equality deduction for tra... |
trpredeq3d 30969 | Equality deduction for tra... |
eltrpred 30970 | A class is a transitive pr... |
trpredlem1 30971 | Technical lemma for transi... |
trpredpred 30972 | Assuming it exists, the pr... |
trpredss 30973 | The transitive predecessor... |
trpredtr 30974 | The transitive predecessor... |
trpredmintr 30975 | The transitive predecessor... |
trpredelss 30976 | Given a transitive predece... |
dftrpred3g 30977 | The transitive predecessor... |
dftrpred4g 30978 | Another recursive expressi... |
trpredpo 30979 | If ` R ` partially orders ... |
trpred0 30980 | The class of transitive pr... |
trpredex 30981 | The transitive predecessor... |
trpredrec 30982 | If ` Y ` is an ` R ` , ` A... |
frmin 30983 | Every (possibly proper) su... |
frind 30984 | The principle of founded i... |
frindi 30985 | The principle of founded i... |
frinsg 30986 | Founded Induction Schema. ... |
frins 30987 | Founded Induction Schema. ... |
frins2fg 30988 | Founded Induction schema, ... |
frins2f 30989 | Founded Induction schema, ... |
frins2g 30990 | Founded Induction schema, ... |
frins2 30991 | Founded Induction schema, ... |
frins3 30992 | Founded Induction schema, ... |
orderseqlem 30993 | Lemma for ~ poseq and ~ so... |
poseq 30994 | A partial ordering of sequ... |
soseq 30995 | A linear ordering of seque... |
wsuceq123 31004 | Equality theorem for well-... |
wsuceq1 31005 | Equality theorem for well-... |
wsuceq2 31006 | Equality theorem for well-... |
wsuceq3 31007 | Equality theorem for well-... |
nfwsuc 31008 | Bound-variable hypothesis ... |
wlimeq12 31009 | Equality theorem for the l... |
wlimeq1 31010 | Equality theorem for the l... |
wlimeq2 31011 | Equality theorem for the l... |
nfwlim 31012 | Bound-variable hypothesis ... |
elwlim 31013 | Membership in the limit cl... |
elwlimOLD 31014 | Membership in the limit cl... |
wzel 31015 | The zero of a well-founded... |
wzelOLD 31016 | The zero of a well-founded... |
wsuclem 31017 | Lemma for the supremum pro... |
wsuclemOLD 31018 | Obsolete version of ~ wsuc... |
wsucex 31019 | Existence theorem for well... |
wsuccl 31020 | If ` X ` is a set with an ... |
wsuclb 31021 | A well-founded successor i... |
wlimss 31022 | The class of limit points ... |
frr3g 31023 | Functions defined by found... |
frrlem1 31024 | Lemma for founded recursio... |
frrlem2 31025 | Lemma for founded recursio... |
frrlem3 31026 | Lemma for founded recursio... |
frrlem4 31027 | Lemma for founded recursio... |
frrlem5 31028 | Lemma for founded recursio... |
frrlem5b 31029 | Lemma for founded recursio... |
frrlem5c 31030 | Lemma for founded recursio... |
frrlem5d 31031 | Lemma for founded recursio... |
frrlem5e 31032 | Lemma for founded recursio... |
frrlem6 31033 | Lemma for founded recursio... |
frrlem7 31034 | Lemma for founded recursio... |
frrlem10 31035 | Lemma for founded recursio... |
frrlem11 31036 | Lemma for founded recursio... |
elno 31043 | Membership in the surreals... |
sltval 31044 | The value of the surreal l... |
bdayval 31045 | The value of the birthday ... |
nofun 31046 | A surreal is a function. ... |
nodmon 31047 | The domain of a surreal is... |
norn 31048 | The range of a surreal is ... |
nofnbday 31049 | A surreal is a function ov... |
nodmord 31050 | The domain of a surreal ha... |
elno2 31051 | An alternative condition f... |
elno3 31052 | Another condition for memb... |
sltval2 31053 | Alternate expression for s... |
nofv 31054 | The function value of a su... |
nosgnn0 31055 | ` (/) ` is not a surreal s... |
nosgnn0i 31056 | If ` X ` is a surreal sign... |
noreson 31057 | The restriction of a surre... |
sltsgn1 31058 |
If ` A |
sltsgn2 31059 |
If ` A |
sltintdifex 31060 |
If ` A |
sltres 31061 | If the restrictions of two... |
noxpsgn 31062 | The Cartesian product of a... |
noxp1o 31063 | The Cartesian product of a... |
noxp2o 31064 | The Cartesian product of a... |
noseponlem 31065 | Lemma for ~ nosepon . Con... |
nosepon 31066 | Given two unequal surreals... |
sltsolem1 31067 | Lemma for ~ sltso . The s... |
sltso 31068 | Surreal less than totally ... |
sltirr 31069 | Surreal less than is irref... |
slttr 31070 | Surreal less than is trans... |
sltasym 31071 | Surreal less than is asymm... |
slttri 31072 | Surreal less than obeys tr... |
slttrieq2 31073 | Trichotomy law for surreal... |
bdayfo 31074 | The birthday function maps... |
bdayfun 31075 | The birthday function is a... |
bdayrn 31076 | The birthday function's ra... |
bdaydm 31077 | The birthday function's do... |
bdayfn 31078 | The birthday function is a... |
bdayelon 31079 | The value of the birthday ... |
noprc 31080 | The surreal numbers are a ... |
fvnobday 31081 | The value of a surreal at ... |
nodenselem3 31082 | Lemma for ~ nodense . If ... |
nodenselem4 31083 | Lemma for ~ nodense . Sho... |
nodenselem5 31084 | Lemma for ~ nodense . If ... |
nodenselem6 31085 | The restriction of a surre... |
nodenselem7 31086 | Lemma for ~ nodense . ` A ... |
nodenselem8 31087 | Lemma for ~ nodense . Giv... |
nodense 31088 | Given two distinct surreal... |
nocvxminlem 31089 | Lemma for ~ nocvxmin . Gi... |
nocvxmin 31090 | Given a nonempty convex cl... |
nobndlem1 31091 | Lemma for ~ nobndup and ~ ... |
nobndlem2 31092 | Lemma for ~ nobndup and ~ ... |
nobndlem3 31093 | Lemma for ~ nobndup and ~ ... |
nobndlem4 31094 | Lemma for ~ nobndup and ~ ... |
nobndlem5 31095 | Lemma for ~ nobndup and ~ ... |
nobndlem6 31096 | Lemma for ~ nobndup and ~ ... |
nobndlem7 31097 | Lemma for ~ nobndup and ~ ... |
nobndlem8 31098 | Lemma for ~ nobndup and ~ ... |
nobndup 31099 | Any set of surreals is bou... |
nobnddown 31100 | Any set of surreals is bou... |
nofulllem1 31101 | Lemma for nofull (future) ... |
nofulllem2 31102 | Lemma for nofull (future) ... |
nofulllem3 31103 | Lemma for nofull (future) ... |
nofulllem4 31104 | Lemma for nofull (future) ... |
nofulllem5 31105 | Lemma for nofull (future) ... |
brv 31154 | The binary relationship ov... |
txpss3v 31155 | A tail Cartesian product i... |
txprel 31156 | A tail Cartesian product i... |
brtxp 31157 | Characterize a trinary rel... |
brtxp2 31158 | The binary relationship ov... |
dfpprod2 31159 | Expanded definition of par... |
pprodcnveq 31160 | A converse law for paralle... |
pprodss4v 31161 | The parallel product is a ... |
brpprod 31162 | Characterize a quatary rel... |
brpprod3a 31163 | Condition for parallel pro... |
brpprod3b 31164 | Condition for parallel pro... |
relsset 31165 | The subset class is a rela... |
brsset 31166 | For sets, the ` SSet ` bin... |
idsset 31167 | ` _I ` is equal to ` SSet ... |
eltrans 31168 | Membership in the class of... |
dfon3 31169 | A quantifier-free definiti... |
dfon4 31170 | Another quantifier-free de... |
brtxpsd 31171 | Expansion of a common form... |
brtxpsd2 31172 | Another common abbreviatio... |
brtxpsd3 31173 | A third common abbreviatio... |
relbigcup 31174 | The ` Bigcup ` relationshi... |
brbigcup 31175 | Binary relationship over `... |
dfbigcup2 31176 | ` Bigcup ` using maps-to n... |
fobigcup 31177 | ` Bigcup ` maps the univer... |
fnbigcup 31178 | ` Bigcup ` is a function o... |
fvbigcup 31179 | For sets, ` Bigcup ` yield... |
elfix 31180 | Membership in the fixpoint... |
elfix2 31181 | Alternative membership in ... |
dffix2 31182 | The fixpoints of a class i... |
fixssdm 31183 | The fixpoints of a class a... |
fixssrn 31184 | The fixpoints of a class a... |
fixcnv 31185 | The fixpoints of a class a... |
fixun 31186 | The fixpoint operator dist... |
ellimits 31187 | Membership in the class of... |
limitssson 31188 | The class of all limit ord... |
dfom5b 31189 | A quantifier-free definiti... |
sscoid 31190 | A condition for subset and... |
dffun10 31191 | Another potential definiti... |
elfuns 31192 | Membership in the class of... |
elfunsg 31193 | Closed form of ~ elfuns . ... |
brsingle 31194 | The binary relationship fo... |
elsingles 31195 | Membership in the class of... |
fnsingle 31196 | The singleton relationship... |
fvsingle 31197 | The value of the singleton... |
dfsingles2 31198 | Alternate definition of th... |
snelsingles 31199 | A singleton is a member of... |
dfiota3 31200 | A definiton of iota using ... |
dffv5 31201 | Another quantifier free de... |
unisnif 31202 | Express union of singleton... |
brimage 31203 | Binary relationship form o... |
brimageg 31204 | Closed form of ~ brimage .... |
funimage 31205 | ` Image A ` is a function.... |
fnimage 31206 | ` Image R ` is a function ... |
imageval 31207 | The image functor in maps-... |
fvimage 31208 | The value of the image fun... |
brcart 31209 | Binary relationship form o... |
brdomain 31210 | The binary relationship fo... |
brrange 31211 | The binary relationship fo... |
brdomaing 31212 | Closed form of ~ brdomain ... |
brrangeg 31213 | Closed form of ~ brrange .... |
brimg 31214 | The binary relationship fo... |
brapply 31215 | The binary relationship fo... |
brcup 31216 | Binary relationship form o... |
brcap 31217 | Binary relationship form o... |
brsuccf 31218 | Binary relationship form o... |
funpartlem 31219 | Lemma for ~ funpartfun . ... |
funpartfun 31220 | The functional part of ` F... |
funpartss 31221 | The functional part of ` F... |
funpartfv 31222 | The function value of the ... |
fullfunfnv 31223 | The full functional part o... |
fullfunfv 31224 | The function value of the ... |
brfullfun 31225 | A binary relationship form... |
brrestrict 31226 | The binary relationship fo... |
dfrecs2 31227 | A quantifier-free definiti... |
dfrdg4 31228 | A quantifier-free definiti... |
dfint3 31229 | Quantifier-free definition... |
imagesset 31230 | The Image functor applied ... |
brub 31231 | Binary relationship form o... |
brlb 31232 | Binary relationship form o... |
altopex 31237 | Alternative ordered pairs ... |
altopthsn 31238 | Two alternate ordered pair... |
altopeq12 31239 | Equality for alternate ord... |
altopeq1 31240 | Equality for alternate ord... |
altopeq2 31241 | Equality for alternate ord... |
altopth1 31242 | Equality of the first memb... |
altopth2 31243 | Equality of the second mem... |
altopthg 31244 | Alternate ordered pair the... |
altopthbg 31245 | Alternate ordered pair the... |
altopth 31246 | The alternate ordered pair... |
altopthb 31247 | Alternate ordered pair the... |
altopthc 31248 | Alternate ordered pair the... |
altopthd 31249 | Alternate ordered pair the... |
altxpeq1 31250 | Equality for alternate Car... |
altxpeq2 31251 | Equality for alternate Car... |
elaltxp 31252 | Membership in alternate Ca... |
altopelaltxp 31253 | Alternate ordered pair mem... |
altxpsspw 31254 | An inclusion rule for alte... |
altxpexg 31255 | The alternate Cartesian pr... |
rankaltopb 31256 | Compute the rank of an alt... |
nfaltop 31257 | Bound-variable hypothesis ... |
sbcaltop 31258 | Distribution of class subs... |
cgrrflx2d 31261 | Deduction form of ~ axcgrr... |
cgrtr4d 31262 | Deduction form of ~ axcgrt... |
cgrtr4and 31263 | Deduction form of ~ axcgrt... |
cgrrflx 31264 | Reflexivity law for congru... |
cgrrflxd 31265 | Deduction form of ~ cgrrfl... |
cgrcomim 31266 | Congruence commutes on the... |
cgrcom 31267 | Congruence commutes betwee... |
cgrcomand 31268 | Deduction form of ~ cgrcom... |
cgrtr 31269 | Transitivity law for congr... |
cgrtrand 31270 | Deduction form of ~ cgrtr ... |
cgrtr3 31271 | Transitivity law for congr... |
cgrtr3and 31272 | Deduction form of ~ cgrtr3... |
cgrcoml 31273 | Congruence commutes on the... |
cgrcomr 31274 | Congruence commutes on the... |
cgrcomlr 31275 | Congruence commutes on bot... |
cgrcomland 31276 | Deduction form of ~ cgrcom... |
cgrcomrand 31277 | Deduction form of ~ cgrcom... |
cgrcomlrand 31278 | Deduction form of ~ cgrcom... |
cgrtriv 31279 | Degenerate segments are co... |
cgrid2 31280 | Identity law for congruenc... |
cgrdegen 31281 | Two congruent segments are... |
brofs 31282 | Binary relationship form o... |
5segofs 31283 | Rephrase ~ ax5seg using th... |
ofscom 31284 | The outer five segment pre... |
cgrextend 31285 | Link congruence over a pai... |
cgrextendand 31286 | Deduction form of ~ cgrext... |
segconeq 31287 | Two points that satsify th... |
segconeu 31288 | Existential uniqueness ver... |
btwntriv2 31289 | Betweenness always holds f... |
btwncomim 31290 | Betweenness commutes. Imp... |
btwncom 31291 | Betweenness commutes. (Co... |
btwncomand 31292 | Deduction form of ~ btwnco... |
btwntriv1 31293 | Betweenness always holds f... |
btwnswapid 31294 | If you can swap the first ... |
btwnswapid2 31295 | If you can swap arguments ... |
btwnintr 31296 | Inner transitivity law for... |
btwnexch3 31297 | Exchange the first endpoin... |
btwnexch3and 31298 | Deduction form of ~ btwnex... |
btwnouttr2 31299 | Outer transitivity law for... |
btwnexch2 31300 | Exchange the outer point o... |
btwnouttr 31301 | Outer transitivity law for... |
btwnexch 31302 | Outer transitivity law for... |
btwnexchand 31303 | Deduction form of ~ btwnex... |
btwndiff 31304 | There is always a ` c ` di... |
trisegint 31305 | A line segment between two... |
funtransport 31308 | The ` TransportTo ` relati... |
fvtransport 31309 | Calculate the value of the... |
transportcl 31310 | Closure law for segment tr... |
transportprops 31311 | Calculate the defining pro... |
brifs 31320 | Binary relationship form o... |
ifscgr 31321 | Inner five segment congrue... |
cgrsub 31322 | Removing identical parts f... |
brcgr3 31323 | Binary relationship form o... |
cgr3permute3 31324 | Permutation law for three-... |
cgr3permute1 31325 | Permutation law for three-... |
cgr3permute2 31326 | Permutation law for three-... |
cgr3permute4 31327 | Permutation law for three-... |
cgr3permute5 31328 | Permutation law for three-... |
cgr3tr4 31329 | Transitivity law for three... |
cgr3com 31330 | Commutativity law for thre... |
cgr3rflx 31331 | Identity law for three-pla... |
cgrxfr 31332 | A line segment can be divi... |
btwnxfr 31333 | A condition for extending ... |
colinrel 31334 | Colinearity is a relations... |
brcolinear2 31335 | Alternate colinearity bina... |
brcolinear 31336 | The binary relationship fo... |
colinearex 31337 | The colinear predicate exi... |
colineardim1 31338 | If ` A ` is colinear with ... |
colinearperm1 31339 | Permutation law for coline... |
colinearperm3 31340 | Permutation law for coline... |
colinearperm2 31341 | Permutation law for coline... |
colinearperm4 31342 | Permutation law for coline... |
colinearperm5 31343 | Permutation law for coline... |
colineartriv1 31344 | Trivial case of colinearit... |
colineartriv2 31345 | Trivial case of colinearit... |
btwncolinear1 31346 | Betweenness implies coline... |
btwncolinear2 31347 | Betweenness implies coline... |
btwncolinear3 31348 | Betweenness implies coline... |
btwncolinear4 31349 | Betweenness implies coline... |
btwncolinear5 31350 | Betweenness implies coline... |
btwncolinear6 31351 | Betweenness implies coline... |
colinearxfr 31352 | Transfer law for colineari... |
lineext 31353 | Extend a line with a missi... |
brofs2 31354 | Change some conditions for... |
brifs2 31355 | Change some conditions for... |
brfs 31356 | Binary relationship form o... |
fscgr 31357 | Congruence law for the gen... |
linecgr 31358 | Congruence rule for lines.... |
linecgrand 31359 | Deduction form of ~ linecg... |
lineid 31360 | Identity law for points on... |
idinside 31361 | Law for finding a point in... |
endofsegid 31362 | If ` A ` , ` B ` , and ` C... |
endofsegidand 31363 | Deduction form of ~ endofs... |
btwnconn1lem1 31364 | Lemma for ~ btwnconn1 . T... |
btwnconn1lem2 31365 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem3 31366 | Lemma for ~ btwnconn1 . E... |
btwnconn1lem4 31367 | Lemma for ~ btwnconn1 . A... |
btwnconn1lem5 31368 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem6 31369 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem7 31370 | Lemma for ~ btwnconn1 . U... |
btwnconn1lem8 31371 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem9 31372 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem10 31373 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem11 31374 | Lemma for ~ btwnconn1 . N... |
btwnconn1lem12 31375 | Lemma for ~ btwnconn1 . U... |
btwnconn1lem13 31376 | Lemma for ~ btwnconn1 . B... |
btwnconn1lem14 31377 | Lemma for ~ btwnconn1 . F... |
btwnconn1 31378 | Connectitivy law for betwe... |
btwnconn2 31379 | Another connectivity law f... |
btwnconn3 31380 | Inner connectivity law for... |
midofsegid 31381 | If two points fall in the ... |
segcon2 31382 | Generalization of ~ axsegc... |
brsegle 31385 | Binary relationship form o... |
brsegle2 31386 | Alternate characterization... |
seglecgr12im 31387 | Substitution law for segme... |
seglecgr12 31388 | Substitution law for segme... |
seglerflx 31389 | Segment comparison is refl... |
seglemin 31390 | Any segment is at least as... |
segletr 31391 | Segment less than is trans... |
segleantisym 31392 | Antisymmetry law for segme... |
seglelin 31393 | Linearity law for segment ... |
btwnsegle 31394 | If ` B ` falls between ` A... |
colinbtwnle 31395 | Given three colinear point... |
broutsideof 31398 | Binary relationship form o... |
broutsideof2 31399 | Alternate form of ` Outsid... |
outsidene1 31400 | Outsideness implies inequa... |
outsidene2 31401 | Outsideness implies inequa... |
btwnoutside 31402 | A principle linking outsid... |
broutsideof3 31403 | Characterization of outsid... |
outsideofrflx 31404 | Reflexitivity of outsidene... |
outsideofcom 31405 | Commutitivity law for outs... |
outsideoftr 31406 | Transitivity law for outsi... |
outsideofeq 31407 | Uniqueness law for ` Outsi... |
outsideofeu 31408 | Given a non-degenerate ray... |
outsidele 31409 | Relate ` OutsideOf ` to ` ... |
outsideofcol 31410 | Outside of implies colinea... |
funray 31417 | Show that the ` Ray ` rela... |
fvray 31418 | Calculate the value of the... |
funline 31419 | Show that the ` Line ` rel... |
linedegen 31420 | When ` Line ` is applied w... |
fvline 31421 | Calculate the value of the... |
liness 31422 | A line is a subset of the ... |
fvline2 31423 | Alternate definition of a ... |
lineunray 31424 | A line is composed of a po... |
lineelsb2 31425 | If ` S ` lies on ` P Q ` ,... |
linerflx1 31426 | Reflexivity law for line m... |
linecom 31427 | Commutativity law for line... |
linerflx2 31428 | Reflexivity law for line m... |
ellines 31429 | Membership in the set of a... |
linethru 31430 | If ` A ` is a line contain... |
hilbert1.1 31431 | There is a line through an... |
hilbert1.2 31432 | There is at most one line ... |
linethrueu 31433 | There is a unique line goi... |
lineintmo 31434 | Two distinct lines interse... |
fwddifval 31439 | Calculate the value of the... |
fwddifnval 31440 | The value of the forward d... |
fwddifn0 31441 | The value of the n-iterate... |
fwddifnp1 31442 | The value of the n-iterate... |
rankung 31443 | The rank of the union of t... |
ranksng 31444 | The rank of a singleton. ... |
rankelg 31445 | The membership relation is... |
rankpwg 31446 | The rank of a power set. ... |
rank0 31447 | The rank of the empty set ... |
rankeq1o 31448 | The only set with rank ` 1... |
elhf 31451 | Membership in the heredita... |
elhf2 31452 | Alternate form of membersh... |
elhf2g 31453 | Hereditarily finiteness vi... |
0hf 31454 | The empty set is a heredit... |
hfun 31455 | The union of two HF sets i... |
hfsn 31456 | The singleton of an HF set... |
hfadj 31457 | Adjoining one HF element t... |
hfelhf 31458 | Any member of an HF set is... |
hftr 31459 | The class of all hereditar... |
hfext 31460 | Extensionality for HF sets... |
hfuni 31461 | The union of an HF set is ... |
hfpw 31462 | The power class of an HF s... |
hfninf 31463 | ` _om ` is not hereditaril... |
a1i14 31464 | Add two antecedents to a w... |
a1i24 31465 | Add two antecedents to a w... |
exp5d 31466 | An exportation inference. ... |
exp5g 31467 | An exportation inference. ... |
exp5k 31468 | An exportation inference. ... |
exp56 31469 | An exportation inference. ... |
exp58 31470 | An exportation inference. ... |
exp510 31471 | An exportation inference. ... |
exp511 31472 | An exportation inference. ... |
exp512 31473 | An exportation inference. ... |
3com12d 31474 | Commutation in consequent.... |
imp5p 31475 | A triple importation infer... |
imp5q 31476 | A triple importation infer... |
ecase13d 31477 | Deduction for elimination ... |
subtr 31478 | Transitivity of implicit s... |
subtr2 31479 | Transitivity of implicit s... |
trer 31480 | A relation intersected wit... |
elicc3 31481 | An equivalent membership c... |
finminlem 31482 | A useful lemma about finit... |
gtinf 31483 | Any number greater than an... |
gtinfOLD 31484 | Any number greater than an... |
opnrebl 31485 | A set is open in the stand... |
opnrebl2 31486 | A set is open in the stand... |
nn0prpwlem 31487 | Lemma for ~ nn0prpw . Use... |
nn0prpw 31488 | Two nonnegative integers a... |
topbnd 31489 | Two equivalent expressions... |
opnbnd 31490 | A set is open iff it is di... |
cldbnd 31491 | A set is closed iff it con... |
ntruni 31492 | A union of interiors is a ... |
clsun 31493 | A pairwise union of closur... |
clsint2 31494 | The closure of an intersec... |
opnregcld 31495 | A set is regularly closed ... |
cldregopn 31496 | A set if regularly open if... |
neiin 31497 | Two neighborhoods intersec... |
hmeoclda 31498 | Homeomorphisms preserve cl... |
hmeocldb 31499 | Homeomorphisms preserve cl... |
ivthALT 31500 | An alternate proof of the ... |
fnerel 31503 | Fineness is a relation. (... |
isfne 31504 | The predicate " ` B ` is f... |
isfne4 31505 | The predicate " ` B ` is f... |
isfne4b 31506 | A condition for a topology... |
isfne2 31507 | The predicate " ` B ` is f... |
isfne3 31508 | The predicate " ` B ` is f... |
fnebas 31509 | A finer cover covers the s... |
fnetg 31510 | A finer cover generates a ... |
fnessex 31511 | If ` B ` is finer than ` A... |
fneuni 31512 | If ` B ` is finer than ` A... |
fneint 31513 | If a cover is finer than a... |
fness 31514 | A cover is finer than its ... |
fneref 31515 | Reflexivity of the finenes... |
fnetr 31516 | Transitivity of the finene... |
fneval 31517 | Two covers are finer than ... |
fneer 31518 | Fineness intersected with ... |
topfne 31519 | Fineness for covers corres... |
topfneec 31520 | A cover is equivalent to a... |
topfneec2 31521 | A topology is precisely id... |
fnessref 31522 | A cover is finer iff it ha... |
refssfne 31523 | A cover is a refinement if... |
neibastop1 31524 | A collection of neighborho... |
neibastop2lem 31525 | Lemma for ~ neibastop2 . ... |
neibastop2 31526 | In the topology generated ... |
neibastop3 31527 | The topology generated by ... |
topmtcl 31528 | The meet of a collection o... |
topmeet 31529 | Two equivalent formulation... |
topjoin 31530 | Two equivalent formulation... |
fnemeet1 31531 | The meet of a collection o... |
fnemeet2 31532 | The meet of equivalence cl... |
fnejoin1 31533 | Join of equivalence classe... |
fnejoin2 31534 | Join of equivalence classe... |
fgmin 31535 | Minimality property of a g... |
neifg 31536 | The neighborhood filter of... |
tailfval 31537 | The tail function for a di... |
tailval 31538 | The tail of an element in ... |
eltail 31539 | An element of a tail. (Co... |
tailf 31540 | The tail function of a dir... |
tailini 31541 | A tail contains its initia... |
tailfb 31542 | The collection of tails of... |
filnetlem1 31543 | Lemma for ~ filnet . Chan... |
filnetlem2 31544 | Lemma for ~ filnet . The ... |
filnetlem3 31545 | Lemma for ~ filnet . (Con... |
filnetlem4 31546 | Lemma for ~ filnet . (Con... |
filnet 31547 | A filter has the same conv... |
tb-ax1 31548 | The first of three axioms ... |
tb-ax2 31549 | The second of three axioms... |
tb-ax3 31550 | The third of three axioms ... |
tbsyl 31551 | The weak syllogism from Ta... |
re1ax2lem 31552 | Lemma for ~ re1ax2 . (Con... |
re1ax2 31553 | ~ ax-2 rederived from the ... |
naim1 31554 | Constructor theorem for ` ... |
naim2 31555 | Constructor theorem for ` ... |
naim1i 31556 | Constructor rule for ` -/\... |
naim2i 31557 | Constructor rule for ` -/\... |
naim12i 31558 | Constructor rule for ` -/\... |
nabi1 31559 | Constructor theorem for ` ... |
nabi2 31560 | Constructor theorem for ` ... |
nabi1i 31561 | Constructor rule for ` -/\... |
nabi2i 31562 | Constructor rule for ` -/\... |
nabi12i 31563 | Constructor rule for ` -/\... |
df3nandALT1 31566 | The double nand expressed ... |
df3nandALT2 31567 | The double nand expressed ... |
andnand1 31568 | Double and in terms of dou... |
imnand2 31569 | An ` -> ` nand relation. ... |
allt 31570 | For all sets, ` T. ` is tr... |
alnof 31571 | For all sets, ` F. ` is no... |
nalf 31572 | Not all sets hold ` F. ` a... |
extt 31573 | There exists a set that ho... |
nextnt 31574 | There does not exist a set... |
nextf 31575 | There does not exist a set... |
unnf 31576 | There does not exist exact... |
unnt 31577 | There does not exist exact... |
mont 31578 | There does not exist at mo... |
mof 31579 | There exist at most one se... |
meran1 31580 | A single axiom for proposi... |
meran2 31581 | A single axiom for proposi... |
meran3 31582 | A single axiom for proposi... |
waj-ax 31583 | A single axiom for proposi... |
lukshef-ax2 31584 | A single axiom for proposi... |
arg-ax 31585 | ? (Contributed by Anthony... |
negsym1 31586 | In the paper "On Variable ... |
imsym1 31587 | A symmetry with ` -> ` . ... |
bisym1 31588 | A symmetry with ` <-> ` . ... |
consym1 31589 | A symmetry with ` /\ ` . ... |
dissym1 31590 | A symmetry with ` \/ ` . ... |
nandsym1 31591 | A symmetry with ` -/\ ` . ... |
unisym1 31592 | A symmetry with ` A. ` . ... |
exisym1 31593 | A symmetry with ` E. ` . ... |
unqsym1 31594 | A symmetry with ` E! ` . ... |
amosym1 31595 | A symmetry with ` E* ` . ... |
subsym1 31596 | A symmetry with ` [ x / y ... |
ontopbas 31597 | An ordinal number is a top... |
onsstopbas 31598 | The class of ordinal numbe... |
onpsstopbas 31599 | The class of ordinal numbe... |
ontgval 31600 | The topology generated fro... |
ontgsucval 31601 | The topology generated fro... |
onsuctop 31602 | A successor ordinal number... |
onsuctopon 31603 | One of the topologies on a... |
ordtoplem 31604 | Membership of the class of... |
ordtop 31605 | An ordinal is a topology i... |
onsucconi 31606 | A successor ordinal number... |
onsuccon 31607 | A successor ordinal number... |
ordtopcon 31608 | An ordinal topology is con... |
onintopsscon 31609 | An ordinal topology is con... |
onsuct0 31610 | A successor ordinal number... |
ordtopt0 31611 | An ordinal topology is T_0... |
onsucsuccmpi 31612 | The successor of a success... |
onsucsuccmp 31613 | The successor of a success... |
limsucncmpi 31614 | The successor of a limit o... |
limsucncmp 31615 | The successor of a limit o... |
ordcmp 31616 | An ordinal topology is com... |
ssoninhaus 31617 | The ordinal topologies ` 1... |
onint1 31618 | The ordinal T_1 spaces are... |
oninhaus 31619 | The ordinal Hausdorff spac... |
fveleq 31620 | Please add description her... |
findfvcl 31621 | Please add description her... |
findreccl 31622 | Please add description her... |
findabrcl 31623 | Please add description her... |
nnssi2 31624 | Convert a theorem for real... |
nnssi3 31625 | Convert a theorem for real... |
nndivsub 31626 | Please add description her... |
nndivlub 31627 | A factor of a positive int... |
ee7.2aOLD 31630 | Lemma for Euclid's Element... |
dnival 31631 | Value of the "distance to ... |
dnicld1 31632 | Closure theorem for the "d... |
dnicld2 31633 | Closure theorem for the "d... |
dnif 31634 | The "distance to nearest i... |
dnizeq0 31635 | The distance to nearest in... |
dnizphlfeqhlf 31636 | The distance to nearest in... |
rddif2 31637 | Variant of ~ rddif . (Con... |
dnibndlem1 31638 | Lemma for ~ dnibnd . (Con... |
dnibndlem2 31639 | Lemma for ~ dnibnd . (Con... |
dnibndlem3 31640 | Lemma for ~ dnibnd . (Con... |
dnibndlem4 31641 | Lemma for ~ dnibnd . (Con... |
dnibndlem5 31642 | Lemma for ~ dnibnd . (Con... |
dnibndlem6 31643 | Lemma for ~ dnibnd . (Con... |
dnibndlem7 31644 | Lemma for ~ dnibnd . (Con... |
dnibndlem8 31645 | Lemma for ~ dnibnd . (Con... |
dnibndlem9 31646 | Lemma for ~ dnibnd . (Con... |
dnibndlem10 31647 | Lemma for ~ dnibnd . (Con... |
dnibndlem11 31648 | Lemma for ~ dnibnd . (Con... |
dnibndlem12 31649 | Lemma for ~ dnibnd . (Con... |
dnibndlem13 31650 | Lemma for ~ dnibnd . (Con... |
dnibnd 31651 | The "distance to nearest i... |
dnicn 31652 | The "distance to nearest i... |
knoppcnlem1 31653 | Lemma for ~ knoppcn . (Co... |
knoppcnlem2 31654 | Lemma for ~ knoppcn . (Co... |
knoppcnlem3 31655 | Lemma for ~ knoppcn . (Co... |
knoppcnlem4 31656 | Lemma for ~ knoppcn . (Co... |
knoppcnlem5 31657 | Lemma for ~ knoppcn . (Co... |
knoppcnlem6 31658 | Lemma for ~ knoppcn . (Co... |
knoppcnlem7 31659 | Lemma for ~ knoppcn . (Co... |
knoppcnlem8 31660 | Lemma for ~ knoppcn . (Co... |
knoppcnlem9 31661 | Lemma for ~ knoppcn . (Co... |
knoppcnlem10 31662 | Lemma for ~ knoppcn . (Co... |
knoppcnlem11 31663 | Lemma for ~ knoppcn . (Co... |
knoppcn 31664 | The continuous nowhere dif... |
knoppcld 31665 | Closure theorem for Knopp'... |
addgtge0d 31666 | Addition of positive and n... |
unblimceq0lem 31667 | Lemma for ~ unblimceq0 . ... |
unblimceq0 31668 | If ` F ` is unbounded near... |
unbdqndv1 31669 | If the difference quotient... |
unbdqndv2lem1 31670 | Lemma for ~ unbdqndv2 . (... |
unbdqndv2lem2 31671 | Lemma for ~ unbdqndv2 . (... |
unbdqndv2 31672 | Variant of ~ unbdqndv1 wit... |
knoppndvlem1 31673 | Lemma for ~ knoppndv . (C... |
knoppndvlem2 31674 | Lemma for ~ knoppndv . (C... |
knoppndvlem3 31675 | Lemma for ~ knoppndv . (C... |
knoppndvlem4 31676 | Lemma for ~ knoppndv . (C... |
knoppndvlem5 31677 | Lemma for ~ knoppndv . (C... |
knoppndvlem6 31678 | Lemma for ~ knoppndv . (C... |
knoppndvlem7 31679 | Lemma for ~ knoppndv . (C... |
knoppndvlem8 31680 | Lemma for ~ knoppndv . (C... |
knoppndvlem9 31681 | Lemma for ~ knoppndv . (C... |
knoppndvlem10 31682 | Lemma for ~ knoppndv . (C... |
knoppndvlem11 31683 | Lemma for ~ knoppndv . (C... |
knoppndvlem12 31684 | Lemma for ~ knoppndv . (C... |
knoppndvlem13 31685 | Lemma for ~ knoppndv . (C... |
knoppndvlem14 31686 | Lemma for ~ knoppndv . (C... |
knoppndvlem15 31687 | Lemma for ~ knoppndv . (C... |
knoppndvlem16 31688 | Lemma for ~ knoppndv . (C... |
knoppndvlem17 31689 | Lemma for ~ knoppndv . (C... |
knoppndvlem18 31690 | Lemma for ~ knoppndv . (C... |
knoppndvlem19 31691 | Lemma for ~ knoppndv . (C... |
knoppndvlem20 31692 | Lemma for ~ knoppndv . (C... |
knoppndvlem21 31693 | Lemma for ~ knoppndv . (C... |
knoppndvlem22 31694 | Lemma for ~ knoppndv . (C... |
knoppndv 31695 | The continuous nowhere dif... |
knoppf 31696 | Knopp's function is a func... |
knoppcn2 31697 | Variant of ~ knoppcn with ... |
cnndvlem1 31698 | Lemma for ~ cnndv . (Cont... |
cnndvlem2 31699 | Lemma for ~ cnndv . (Cont... |
cnndv 31700 | There exists a continuous ... |
bj-mp2c 31701 | A double modus ponens infe... |
bj-mp2d 31702 | A double modus ponens infe... |
bj-0 31703 | A syntactic theorem. See ... |
bj-1 31704 | In this proof, the use of ... |
bj-a1k 31705 | Weakening of ~ ax-1 . Thi... |
bj-jarri 31706 | Inference associated with ... |
bj-jarrii 31707 | Inference associated with ... |
bj-imim2ALT 31708 | More direct proof of ~ imi... |
bj-imim21 31709 | The propositional function... |
bj-imim21i 31710 | Inference associated with ... |
bj-orim2 31711 | Proof of ~ orim2 from the ... |
bj-curry 31712 | A non-intuitionistic posit... |
bj-peirce 31713 | Proof of ~ peirce from min... |
bj-currypeirce 31714 | Curry's axiom (a non-intui... |
bj-peircecurry 31715 | Peirce's axiom ~ peirce im... |
pm4.81ALT 31716 | Alternate proof of ~ pm4.8... |
bj-con4iALT 31717 | Alternate proof of ~ con4i... |
bj-con2com 31718 | A commuted form of the con... |
bj-con2comi 31719 | Inference associated with ... |
bj-pm2.01i 31720 | Inference associated with ... |
bj-nimn 31721 | If a formula is true, then... |
bj-nimni 31722 | Inference associated with ... |
bj-peircei 31723 | Inference associated with ... |
bj-looinvi 31724 | Inference associated with ... |
bj-looinvii 31725 | Inference associated with ... |
bj-jaoi1 31726 | Shortens 11 proofs by a to... |
bj-jaoi2 31727 | Shortens 9 proofs by a tot... |
bj-dfbi4 31728 | Alternate definition of th... |
bj-dfbi5 31729 | Alternate definition of th... |
bj-dfbi6 31730 | Alternate definition of th... |
bj-bijust0 31731 | The general statement that... |
bj-consensus 31732 | Version of ~ consensus exp... |
bj-consensusALT 31733 | Alternate proof of ~ bj-co... |
bj-dfifc2 31734 | This should be the alterna... |
bj-df-ifc 31735 | The definition of "ifc" if... |
bj-ififc 31736 | A theorem linking ` if- ` ... |
sylancl2 31737 | Shortens 5 proofs. (Contr... |
sylancl3 31738 | Shortens 11 proofs by a to... |
bj-imbi12 31739 | Imported form (uncurried f... |
bj-trut 31740 | A proposition is equivalen... |
bj-biorfi 31741 | This should be labeled "bi... |
bj-falor 31742 | Dual of ~ truan (which has... |
bj-falor2 31743 | Dual of ~ truan . (Contri... |
bj-bibibi 31744 | A property of the bicondit... |
bj-imn3ani 31745 | Duplication of ~ bnj1224 .... |
bj-andnotim 31746 | Two ways of expressing a c... |
bj-bi3ant 31747 | This used to be in the mai... |
bj-bisym 31748 | This used to be in the mai... |
bj-axdd2 31749 | This implication, proved u... |
bj-axd2d 31750 | This implication, proved u... |
bj-axtd 31751 | This implication, proved f... |
bj-gl4lem 31752 | Lemma for ~ bj-gl4 . Note... |
bj-gl4 31753 | In a normal modal logic, t... |
bj-axc4 31754 | Over minimal calculus, the... |
prvlem1 31759 | An elementary property of ... |
prvlem2 31760 | An elementary property of ... |
bj-babygodel 31761 | See the section header com... |
bj-babylob 31762 | See the section header com... |
bj-godellob 31763 | Proof of Gödel's theo... |
bj-nf2 31766 | Alternate definition of ~ ... |
bj-nf3 31767 | Alternate definition of ~ ... |
bj-nf4 31768 | Alternate definition of ~ ... |
bj-nftht 31769 | Closed form of ~ nfth . (... |
bj-nfntht 31770 | Closed form of ~ nfnth . ... |
bj-nfntht2 31771 | Closed form of ~ nfnth . ... |
bj-nfth 31772 | Any variable is not free i... |
bj-nftru 31773 | The true constant has no f... |
bj-nfnth 31774 | Any variable is not free i... |
bj-nffal 31775 | The false constant has no ... |
bj-genr 31776 | Generalization rule on the... |
bj-genl 31777 | Generalization rule on the... |
bj-genan 31778 | Generalization rule on a c... |
bj-2alim 31779 | Closed form of ~ 2alimi . ... |
bj-2exim 31780 | Closed form of ~ 2eximi . ... |
bj-alanim 31781 | Closed form of ~ alanimi .... |
bj-2albi 31782 | Closed form of ~ 2albii . ... |
bj-notalbii 31783 | Equivalence of universal q... |
bj-2exbi 31784 | Closed form of ~ 2exbii . ... |
bj-3exbi 31785 | Closed form of ~ 3exbii . ... |
bj-sylgt2 31786 | Uncurried form of ~ sylgt ... |
bj-exlimh 31787 | Closed form of close to ~ ... |
bj-exlimh2 31788 | Uncurried form of ~ bj-exl... |
bj-alrimhi 31789 | An inference associated wi... |
bj-nexdh 31790 | Closed form of ~ nexdh (an... |
bj-nexdh2 31791 | Uncurried form of ~ bj-nex... |
bj-hbxfrbi 31792 | Closed form of ~ hbxfrbi .... |
bj-nfbi 31793 | Closed form of ~ nfbii (wi... |
bj-nfxfr 31794 | Proof of ~ nfxfr from ~ bj... |
bj-nfn 31795 | A variable is non-free in ... |
bj-exlime 31796 | Variant of ~ exlimih where... |
bj-exnalimn 31797 | A transformation of quanti... |
bj-nalnaleximiOLD 31798 | An inference for distribut... |
bj-nalnalimiOLD 31799 | An inference for distribut... |
bj-exaleximi 31800 | An inference for distribut... |
bj-exalimi 31801 | An inference for distribut... |
bj-ax12ig 31802 | A lemma used to prove a we... |
bj-ax12i 31803 | A weakening of ~ bj-ax12ig... |
bj-ax12iOLD 31804 | Old proof of ~ bj-ax12i . ... |
bj-ax5ea 31805 | If a formula holds for som... |
bj-nfv 31806 | A non-occurring variable i... |
bj-ax12wlem 31807 | A lemma used to prove a we... |
bj-ssbim 31810 | Distribute substitution ov... |
bj-ssbbi 31811 | Biconditional property for... |
bj-ssbimi 31812 | Distribute substitution ov... |
bj-ssbbii 31813 | Biconditional property for... |
bj-alsb 31814 | If a proposition is true f... |
bj-sbex 31815 | If a proposition is true f... |
bj-ssbeq 31816 | Substitution in an equalit... |
bj-ssb0 31817 | Substitution for a variabl... |
bj-ssbequ 31818 | Equality property for subs... |
bj-ssblem1 31819 | A lemma for the definiens ... |
bj-ssblem2 31820 | The converse may not be pr... |
bj-ssb1a 31821 | One direction of a simplif... |
bj-ssb1 31822 | A simplified definition of... |
bj-ax12 31823 | A weaker form of ~ ax-12 a... |
bj-ax12ssb 31824 | The axiom ~ bj-ax12 expres... |
bj-modal5e 31825 | Dual statement of ~ hbe1 (... |
bj-19.41al 31826 | Special case of ~ 19.41 pr... |
bj-equsexval 31827 | Special case of ~ equsexv ... |
bj-sb56 31828 | Proof of ~ sb56 from Tarsk... |
bj-dfssb2 31829 | An alternate definition of... |
bj-ssbn 31830 | The result of a substituti... |
bj-ssbft 31831 | See ~ sbft . This proof i... |
bj-ssbequ2 31832 | Note that ~ ax-12 is used ... |
bj-ssbequ1 31833 | This uses ~ ax-12 with a d... |
bj-ssbid2 31834 | A special case of ~ bj-ssb... |
bj-ssbid2ALT 31835 | Alternate proof of ~ bj-ss... |
bj-ssbid1 31836 | A special case of ~ bj-ssb... |
bj-ssbid1ALT 31837 | Alternate proof of ~ bj-ss... |
bj-ssbssblem 31838 | Composition of two substit... |
bj-ssbcom3lem 31839 | Lemma for bj-ssbcom3 when ... |
bj-ax6elem1 31840 | Lemma for ~ bj-ax6e . (Co... |
bj-ax6elem2 31841 | Lemma for ~ bj-ax6e . (Co... |
bj-ax6e 31842 | Proof of ~ ax6e (hence ~ a... |
bj-extru 31843 | There exists a variable su... |
bj-spimevw 31844 | Existential introduction, ... |
bj-spnfw 31845 | Theorem close to a closed ... |
bj-cbvexiw 31846 | Change bound variable. Th... |
bj-cbvexivw 31847 | Change bound variable. Th... |
bj-modald 31848 | A short form of the axiom ... |
bj-denot 31849 | A weakening of ~ ax-6 and ... |
bj-eqs 31850 | A lemma for substitutions,... |
bj-cbvexw 31851 | Change bound variable. Th... |
bj-ax12w 31852 | The general statement that... |
bj-elequ2g 31853 | A form of ~ elequ2 with a ... |
bj-ax89 31854 | A theorem which could be u... |
bj-elequ12 31855 | An identity law for the no... |
bj-cleljusti 31856 | One direction of ~ cleljus... |
bj-alcomexcom 31857 | Commutation of universal q... |
bj-hbalt 31858 | Closed form of ~ hbal . W... |
axc11n11 31859 | Proof of ~ axc11n from { ~... |
axc11n11r 31860 | Proof of ~ axc11n from { ~... |
bj-axc16g16 31861 | Proof of ~ axc16g from { ~... |
bj-ax12v3 31862 | A weak version of ~ ax-12 ... |
bj-ax12v3ALT 31863 | Alternate proof of ~ bj-ax... |
bj-sb 31864 | A weak variant of ~ sbid2 ... |
bj-modalbe 31865 | The predicate-calculus ver... |
bj-spst 31866 | Closed form of ~ sps . On... |
bj-19.21bit 31867 | Closed form of ~ 19.21bi .... |
bj-19.23bit 31868 | Closed form of ~ 19.23bi .... |
bj-nexrt 31869 | Closed form of ~ nexr . C... |
bj-alrim 31870 | Closed form of ~ alrimi . ... |
bj-alrim2 31871 | Imported form (uncurried f... |
bj-nfdt0 31872 | A theorem close to a close... |
bj-nfdt 31873 | Closed form of ~ nf5d and ... |
bj-nexdt 31874 | Closed form of ~ nexd . (... |
bj-nexdvt 31875 | Closed form of ~ nexdv . ... |
bj-19.3t 31876 | Closed form of ~ 19.3 . (... |
bj-alexbiex 31877 | Adding a second quantifier... |
bj-exexbiex 31878 | Adding a second quantifier... |
bj-alalbial 31879 | Adding a second quantifier... |
bj-exalbial 31880 | Adding a second quantifier... |
bj-19.9htbi 31881 | Strengthening ~ 19.9ht by ... |
bj-hbntbi 31882 | Strengthening ~ hbnt by re... |
bj-biexal1 31883 | A general FOL biconditiona... |
bj-biexal2 31884 | A general FOL biconditiona... |
bj-biexal3 31885 | A general FOL biconditiona... |
bj-bialal 31886 | A general FOL biconditiona... |
bj-biexex 31887 | A general FOL biconditiona... |
bj-hbext 31888 | Closed form of ~ hbex . (... |
bj-nfalt 31889 | Closed form of ~ nfal . (... |
bj-nfext 31890 | Closed form of ~ nfex . (... |
bj-eeanvw 31891 | Version of ~ eeanv with a ... |
bj-modal4e 31892 | Dual statement of ~ hba1 (... |
bj-modalb 31893 | A short form of the axiom ... |
bj-axc10 31894 | Alternate (shorter) proof ... |
bj-alequex 31895 | A fol lemma. Can be used ... |
bj-spimt2 31896 | A step in the proof of ~ s... |
bj-cbv3ta 31897 | Closed form of ~ cbv3 . (... |
bj-cbv3tb 31898 | Closed form of ~ cbv3 . (... |
bj-hbsb3t 31899 | A theorem close to a close... |
bj-hbsb3 31900 | Shorter proof of ~ hbsb3 .... |
bj-nfs1t 31901 | A theorem close to a close... |
bj-nfs1t2 31902 | A theorem close to a close... |
bj-nfs1 31903 | Shorter proof of ~ nfs1 (t... |
bj-axc10v 31904 | Version of ~ axc10 with a ... |
bj-spimtv 31905 | Version of ~ spimt with a ... |
bj-spimedv 31906 | Version of ~ spimed with a... |
bj-spimev 31907 | Version of ~ spime with a ... |
bj-spimvv 31908 | Version of ~ spimv and ~ s... |
bj-spimevv 31909 | Version of ~ spimev with a... |
bj-spvv 31910 | Version of ~ spv with a dv... |
bj-speiv 31911 | Version of ~ spei with a d... |
bj-chvarv 31912 | Version of ~ chvar with a ... |
bj-chvarvv 31913 | Version of ~ chvarv with a... |
bj-cbv3v2 31914 | Version of ~ cbv3 with two... |
bj-cbv3hv2 31915 | Version of ~ cbv3h with tw... |
bj-cbv1v 31916 | Version of ~ cbv1 with a d... |
bj-cbv1hv 31917 | Version of ~ cbv1h with a ... |
bj-cbv2hv 31918 | Version of ~ cbv2h with a ... |
bj-cbv2v 31919 | Version of ~ cbv2 with a d... |
bj-cbvalvv 31920 | Version of ~ cbvalv with a... |
bj-cbvexvv 31921 | Version of ~ cbvexv with a... |
bj-cbvaldv 31922 | Version of ~ cbvald with a... |
bj-cbvexdv 31923 | Version of ~ cbvexd with a... |
bj-cbval2v 31924 | Version of ~ cbval2 with a... |
bj-cbvex2v 31925 | Version of ~ cbvex2 with a... |
bj-cbval2vv 31926 | Version of ~ cbval2v with ... |
bj-cbvex2vv 31927 | Version of ~ cbvex2v with ... |
bj-cbvaldvav 31928 | Version of ~ cbvaldva with... |
bj-cbvexdvav 31929 | Version of ~ cbvexdva with... |
bj-cbvex4vv 31930 | Version of ~ cbvex4v with ... |
bj-equsalv 31931 | Version of ~ equsal with a... |
bj-equsalhv 31932 | Version of ~ equsalh with ... |
bj-axc11nv 31933 | Version of ~ axc11n with a... |
bj-aecomsv 31934 | Version of ~ aecoms with a... |
bj-axc11v 31935 | Version of ~ axc11 with a ... |
bj-dral1v 31936 | Version of ~ dral1 with a ... |
bj-drex1v 31937 | Version of ~ drex1 with a ... |
bj-drnf1v 31938 | Version of ~ drnf1 with a ... |
bj-drnf2v 31939 | Version of ~ drnf2 with a ... |
bj-equs45fv 31940 | Version of ~ equs45f with ... |
bj-sb2v 31941 | Version of ~ sb2 with a dv... |
bj-stdpc4v 31942 | Version of ~ stdpc4 with a... |
bj-2stdpc4v 31943 | Version of ~ 2stdpc4 with ... |
bj-sb3v 31944 | Version of ~ sb3 with a dv... |
bj-sb4v 31945 | Version of ~ sb4 with a dv... |
bj-hbs1 31946 | Version of ~ hbsb2 with a ... |
bj-nfs1v 31947 | Version of ~ nfsb2 with a ... |
bj-hbsb2av 31948 | Version of ~ hbsb2a with a... |
bj-hbsb3v 31949 | Version of ~ hbsb3 with a ... |
bj-equsb1v 31950 | Version of ~ equsb1 with a... |
bj-sbftv 31951 | Version of ~ sbft with a d... |
bj-sbfv 31952 | Version of ~ sbf with a dv... |
bj-sbfvv 31953 | Version of ~ sbf with two ... |
bj-sbtv 31954 | Version of ~ sbt with a dv... |
bj-sb6 31955 | Remove dependency on ~ ax-... |
bj-sb5 31956 | Remove dependency on ~ ax-... |
bj-axext3 31957 | Remove dependency on ~ ax-... |
bj-axext4 31958 | Remove dependency on ~ ax-... |
bj-hbab1 31959 | Remove dependency on ~ ax-... |
bj-nfsab1 31960 | Remove dependency on ~ ax-... |
bj-abeq2 31961 | Remove dependency on ~ ax-... |
bj-abeq1 31962 | Remove dependency on ~ ax-... |
bj-abbi 31963 | Remove dependency on ~ ax-... |
bj-abbi2i 31964 | Remove dependency on ~ ax-... |
bj-abbii 31965 | Remove dependency on ~ ax-... |
bj-abbid 31966 | Remove dependency on ~ ax-... |
bj-abbidv 31967 | Remove dependency on ~ ax-... |
bj-abbi2dv 31968 | Remove dependency on ~ ax-... |
bj-abbi1dv 31969 | Remove dependency on ~ ax-... |
bj-abid2 31970 | Remove dependency on ~ ax-... |
bj-clabel 31971 | Remove dependency on ~ ax-... |
bj-sbab 31972 | Remove dependency on ~ ax-... |
bj-nfab1 31973 | Remove dependency on ~ ax-... |
bj-vjust 31974 | Remove dependency on ~ ax-... |
bj-cdeqab 31975 | Remove dependency on ~ ax-... |
bj-axrep1 31976 | Remove dependency on ~ ax-... |
bj-axrep2 31977 | Remove dependency on ~ ax-... |
bj-axrep3 31978 | Remove dependency on ~ ax-... |
bj-axrep4 31979 | Remove dependency on ~ ax-... |
bj-axrep5 31980 | Remove dependency on ~ ax-... |
bj-axsep 31981 | Remove dependency on ~ ax-... |
bj-nalset 31982 | Remove dependency on ~ ax-... |
bj-zfpow 31983 | Remove dependency on ~ ax-... |
bj-el 31984 | Remove dependency on ~ ax-... |
bj-dtru 31985 | Remove dependency on ~ ax-... |
bj-axc16b 31986 | Remove dependency on ~ ax-... |
bj-eunex 31987 | Remove dependency on ~ ax-... |
bj-dtrucor 31988 | Remove dependency on ~ ax-... |
bj-dtrucor2v 31989 | Version of ~ dtrucor2 with... |
bj-dvdemo1 31990 | Remove dependency on ~ ax-... |
bj-dvdemo2 31991 | Remove dependency on ~ ax-... |
bj-sb3b 31992 | Simplified definition of s... |
bj-hbaeb2 31993 | Biconditional version of a... |
bj-hbaeb 31994 | Biconditional version of ~... |
bj-hbnaeb 31995 | Biconditional version of ~... |
bj-dvv 31996 | A special instance of ~ bj... |
bj-equsal1t 31997 | Duplication of ~ wl-equsal... |
bj-equsal1ti 31998 | Inference associated with ... |
bj-equsal1 31999 | One direction of ~ equsal ... |
bj-equsal2 32000 | One direction of ~ equsal ... |
bj-equsal 32001 | Shorter proof of ~ equsal ... |
stdpc5t 32002 | Closed form of ~ stdpc5 . ... |
bj-stdpc5 32003 | More direct proof of ~ std... |
2stdpc5 32004 | A double ~ stdpc5 (one dir... |
bj-19.21t 32005 | Proof of ~ 19.21t from ~ s... |
exlimii 32006 | Inference associated with ... |
ax11-pm 32007 | Proof of ~ ax-11 similar t... |
ax6er 32008 | Another form of ~ ax6e . ... |
exlimiieq1 32009 | Inferring a theorem when i... |
exlimiieq2 32010 | Inferring a theorem when i... |
ax11-pm2 32011 | Proof of ~ ax-11 from the ... |
bj-sbsb 32012 | Biconditional showing two ... |
bj-dfsb2 32013 | Alternate (dual) definitio... |
bj-sbf3 32014 | Substitution has no effect... |
bj-sbf4 32015 | Substitution has no effect... |
bj-sbnf 32016 | Move non-free predicate in... |
bj-eu3f 32017 | Version of ~ eu3v where th... |
bj-eumo0 32018 | Existential uniqueness imp... |
bj-nfdiOLD 32019 | Obsolete proof temporarily... |
bj-sbieOLD 32020 | Obsolete proof temporarily... |
bj-sbidmOLD 32021 | Obsolete proof temporarily... |
bj-mo3OLD 32022 | Obsolete proof temporarily... |
bj-syl66ib 32023 | A mixed syllogism inferenc... |
bj-nfbiit 32024 | Closed form of ~ nfbii (th... |
bj-nfimt 32025 | Closed form of ~ nfim . (... |
bj-nfimt2 32026 | Uncurried form of ~ bj-nfi... |
bj-dvelimdv 32027 | Deduction form of ~ dvelim... |
bj-dvelimdv1 32028 | Curried form (exported for... |
bj-dvelimv 32029 | A version of ~ dvelim usin... |
bj-nfeel2 32030 | Non-freeness in an equalit... |
bj-axc14nf 32031 | Proof of a version of ~ ax... |
bj-axc14 32032 | Alternate proof of ~ axc14... |
eliminable1 32033 | A theorem used to prove th... |
eliminable2a 32034 | A theorem used to prove th... |
eliminable2b 32035 | A theorem used to prove th... |
eliminable2c 32036 | A theorem used to prove th... |
eliminable3a 32037 | A theorem used to prove th... |
eliminable3b 32038 | A theorem used to prove th... |
bj-termab 32039 | Every class can be written... |
bj-eleq1w 32040 | Weaker version of ~ eleq1 ... |
bj-eleq2w 32041 | Weaker version of ~ eleq2 ... |
bj-clelsb3 32042 | Remove dependency on ~ ax-... |
bj-hblem 32043 | Remove dependency on ~ ax-... |
bj-nfcjust 32044 | Remove dependency on ~ ax-... |
bj-nfcrii 32045 | Remove dependency on ~ ax-... |
bj-nfcri 32046 | Remove dependency on ~ ax-... |
bj-nfnfc 32047 | Remove dependency on ~ ax-... |
bj-vexwt 32048 | Closed form of ~ bj-vexw .... |
bj-vexw 32049 | If ` ph ` is a theorem, th... |
bj-vexwvt 32050 | Closed form of ~ bj-vexwv ... |
bj-vexwv 32051 | Version of ~ bj-vexw with ... |
bj-denotes 32052 | This would be the justific... |
bj-issetwt 32053 | Closed form of ~ bj-issetw... |
bj-issetw 32054 | The closest one can get to... |
bj-elissetv 32055 | Version of ~ bj-elisset wi... |
bj-elisset 32056 | Remove from ~ elisset depe... |
bj-issetiv 32057 | Version of ~ bj-isseti wit... |
bj-isseti 32058 | Remove from ~ isseti depen... |
bj-ralvw 32059 | A weak version of ~ ralv n... |
bj-rexvwv 32060 | A weak version of ~ rexv n... |
bj-rababwv 32061 | A weak version of ~ rabab ... |
bj-ralcom4 32062 | Remove from ~ ralcom4 depe... |
bj-rexcom4 32063 | Remove from ~ rexcom4 depe... |
bj-rexcom4a 32064 | Remove from ~ rexcom4a dep... |
bj-rexcom4bv 32065 | Version of ~ bj-rexcom4b w... |
bj-rexcom4b 32066 | Remove from ~ rexcom4b dep... |
bj-ceqsalt0 32067 | The FOL content of ~ ceqsa... |
bj-ceqsalt1 32068 | The FOL content of ~ ceqsa... |
bj-ceqsalt 32069 | Remove from ~ ceqsalt depe... |
bj-ceqsaltv 32070 | Version of ~ bj-ceqsalt wi... |
bj-ceqsalg0 32071 | The FOL content of ~ ceqsa... |
bj-ceqsalg 32072 | Remove from ~ ceqsalg depe... |
bj-ceqsalgALT 32073 | Alternate proof of ~ bj-ce... |
bj-ceqsalgv 32074 | Version of ~ bj-ceqsalg wi... |
bj-ceqsalgvALT 32075 | Alternate proof of ~ bj-ce... |
bj-ceqsal 32076 | Remove from ~ ceqsal depen... |
bj-ceqsalv 32077 | Remove from ~ ceqsalv depe... |
bj-spcimdv 32078 | Remove from ~ spcimdv depe... |
bj-nfcsym 32079 | The class-form not-free pr... |
bj-ax8 32080 | Proof of ~ ax-8 from ~ df-... |
bj-df-clel 32081 | Candidate definition for ~... |
bj-dfclel 32082 | Characterization of the el... |
bj-ax9 32083 | Proof of ~ ax-9 from ~ ax-... |
bj-cleqhyp 32084 | The hypothesis of ~ bj-df-... |
bj-df-cleq 32085 | Candidate definition for ~... |
bj-dfcleq 32086 | Proof of class extensional... |
bj-sbeqALT 32087 | Substitution in an equalit... |
bj-sbeq 32088 | Distribute proper substitu... |
bj-sbceqgALT 32089 | Distribute proper substitu... |
bj-csbsnlem 32090 | Lemma for ~ bj-csbsn (in t... |
bj-csbsn 32091 | Substitution in a singleto... |
bj-sbel1 32092 | Version of ~ sbcel1g when ... |
bj-abv 32093 | The class of sets verifyin... |
bj-ab0 32094 | The class of sets verifyin... |
bj-abf 32095 | Shorter proof of ~ abf (wh... |
bj-csbprc 32096 | More direct proof of ~ csb... |
bj-exlimmpi 32097 | Lemma for ~ bj-vtoclg1f1 (... |
bj-exlimmpbi 32098 | Lemma for theorems of the ... |
bj-exlimmpbir 32099 | Lemma for theorems of the ... |
bj-vtoclf 32100 | Remove dependency on ~ ax-... |
bj-vtocl 32101 | Remove dependency on ~ ax-... |
bj-vtoclg1f1 32102 | The FOL content of ~ vtocl... |
bj-vtoclg1f 32103 | Reprove ~ vtoclg1f from ~ ... |
bj-vtoclg1fv 32104 | Version of ~ bj-vtoclg1f w... |
bj-rabbida2 32105 | Version of ~ rabbidva2 wit... |
bj-rabbida 32106 | Version of ~ rabbidva with... |
bj-rabbid 32107 | Version of ~ rabbidv with ... |
bj-rabeqd 32108 | Deduction form of ~ rabeq ... |
bj-rabeqbid 32109 | Version of ~ rabeqbidv wit... |
bj-rabeqbida 32110 | Version of ~ rabeqbidva wi... |
bj-seex 32111 | Version of ~ seex with a d... |
bj-nfcf 32112 | Version of ~ df-nfc with a... |
bj-axsep2 32113 | Remove dependency on ~ ax-... |
bj-unrab 32114 | Generalization of ~ unrab ... |
bj-inrab 32115 | Generalization of ~ inrab ... |
bj-inrab2 32116 | Shorter proof of ~ inrab .... |
bj-inrab3 32117 | Generalization of ~ dfrab3... |
bj-rabtr 32118 | Restricted class abstracti... |
bj-rabtrALT 32119 | Alternate proof of ~ bj-ra... |
bj-rabtrALTALT 32120 | Alternate proof of ~ bj-ra... |
bj-rabtrAUTO 32121 | Proof of ~ bj-rabtr found ... |
bj-ru0 32124 | The FOL part of Russell's ... |
bj-ru1 32125 | A version of Russell's par... |
bj-ru 32126 | Remove dependency on ~ ax-... |
bj-n0i 32127 | Inference associated with ... |
bj-nel0 32128 | From the general negation ... |
bj-disjcsn 32129 | A class is disjoint from i... |
bj-disjsn01 32130 | Disjointness of the single... |
bj-1ex 32131 | ` 1o ` is a set. (Contrib... |
bj-2ex 32132 | ` 2o ` is a set. (Contrib... |
bj-0nel1 32133 | The empty set does not bel... |
bj-1nel0 32134 | ` 1o ` does not belong to ... |
bj-xpimasn 32135 | The image of a singleton, ... |
bj-xpima1sn 32136 | The image of a singleton b... |
bj-xpima1snALT 32137 | Alternate proof of ~ bj-xp... |
bj-xpima2sn 32138 | The image of a singleton b... |
bj-xpnzex 32139 | If the first factor of a p... |
bj-xpexg2 32140 | Exported form (curried for... |
bj-xpnzexb 32141 | If the first factor of a p... |
bj-cleq 32142 | Substitution property for ... |
bj-sels 32143 | If a class is a set, then ... |
bj-snsetex 32144 | The class of sets "whose s... |
bj-clex 32145 | Sethood of certain classes... |
bj-sngleq 32148 | Substitution property for ... |
bj-elsngl 32149 | Characterization of the el... |
bj-snglc 32150 | Characterization of the el... |
bj-snglss 32151 | The singletonization of a ... |
bj-0nelsngl 32152 | The empty set is not a mem... |
bj-snglinv 32153 | Inverse of singletonizatio... |
bj-snglex 32154 | A class is a set if and on... |
bj-tageq 32157 | Substitution property for ... |
bj-eltag 32158 | Characterization of the el... |
bj-0eltag 32159 | The empty set belongs to t... |
bj-tagn0 32160 | The tagging of a class is ... |
bj-tagss 32161 | The tagging of a class is ... |
bj-snglsstag 32162 | The singletonization is in... |
bj-sngltagi 32163 | The singletonization is in... |
bj-sngltag 32164 | The singletonization and t... |
bj-tagci 32165 | Characterization of the el... |
bj-tagcg 32166 | Characterization of the el... |
bj-taginv 32167 | Inverse of tagging. (Cont... |
bj-tagex 32168 | A class is a set if and on... |
bj-xtageq 32169 | The products of a given cl... |
bj-xtagex 32170 | The product of a set and t... |
bj-projeq 32173 | Substitution property for ... |
bj-projeq2 32174 | Substitution property for ... |
bj-projun 32175 | The class projection on a ... |
bj-projex 32176 | Sethood of the class proje... |
bj-projval 32177 | Value of the class project... |
bj-1upleq 32180 | Substitution property for ... |
bj-pr1eq 32183 | Substitution property for ... |
bj-pr1un 32184 | The first projection prese... |
bj-pr1val 32185 | Value of the first project... |
bj-pr11val 32186 | Value of the first project... |
bj-pr1ex 32187 | Sethood of the first proje... |
bj-1uplth 32188 | The characteristic propert... |
bj-1uplex 32189 | A monuple is a set if and ... |
bj-1upln0 32190 | A monuple is nonempty. (C... |
bj-2upleq 32193 | Substitution property for ... |
bj-pr21val 32194 | Value of the first project... |
bj-pr2eq 32197 | Substitution property for ... |
bj-pr2un 32198 | The second projection pres... |
bj-pr2val 32199 | Value of the second projec... |
bj-pr22val 32200 | Value of the second projec... |
bj-pr2ex 32201 | Sethood of the second proj... |
bj-2uplth 32202 | The characteristic propert... |
bj-2uplex 32203 | A couple is a set if and o... |
bj-2upln0 32204 | A couple is nonempty. (Co... |
bj-2upln1upl 32205 | A couple is never equal to... |
bj-vjust2 32206 | Justification theorem for ... |
bj-df-v 32207 | Alternate definition of th... |
bj-df-nul 32208 | Alternate definition of th... |
bj-nul 32209 | Two formulations of the ax... |
bj-nuliota 32210 | Definition of the empty se... |
bj-nuliotaALT 32211 | Alternate proof of ~ bj-nu... |
bj-vtoclgfALT 32212 | Alternate proof of ~ vtocl... |
bj-pwcfsdom 32213 | Remove hypothesis from ~ p... |
bj-grur1 32214 | Remove hypothesis from ~ g... |
bj-rest00 32215 | An elementwise intersectio... |
bj-restsn 32216 | An elementwise intersectio... |
bj-restsnss 32217 | Special case of ~ bj-rests... |
bj-restsnss2 32218 | Special case of ~ bj-rests... |
bj-restsn0 32219 | An elementwise intersectio... |
bj-restsn10 32220 | Special case of ~ bj-rests... |
bj-restsnid 32221 | The elementwise intersecti... |
bj-rest10 32222 | An elementwise intersectio... |
bj-rest10b 32223 | Alternate version of ~ bj-... |
bj-restn0 32224 | An elementwise intersectio... |
bj-restn0b 32225 | Alternate version of ~ bj-... |
bj-restpw 32226 | The elementwise intersecti... |
bj-rest0 32227 | An elementwise intersectio... |
bj-restb 32228 | An elementwise intersectio... |
bj-restv 32229 | An elementwise intersectio... |
bj-resta 32230 | An elementwise intersectio... |
bj-restuni 32231 | The union of an elementwis... |
bj-restuni2 32232 | The union of an elementwis... |
bj-restreg 32233 | A reformulation of the axi... |
bj-toptopon2 32234 | A topology is the same thi... |
bj-topontopon 32235 | A topology on a set is a t... |
bj-funtopon 32236 | ` TopOn ` is a function. ... |
bj-elpw3 32237 | A variant of ~ elpwg . (C... |
bj-sspwpw 32238 | The union of a set is incl... |
bj-sspwpwab 32239 | The class of families whos... |
bj-sspwpweq 32240 | The class of families whos... |
bj-toponss 32241 | The set of topologies on a... |
bj-dmtopon 32242 | The domain of ` TopOn ` is... |
bj-fntopon 32243 | ` TopOn ` is a function wi... |
bj-toprntopon 32244 | A topology is the same thi... |
bj-xnex 32245 | Lemma for ~ snnex and ~ bj... |
bj-pwnex 32246 | The class of all power set... |
bj-topnex 32247 | The class of all topologie... |
bj-0nelmpt 32250 | The empty set is not an el... |
bj-mptval 32251 | Value of a function given ... |
bj-dfmpt2a 32252 | An equivalent definition o... |
bj-mpt2mptALT 32253 | Alternate proof of ~ mpt2m... |
bj-elid 32262 | Characterization of the el... |
bj-elid2 32263 | Characterization of the el... |
bj-elid3 32264 | Characterization of the el... |
bj-diagval 32267 | Value of the diagonal. (C... |
bj-eldiag 32268 | Characterization of the el... |
bj-eldiag2 32269 | Characterization of the el... |
bj-inftyexpiinv 32272 | Utility theorem for the in... |
bj-inftyexpiinj 32273 | Injectivity of the paramet... |
bj-inftyexpidisj 32274 | An element of the circle a... |
bj-ccinftydisj 32277 | The circle at infinity is ... |
bj-elccinfty 32278 | A lemma for infinite exten... |
bj-ccssccbar 32281 | Complex numbers are extend... |
bj-ccinftyssccbar 32282 | Infinite extended complex ... |
bj-pinftyccb 32285 | The class ` pinfty ` is an... |
bj-pinftynrr 32286 | The extended complex numbe... |
bj-minftyccb 32289 | The class ` minfty ` is an... |
bj-minftynrr 32290 | The extended complex numbe... |
bj-pinftynminfty 32291 | The extended complex numbe... |
bj-rrhatsscchat 32300 | The real projective line i... |
bj-cmnssmnd 32313 | Commutative monoids are mo... |
bj-cmnssmndel 32314 | Commutative monoids are mo... |
bj-ablssgrp 32315 | Abelian groups are groups.... |
bj-ablssgrpel 32316 | Abelian groups are groups ... |
bj-ablsscmn 32317 | Abelian groups are commuta... |
bj-ablsscmnel 32318 | Abelian groups are commuta... |
bj-modssabl 32319 | (The additive groups of) m... |
bj-vecssmod 32320 | Vector spaces are modules.... |
bj-vecssmodel 32321 | Vector spaces are modules ... |
bj-finsumval0 32324 | Value of a finite sum. (C... |
bj-rrvecssvec 32327 | Real vector spaces are vec... |
bj-rrvecssvecel 32328 | Real vector spaces are vec... |
bj-rrvecsscmn 32329 | (The additive groups of) r... |
bj-rrvecsscmnel 32330 | (The additive groups of) r... |
bj-subcom 32331 | A consequence of commutati... |
bj-ldiv 32332 | Left-division. (Contribut... |
bj-rdiv 32333 | Right-division. (Contribu... |
bj-mdiv 32334 | A division law. (Contribu... |
bj-lineq 32335 | Solution of a (scalar) lin... |
bj-lineqi 32336 | Solution of a (scalar) lin... |
bj-bary1lem 32337 | A lemma for barycentric co... |
bj-bary1lem1 32338 | Existence and uniqueness (... |
bj-bary1 32339 | Barycentric coordinates in... |
taupilem3 32342 | Lemma for tau-related theo... |
taupilemrplb 32343 | A set of positive reals ha... |
taupilem1 32344 | Lemma for ~ taupi . A pos... |
taupilem2 32345 | Lemma for ~ taupi . The s... |
taupi 32346 | Relationship between ` _ta... |
csbdif 32347 | Distribution of class subs... |
csbpredg 32348 | Move class substitution in... |
csbwrecsg 32349 | Move class substitution in... |
csbrecsg 32350 | Move class substitution in... |
csbrdgg 32351 | Move class substitution in... |
csboprabg 32352 | Move class substitution in... |
csbmpt22g 32353 | Move class substitution in... |
mpnanrd 32354 | Eliminate the right side o... |
con1bii2 32355 | A contraposition inference... |
con2bii2 32356 | A contraposition inference... |
vtoclefex 32357 | Implicit substitution of a... |
rnmptsn 32358 | The range of a function ma... |
f1omptsnlem 32359 | This is the core of the pr... |
f1omptsn 32360 | A function mapping to sing... |
mptsnunlem 32361 | This is the core of the pr... |
mptsnun 32362 | A class ` B ` is equal to ... |
dissneqlem 32363 | This is the core of the pr... |
dissneq 32364 | Any topology that contains... |
exlimim 32365 | Closed form of ~ exlimimd ... |
exlimimd 32366 | Existential elimination ru... |
exlimimdd 32367 | Existential elimination ru... |
exellim 32368 | Closed form of ~ exellimdd... |
exellimddv 32369 | Eliminate an antecedent wh... |
topdifinfindis 32370 | Part of Exercise 3 of [Mun... |
topdifinffinlem 32371 | This is the core of the pr... |
topdifinffin 32372 | Part of Exercise 3 of [Mun... |
topdifinf 32373 | Part of Exercise 3 of [Mun... |
topdifinfeq 32374 | Two different ways of defi... |
icorempt2 32375 | Closed-below, open-above i... |
icoreresf 32376 | Closed-below, open-above i... |
icoreval 32377 | Value of the closed-below,... |
icoreelrnab 32378 | Elementhood in the set of ... |
isbasisrelowllem1 32379 | Lemma for ~ isbasisrelowl ... |
isbasisrelowllem2 32380 | Lemma for ~ isbasisrelowl ... |
icoreclin 32381 | The set of closed-below, o... |
isbasisrelowl 32382 | The set of all closed-belo... |
icoreunrn 32383 | The union of all closed-be... |
istoprelowl 32384 | The set of all closed-belo... |
icoreelrn 32385 | A class abstraction which ... |
iooelexlt 32386 | An element of an open inte... |
relowlssretop 32387 | The lower limit topology o... |
relowlpssretop 32388 | The lower limit topology o... |
sucneqond 32389 | Inequality of an ordinal s... |
sucneqoni 32390 | Inequality of an ordinal s... |
onsucuni3 32391 | If an ordinal number has a... |
1oequni2o 32392 | The ordinal number ` 1o ` ... |
rdgsucuni 32393 | If an ordinal number has a... |
rdgeqoa 32394 | If a recursive function wi... |
elxp8 32395 | Membership in a Cartesian ... |
dffinxpf 32398 | This theorem is the same a... |
finxpeq1 32399 | Equality theorem for Carte... |
finxpeq2 32400 | Equality theorem for Carte... |
csbfinxpg 32401 | Distribute proper substitu... |
finxpreclem1 32402 | Lemma for ` ^^ ` recursion... |
finxpreclem2 32403 | Lemma for ` ^^ ` recursion... |
finxp0 32404 | The value of Cartesian exp... |
finxp1o 32405 | The value of Cartesian exp... |
finxpreclem3 32406 | Lemma for ` ^^ ` recursion... |
finxpreclem4 32407 | Lemma for ` ^^ ` recursion... |
finxpreclem5 32408 | Lemma for ` ^^ ` recursion... |
finxpreclem6 32409 | Lemma for ` ^^ ` recursion... |
finxpsuclem 32410 | Lemma for ~ finxpsuc . (C... |
finxpsuc 32411 | The value of Cartesian exp... |
finxp2o 32412 | The value of Cartesian exp... |
finxp3o 32413 | The value of Cartesian exp... |
finxpnom 32414 | Cartesian exponentiation w... |
finxp00 32415 | Cartesian exponentiation o... |
wl-section-prop 32416 | Intuitionistic logic is no... |
wl-section-boot 32420 | In this section, I provide... |
wl-imim1i 32421 | Inference adding common co... |
wl-syl 32422 | An inference version of th... |
wl-syl5 32423 | A syllogism rule of infere... |
wl-pm2.18d 32424 | Deduction based on reducti... |
wl-con4i 32425 | Inference rule. Copy of ~... |
wl-pm2.24i 32426 | Inference rule. Copy of ~... |
wl-a1i 32427 | Inference rule. Copy of ~... |
wl-mpi 32428 | A nested modus ponens infe... |
wl-imim2i 32429 | Inference adding common an... |
wl-syl6 32430 | A syllogism rule of infere... |
wl-ax3 32431 | ~ ax-3 proved from Lukasie... |
wl-ax1 32432 | ~ ax-1 proved from Lukasie... |
wl-pm2.27 32433 | This theorem, called "Asse... |
wl-com12 32434 | Inference that swaps (comm... |
wl-pm2.21 32435 | From a wff and its negatio... |
wl-con1i 32436 | A contraposition inference... |
wl-ja 32437 | Inference joining the ante... |
wl-imim2 32438 | A closed form of syllogism... |
wl-a1d 32439 | Deduction introducing an e... |
wl-ax2 32440 | ~ ax-2 proved from Lukasie... |
wl-id 32441 | Principle of identity. Th... |
wl-notnotr 32442 | Converse of double negatio... |
wl-pm2.04 32443 | Swap antecedents. Theorem... |
wl-section-impchain 32444 | An implication like ` ( ps... |
wl-impchain-mp-x 32445 | This series of theorems pr... |
wl-impchain-mp-0 32446 | This theorem is the start ... |
wl-impchain-mp-1 32447 | This theorem is in fact a ... |
wl-impchain-mp-2 32448 | This theorem is in fact a ... |
wl-impchain-com-1.x 32449 | It is often convenient to ... |
wl-impchain-com-1.1 32450 | A degenerate form of antec... |
wl-impchain-com-1.2 32451 | This theorem is in fact a ... |
wl-impchain-com-1.3 32452 | This theorem is in fact a ... |
wl-impchain-com-1.4 32453 | This theorem is in fact a ... |
wl-impchain-com-n.m 32454 | This series of theorems al... |
wl-impchain-com-2.3 32455 | This theorem is in fact a ... |
wl-impchain-com-2.4 32456 | This theorem is in fact a ... |
wl-impchain-com-3.2.1 32457 | This theorem is in fact a ... |
wl-impchain-a1-x 32458 | If an implication chain is... |
wl-impchain-a1-1 32459 | Inference rule, a copy of ... |
wl-impchain-a1-2 32460 | Inference rule, a copy of ... |
wl-impchain-a1-3 32461 | Inference rule, a copy of ... |
wl-section-nf 32462 | The current definition of ... |
wl-nf-nf2 32463 | By ~ ax-10 the definition ... |
wl-nf2-nf 32464 | ~ hba1 is sufficient to le... |
wl-ax13lem1 32466 | A version of ~ ax-wl-13v w... |
wl-jarri 32467 | Dropping a nested antecede... |
wl-jarli 32468 | Dropping a nested conseque... |
wl-mps 32469 | Replacing a nested consequ... |
wl-syls1 32470 | Replacing a nested consequ... |
wl-syls2 32471 | Replacing a nested anteced... |
wl-embant 32472 | A true wff can always be a... |
wl-orel12 32473 | In a conjunctive normal fo... |
wl-cases2-dnf 32474 | A particular instance of ~... |
wl-dfnan2 32475 | An alternative definition ... |
wl-nancom 32476 | The 'nand' operator commut... |
wl-nannan 32477 | Lemma for handling nested ... |
wl-nannot 32478 | Show equivalence between n... |
wl-nanbi1 32479 | Introduce a right anti-con... |
wl-nanbi2 32480 | Introduce a left anti-conj... |
wl-naev 32481 | If some set variables can ... |
wl-hbae1 32482 | This specialization of ~ h... |
wl-naevhba1v 32483 | An instance of ~ hbn1w app... |
wl-hbnaev 32484 | Any variable is free in ` ... |
wl-spae 32485 | Prove an instance of ~ sp ... |
wl-cbv3vv 32486 | Avoiding ~ ax-11 . (Contr... |
wl-speqv 32487 | Under the assumption ` -. ... |
wl-19.8eqv 32488 | Under the assumption ` -. ... |
wl-19.2reqv 32489 | Under the assumption ` -. ... |
wl-dveeq12 32490 | The current form of ~ ax-1... |
wl-nfalv 32491 | If ` x ` is not present in... |
wl-nfimf1 32492 | An antecedent is irrelevan... |
wl-nfnbi 32493 | Being free does not depend... |
wl-nfae1 32494 | Unlike ~ nfae , this speci... |
wl-nfnae1 32495 | Unlike ~ nfnae , this spec... |
wl-aetr 32496 | A transitive law for varia... |
wl-dral1d 32497 | A version of ~ dral1 with ... |
wl-cbvalnaed 32498 | ~ wl-cbvalnae with a conte... |
wl-cbvalnae 32499 | A more general version of ... |
wl-exeq 32500 | The semantics of ` E. x y ... |
wl-aleq 32501 | The semantics of ` A. x y ... |
wl-nfeqfb 32502 | Extend ~ nfeqf to an equiv... |
wl-nfs1t 32503 | If ` y ` is not free in ` ... |
wl-equsald 32504 | Deduction version of ~ equ... |
wl-equsal 32505 | A useful equivalence relat... |
wl-equsal1t 32506 | The expression ` x = y ` i... |
wl-equsalcom 32507 | This simple equivalence ea... |
wl-equsal1i 32508 | The antecedent ` x = y ` i... |
wl-sb6rft 32509 | A specialization of ~ wl-e... |
wl-sbrimt 32510 | Substitution with a variab... |
wl-sblimt 32511 | Substitution with a variab... |
wl-sb8t 32512 | Substitution of variable i... |
wl-sb8et 32513 | Substitution of variable i... |
wl-sbhbt 32514 | Closed form of ~ sbhb . C... |
wl-sbnf1 32515 | Two ways expressing that `... |
wl-equsb3 32516 | ~ equsb3 with a distinctor... |
wl-equsb4 32517 | Substitution applied to an... |
wl-sb6nae 32518 | Version of ~ sb6 suitable ... |
wl-sb5nae 32519 | Version of ~ sb5 suitable ... |
wl-2sb6d 32520 | Version of ~ 2sb6 with a c... |
wl-sbcom2d-lem1 32521 | Lemma used to prove ~ wl-s... |
wl-sbcom2d-lem2 32522 | Lemma used to prove ~ wl-s... |
wl-sbcom2d 32523 | Version of ~ sbcom2 with a... |
wl-sbalnae 32524 | A theorem used in eliminat... |
wl-sbal1 32525 | A theorem used in eliminat... |
wl-sbal2 32526 | Move quantifier in and out... |
wl-lem-exsb 32527 | This theorem provides a ba... |
wl-lem-nexmo 32528 | This theorem provides a ba... |
wl-lem-moexsb 32529 | The antecedent ` A. x ( ph... |
wl-alanbii 32530 | This theorem extends ~ ala... |
wl-mo2df 32531 | Version of ~ mo2 with a co... |
wl-mo2tf 32532 | Closed form of ~ mo2 with ... |
wl-eudf 32533 | Version of ~ df-eu with a ... |
wl-eutf 32534 | Closed form of ~ df-eu wit... |
wl-euequ1f 32535 | ~ euequ1 proved with a dis... |
wl-mo2t 32536 | Closed form of ~ mo2 . (C... |
wl-mo3t 32537 | Closed form of ~ mo3 . (C... |
wl-sb8eut 32538 | Substitution of variable i... |
wl-sb8mot 32539 | Substitution of variable i... |
wl-ax11-lem1 32541 | A transitive law for varia... |
wl-ax11-lem2 32542 | Lemma. (Contributed by Wo... |
wl-ax11-lem3 32543 | Lemma. (Contributed by Wo... |
wl-ax11-lem4 32544 | Lemma. (Contributed by Wo... |
wl-ax11-lem5 32545 | Lemma. (Contributed by Wo... |
wl-ax11-lem6 32546 | Lemma. (Contributed by Wo... |
wl-ax11-lem7 32547 | Lemma. (Contributed by Wo... |
wl-ax11-lem8 32548 | Lemma. (Contributed by Wo... |
wl-ax11-lem9 32549 | The easy part when ` x ` c... |
wl-ax11-lem10 32550 | We now have prepared every... |
wl-sbcom3 32551 | Substituting ` y ` for ` x... |
rabiun 32552 | Abstraction restricted to ... |
iundif1 32553 | Indexed union of class dif... |
imadifss 32554 | The difference of images i... |
cureq 32555 | Equality theorem for curry... |
unceq 32556 | Equality theorem for uncur... |
curf 32557 | Functional property of cur... |
uncf 32558 | Functional property of unc... |
curfv 32559 | Value of currying. (Contr... |
uncov 32560 | Value of uncurrying. (Con... |
curunc 32561 | Currying of uncurrying. (... |
unccur 32562 | Uncurrying of currying. (... |
phpreu 32563 | Theorem related to pigeonh... |
finixpnum 32564 | A finite Cartesian product... |
fin2solem 32565 | Lemma for ~ fin2so . (Con... |
fin2so 32566 | Any totally ordered Tarski... |
ltflcei 32567 | Theorem to move the floor ... |
leceifl 32568 | Theorem to move the floor ... |
sin2h 32569 | Half-angle rule for sine. ... |
cos2h 32570 | Half-angle rule for cosine... |
tan2h 32571 | Half-angle rule for tangen... |
pigt3 32572 | ` _pi ` is greater than 3.... |
lindsdom 32573 | A linearly independent set... |
lindsenlbs 32574 | A maximal linearly indepen... |
matunitlindflem1 32575 | One direction of ~ matunit... |
matunitlindflem2 32576 | One direction of ~ matunit... |
matunitlindf 32577 | A matrix over a field is i... |
ptrest 32578 | Expressing a restriction o... |
ptrecube 32579 | Any point in an open set o... |
poimirlem1 32580 | Lemma for ~ poimir - the v... |
poimirlem2 32581 | Lemma for ~ poimir - conse... |
poimirlem3 32582 | Lemma for ~ poimir to add ... |
poimirlem4 32583 | Lemma for ~ poimir connect... |
poimirlem5 32584 | Lemma for ~ poimir to esta... |
poimirlem6 32585 | Lemma for ~ poimir establi... |
poimirlem7 32586 | Lemma for ~ poimir , simil... |
poimirlem8 32587 | Lemma for ~ poimir , estab... |
poimirlem9 32588 | Lemma for ~ poimir , estab... |
poimirlem10 32589 | Lemma for ~ poimir establi... |
poimirlem11 32590 | Lemma for ~ poimir connect... |
poimirlem12 32591 | Lemma for ~ poimir connect... |
poimirlem13 32592 | Lemma for ~ poimir - for a... |
poimirlem14 32593 | Lemma for ~ poimir - for a... |
poimirlem15 32594 | Lemma for ~ poimir , that ... |
poimirlem16 32595 | Lemma for ~ poimir establi... |
poimirlem17 32596 | Lemma for ~ poimir establi... |
poimirlem18 32597 | Lemma for ~ poimir stating... |
poimirlem19 32598 | Lemma for ~ poimir establi... |
poimirlem20 32599 | Lemma for ~ poimir establi... |
poimirlem21 32600 | Lemma for ~ poimir stating... |
poimirlem22 32601 | Lemma for ~ poimir , that ... |
poimirlem23 32602 | Lemma for ~ poimir , two w... |
poimirlem24 32603 | Lemma for ~ poimir , two w... |
poimirlem25 32604 | Lemma for ~ poimir stating... |
poimirlem26 32605 | Lemma for ~ poimir showing... |
poimirlem27 32606 | Lemma for ~ poimir showing... |
poimirlem28 32607 | Lemma for ~ poimir , a var... |
poimirlem29 32608 | Lemma for ~ poimir connect... |
poimirlem30 32609 | Lemma for ~ poimir combini... |
poimirlem31 32610 | Lemma for ~ poimir , assig... |
poimirlem32 32611 | Lemma for ~ poimir , combi... |
poimir 32612 | Poincare-Miranda theorem. ... |
broucube 32613 | Brouwer - or as Kulpa call... |
heicant 32614 | Heine-Cantor theorem: a co... |
opnmbllem0 32615 | Lemma for ~ ismblfin ; cou... |
mblfinlem1 32616 | Lemma for ~ ismblfin , ord... |
mblfinlem2 32617 | Lemma for ~ ismblfin , eff... |
mblfinlem3 32618 | The difference between two... |
mblfinlem4 32619 | Backward direction of ~ is... |
ismblfin 32620 | Measurability in terms of ... |
ovoliunnfl 32621 | ~ ovoliun is incompatible ... |
ex-ovoliunnfl 32622 | Demonstration of ~ ovoliun... |
voliunnfl 32623 | ~ voliun is incompatible w... |
volsupnfl 32624 | ~ volsup is incompatible w... |
0mbf 32625 | The empty function is meas... |
mbfresfi 32626 | Measurability of a piecewi... |
mbfposadd 32627 | If the sum of two measurab... |
cnambfre 32628 | A real-valued, a.e. contin... |
dvtanlem 32629 | Lemma for ~ dvtan - the do... |
dvtan 32630 | Derivative of tangent. (C... |
itg2addnclem 32631 | An alternate expression fo... |
itg2addnclem2 32632 | Lemma for ~ itg2addnc . T... |
itg2addnclem3 32633 | Lemma incomprehensible in ... |
itg2addnc 32634 | Alternate proof of ~ itg2a... |
itg2gt0cn 32635 | ~ itg2gt0 holds on functio... |
ibladdnclem 32636 | Lemma for ~ ibladdnc ; cf ... |
ibladdnc 32637 | Choice-free analogue of ~ ... |
itgaddnclem1 32638 | Lemma for ~ itgaddnc ; cf.... |
itgaddnclem2 32639 | Lemma for ~ itgaddnc ; cf.... |
itgaddnc 32640 | Choice-free analogue of ~ ... |
iblsubnc 32641 | Choice-free analogue of ~ ... |
itgsubnc 32642 | Choice-free analogue of ~ ... |
iblabsnclem 32643 | Lemma for ~ iblabsnc ; cf.... |
iblabsnc 32644 | Choice-free analogue of ~ ... |
iblmulc2nc 32645 | Choice-free analogue of ~ ... |
itgmulc2nclem1 32646 | Lemma for ~ itgmulc2nc ; c... |
itgmulc2nclem2 32647 | Lemma for ~ itgmulc2nc ; c... |
itgmulc2nc 32648 | Choice-free analogue of ~ ... |
itgabsnc 32649 | Choice-free analogue of ~ ... |
bddiblnc 32650 | Choice-free proof of ~ bdd... |
cnicciblnc 32651 | Choice-free proof of ~ cni... |
itggt0cn 32652 | ~ itggt0 holds for continu... |
ftc1cnnclem 32653 | Lemma for ~ ftc1cnnc ; cf.... |
ftc1cnnc 32654 | Choice-free proof of ~ ftc... |
ftc1anclem1 32655 | Lemma for ~ ftc1anc - the ... |
ftc1anclem2 32656 | Lemma for ~ ftc1anc - rest... |
ftc1anclem3 32657 | Lemma for ~ ftc1anc - the ... |
ftc1anclem4 32658 | Lemma for ~ ftc1anc . (Co... |
ftc1anclem5 32659 | Lemma for ~ ftc1anc , the ... |
ftc1anclem6 32660 | Lemma for ~ ftc1anc - cons... |
ftc1anclem7 32661 | Lemma for ~ ftc1anc . (Co... |
ftc1anclem8 32662 | Lemma for ~ ftc1anc . (Co... |
ftc1anc 32663 | ~ ftc1a holds for function... |
ftc2nc 32664 | Choice-free proof of ~ ftc... |
asindmre 32665 | Real part of domain of dif... |
dvasin 32666 | Derivative of arcsine. (C... |
dvacos 32667 | Derivative of arccosine. ... |
dvreasin 32668 | Real derivative of arcsine... |
dvreacos 32669 | Real derivative of arccosi... |
areacirclem1 32670 | Antiderivative of cross-se... |
areacirclem2 32671 | Endpoint-inclusive continu... |
areacirclem3 32672 | Integrability of cross-sec... |
areacirclem4 32673 | Endpoint-inclusive continu... |
areacirclem5 32674 | Finding the cross-section ... |
areacirc 32675 | The area of a circle of ra... |
anim12da 32676 | Conjoin antecedents and co... |
unirep 32677 | Define a quantity whose de... |
cover2 32678 | Two ways of expressing the... |
cover2g 32679 | Two ways of expressing the... |
brabg2 32680 | Relation by a binary relat... |
opelopab3 32681 | Ordered pair membership in... |
cocanfo 32682 | Cancellation of a surjecti... |
brresi 32683 | Restriction of a binary re... |
fnopabeqd 32684 | Equality deduction for fun... |
fvopabf4g 32685 | Function value of an opera... |
eqfnun 32686 | Two functions on ` A u. B ... |
fnopabco 32687 | Composition of a function ... |
opropabco 32688 | Composition of an operator... |
f1opr 32689 | Condition for an operation... |
cocnv 32690 | Composition with a functio... |
f1ocan1fv 32691 | Cancel a composition by a ... |
f1ocan2fv 32692 | Cancel a composition by th... |
inixp 32693 | Intersection of Cartesian ... |
upixp 32694 | Universal property of the ... |
abrexdom 32695 | An indexed set is dominate... |
abrexdom2 32696 | An indexed set is dominate... |
ac6gf 32697 | Axiom of Choice. (Contrib... |
indexa 32698 | If for every element of an... |
indexdom 32699 | If for every element of an... |
frinfm 32700 | A subset of a well-founded... |
welb 32701 | A nonempty subset of a wel... |
supex2g 32702 | Existence of supremum. (C... |
supclt 32703 | Closure of supremum. (Con... |
supubt 32704 | Upper bound property of su... |
filbcmb 32705 | Combine a finite set of lo... |
rdr 32706 | Two ways of expressing the... |
fzmul 32707 | Membership of a product in... |
sdclem2 32708 | Lemma for ~ sdc . (Contri... |
sdclem1 32709 | Lemma for ~ sdc . (Contri... |
sdc 32710 | Strong dependent choice. ... |
fdc 32711 | Finite version of dependen... |
fdc1 32712 | Variant of ~ fdc with no s... |
seqpo 32713 | Two ways to say that a seq... |
incsequz 32714 | An increasing sequence of ... |
incsequz2 32715 | An increasing sequence of ... |
nnubfi 32716 | A bounded above set of pos... |
nninfnub 32717 | An infinite set of positiv... |
subspopn 32718 | An open set is open in the... |
neificl 32719 | Neighborhoods are closed u... |
lpss2 32720 | Limit points of a subset a... |
metf1o 32721 | Use a bijection with a met... |
blssp 32722 | A ball in the subspace met... |
mettrifi 32723 | Generalized triangle inequ... |
lmclim2 32724 | A sequence in a metric spa... |
geomcau 32725 | If the distance between co... |
caures 32726 | The restriction of a Cauch... |
caushft 32727 | A shifted Cauchy sequence ... |
constcncf 32728 | A constant function is a c... |
idcncf 32729 | The identity function is a... |
sub1cncf 32730 | Subtracting a constant is ... |
sub2cncf 32731 | Subtraction from a constan... |
cnres2 32732 | The restriction of a conti... |
cnresima 32733 | A continuous function is c... |
cncfres 32734 | A continuous function on c... |
istotbnd 32738 | The predicate "is a totall... |
istotbnd2 32739 | The predicate "is a totall... |
istotbnd3 32740 | A metric space is totally ... |
totbndmet 32741 | The predicate "totally bou... |
0totbnd 32742 | The metric (there is only ... |
sstotbnd2 32743 | Condition for a subset of ... |
sstotbnd 32744 | Condition for a subset of ... |
sstotbnd3 32745 | Use a net that is not nece... |
totbndss 32746 | A subset of a totally boun... |
equivtotbnd 32747 | If the metric ` M ` is "st... |
isbnd 32749 | The predicate "is a bounde... |
bndmet 32750 | A bounded metric space is ... |
isbndx 32751 | A "bounded extended metric... |
isbnd2 32752 | The predicate "is a bounde... |
isbnd3 32753 | A metric space is bounded ... |
isbnd3b 32754 | A metric space is bounded ... |
bndss 32755 | A subset of a bounded metr... |
blbnd 32756 | A ball is bounded. (Contr... |
ssbnd 32757 | A subset of a metric space... |
totbndbnd 32758 | A totally bounded metric s... |
equivbnd 32759 | If the metric ` M ` is "st... |
bnd2lem 32760 | Lemma for ~ equivbnd2 and ... |
equivbnd2 32761 | If balls are totally bound... |
prdsbnd 32762 | The product metric over fi... |
prdstotbnd 32763 | The product metric over fi... |
prdsbnd2 32764 | If balls are totally bound... |
cntotbnd 32765 | A subset of the complex nu... |
cnpwstotbnd 32766 | A subset of ` A ^ I ` , wh... |
ismtyval 32769 | The set of isometries betw... |
isismty 32770 | The condition "is an isome... |
ismtycnv 32771 | The inverse of an isometry... |
ismtyima 32772 | The image of a ball under ... |
ismtyhmeolem 32773 | Lemma for ~ ismtyhmeo . (... |
ismtyhmeo 32774 | An isometry is a homeomorp... |
ismtybndlem 32775 | Lemma for ~ ismtybnd . (C... |
ismtybnd 32776 | Isometries preserve bounde... |
ismtyres 32777 | A restriction of an isomet... |
heibor1lem 32778 | Lemma for ~ heibor1 . A c... |
heibor1 32779 | One half of ~ heibor , tha... |
heiborlem1 32780 | Lemma for ~ heibor . We w... |
heiborlem2 32781 | Lemma for ~ heibor . Subs... |
heiborlem3 32782 | Lemma for ~ heibor . Usin... |
heiborlem4 32783 | Lemma for ~ heibor . Usin... |
heiborlem5 32784 | Lemma for ~ heibor . The ... |
heiborlem6 32785 | Lemma for ~ heibor . Sinc... |
heiborlem7 32786 | Lemma for ~ heibor . Sinc... |
heiborlem8 32787 | Lemma for ~ heibor . The ... |
heiborlem9 32788 | Lemma for ~ heibor . Disc... |
heiborlem10 32789 | Lemma for ~ heibor . The ... |
heibor 32790 | Generalized Heine-Borel Th... |
bfplem1 32791 | Lemma for ~ bfp . The seq... |
bfplem2 32792 | Lemma for ~ bfp . Using t... |
bfp 32793 | Banach fixed point theorem... |
rrnval 32796 | The n-dimensional Euclidea... |
rrnmval 32797 | The value of the Euclidean... |
rrnmet 32798 | Euclidean space is a metri... |
rrndstprj1 32799 | The distance between two p... |
rrndstprj2 32800 | Bound on the distance betw... |
rrncmslem 32801 | Lemma for ~ rrncms . (Con... |
rrncms 32802 | Euclidean space is complet... |
repwsmet 32803 | The supremum metric on ` R... |
rrnequiv 32804 | The supremum metric on ` R... |
rrntotbnd 32805 | A set in Euclidean space i... |
rrnheibor 32806 | Heine-Borel theorem for Eu... |
ismrer1 32807 | An isometry between ` RR `... |
reheibor 32808 | Heine-Borel theorem for re... |
iccbnd 32809 | A closed interval in ` RR ... |
icccmpALT 32810 | A closed interval in ` RR ... |
isass 32815 | The predicate "is an assoc... |
isexid 32816 | The predicate ` G ` has a ... |
ismgmOLD 32819 | Obsolete version of ~ ismg... |
clmgmOLD 32820 | Obsolete version of ~ mgmc... |
opidonOLD 32821 | Obsolete version of ~ mndp... |
rngopidOLD 32822 | Obsolete version of ~ mndp... |
opidon2OLD 32823 | Obsolete version of ~ mndp... |
isexid2 32824 | If ` G e. ( Magma i^i ExId... |
exidu1 32825 | Unicity of the left and ri... |
idrval 32826 | The value of the identity ... |
iorlid 32827 | A magma right and left ide... |
cmpidelt 32828 | A magma right and left ide... |
smgrpismgmOLD 32831 | Obsolete version of ~ sgrp... |
issmgrpOLD 32832 | Obsolete version of ~ issg... |
smgrpmgm 32833 | A semi-group is a magma. ... |
smgrpassOLD 32834 | Obsolete version of ~ sgrp... |
mndoissmgrpOLD 32837 | Obsolete version of ~ mnds... |
mndoisexid 32838 | A monoid has an identity e... |
mndoismgmOLD 32839 | Obsolete version of ~ mndm... |
mndomgmid 32840 | A monoid is a magma with a... |
ismndo 32841 | The predicate "is a monoid... |
ismndo1 32842 | The predicate "is a monoid... |
ismndo2 32843 | The predicate "is a monoid... |
grpomndo 32844 | A group is a monoid. (Con... |
exidcl 32845 | Closure of the binary oper... |
exidreslem 32846 | Lemma for ~ exidres and ~ ... |
exidres 32847 | The restriction of a binar... |
exidresid 32848 | The restriction of a binar... |
ablo4pnp 32849 | A commutative/associative ... |
grpoeqdivid 32850 | Two group elements are equ... |
grposnOLD 32851 | The group operation for th... |
elghomlem1OLD 32854 | Obsolete as of 15-Mar-2020... |
elghomlem2OLD 32855 | Obsolete as of 15-Mar-2020... |
elghomOLD 32856 | Obsolete version of ~ isgh... |
ghomlinOLD 32857 | Obsolete version of ~ ghml... |
ghomidOLD 32858 | Obsolete version of ~ ghmi... |
ghomf 32859 | Mapping property of a grou... |
ghomco 32860 | The composition of two gro... |
ghomdiv 32861 | Group homomorphisms preser... |
grpokerinj 32862 | A group homomorphism is in... |
relrngo 32865 | The class of all unital ri... |
isrngo 32866 | The predicate "is a (unita... |
isrngod 32867 | Conditions that determine ... |
rngoi 32868 | The properties of a unital... |
rngosm 32869 | Functionality of the multi... |
rngocl 32870 | Closure of the multiplicat... |
rngoid 32871 | The multiplication operati... |
rngoideu 32872 | The unit element of a ring... |
rngodi 32873 | Distributive law for the m... |
rngodir 32874 | Distributive law for the m... |
rngoass 32875 | Associative law for the mu... |
rngo2 32876 | A ring element plus itself... |
rngoablo 32877 | A ring's addition operatio... |
rngoablo2 32878 | In a unital ring the addit... |
rngogrpo 32879 | A ring's addition operatio... |
rngone0 32880 | The base set of a ring is ... |
rngogcl 32881 | Closure law for the additi... |
rngocom 32882 | The addition operation of ... |
rngoaass 32883 | The addition operation of ... |
rngoa32 32884 | The addition operation of ... |
rngoa4 32885 | Rearrangement of 4 terms i... |
rngorcan 32886 | Right cancellation law for... |
rngolcan 32887 | Left cancellation law for ... |
rngo0cl 32888 | A ring has an additive ide... |
rngo0rid 32889 | The additive identity of a... |
rngo0lid 32890 | The additive identity of a... |
rngolz 32891 | The zero of a unital ring ... |
rngorz 32892 | The zero of a unital ring ... |
rngosn3 32893 | Obsolete as of 25-Jan-2020... |
rngosn4 32894 | Obsolete as of 25-Jan-2020... |
rngosn6 32895 | Obsolete as of 25-Jan-2020... |
rngonegcl 32896 | A ring is closed under neg... |
rngoaddneg1 32897 | Adding the negative in a r... |
rngoaddneg2 32898 | Adding the negative in a r... |
rngosub 32899 | Subtraction in a ring, in ... |
rngmgmbs4 32900 | The range of an internal o... |
rngodm1dm2 32901 | In a unital ring the domai... |
rngorn1 32902 | In a unital ring the range... |
rngorn1eq 32903 | In a unital ring the range... |
rngomndo 32904 | In a unital ring the multi... |
rngoidmlem 32905 | The unit of a ring is an i... |
rngolidm 32906 | The unit of a ring is an i... |
rngoridm 32907 | The unit of a ring is an i... |
rngo1cl 32908 | The unit of a ring belongs... |
rngoueqz 32909 | Obsolete as of 23-Jan-2020... |
rngonegmn1l 32910 | Negation in a ring is the ... |
rngonegmn1r 32911 | Negation in a ring is the ... |
rngoneglmul 32912 | Negation of a product in a... |
rngonegrmul 32913 | Negation of a product in a... |
rngosubdi 32914 | Ring multiplication distri... |
rngosubdir 32915 | Ring multiplication distri... |
zerdivemp1x 32916 | In a unitary ring a left i... |
isdivrngo 32919 | The predicate "is a divisi... |
drngoi 32920 | The properties of a divisi... |
gidsn 32921 | Obsolete as of 23-Jan-2020... |
zrdivrng 32922 | The zero ring is not a div... |
dvrunz 32923 | In a division ring the uni... |
isgrpda 32924 | Properties that determine ... |
isdrngo1 32925 | The predicate "is a divisi... |
divrngcl 32926 | The product of two nonzero... |
isdrngo2 32927 | A division ring is a ring ... |
isdrngo3 32928 | A division ring is a ring ... |
rngohomval 32933 | The set of ring homomorphi... |
isrngohom 32934 | The predicate "is a ring h... |
rngohomf 32935 | A ring homomorphism is a f... |
rngohomcl 32936 | Closure law for a ring hom... |
rngohom1 32937 | A ring homomorphism preser... |
rngohomadd 32938 | Ring homomorphisms preserv... |
rngohommul 32939 | Ring homomorphisms preserv... |
rngogrphom 32940 | A ring homomorphism is a g... |
rngohom0 32941 | A ring homomorphism preser... |
rngohomsub 32942 | Ring homomorphisms preserv... |
rngohomco 32943 | The composition of two rin... |
rngokerinj 32944 | A ring homomorphism is inj... |
rngoisoval 32946 | The set of ring isomorphis... |
isrngoiso 32947 | The predicate "is a ring i... |
rngoiso1o 32948 | A ring isomorphism is a bi... |
rngoisohom 32949 | A ring isomorphism is a ri... |
rngoisocnv 32950 | The inverse of a ring isom... |
rngoisoco 32951 | The composition of two rin... |
isriscg 32953 | The ring isomorphism relat... |
isrisc 32954 | The ring isomorphism relat... |
risc 32955 | The ring isomorphism relat... |
risci 32956 | Determine that two rings a... |
riscer 32957 | Ring isomorphism is an equ... |
iscom2 32964 | A device to add commutativ... |
iscrngo 32965 | The predicate "is a commut... |
iscrngo2 32966 | The predicate "is a commut... |
iscringd 32967 | Conditions that determine ... |
flddivrng 32968 | A field is a division ring... |
crngorngo 32969 | A commutative ring is a ri... |
crngocom 32970 | The multiplication operati... |
crngm23 32971 | Commutative/associative la... |
crngm4 32972 | Commutative/associative la... |
fldcrng 32973 | A field is a commutative r... |
isfld2 32974 | The predicate "is a field"... |
crngohomfo 32975 | The image of a homomorphis... |
idlval 32982 | The class of ideals of a r... |
isidl 32983 | The predicate "is an ideal... |
isidlc 32984 | The predicate "is an ideal... |
idlss 32985 | An ideal of ` R ` is a sub... |
idlcl 32986 | An element of an ideal is ... |
idl0cl 32987 | An ideal contains ` 0 ` . ... |
idladdcl 32988 | An ideal is closed under a... |
idllmulcl 32989 | An ideal is closed under m... |
idlrmulcl 32990 | An ideal is closed under m... |
idlnegcl 32991 | An ideal is closed under n... |
idlsubcl 32992 | An ideal is closed under s... |
rngoidl 32993 | A ring ` R ` is an ` R ` i... |
0idl 32994 | The set containing only ` ... |
1idl 32995 | Two ways of expressing the... |
0rngo 32996 | In a ring, ` 0 = 1 ` iff t... |
divrngidl 32997 | The only ideals in a divis... |
intidl 32998 | The intersection of a none... |
inidl 32999 | The intersection of two id... |
unichnidl 33000 | The union of a nonempty ch... |
keridl 33001 | The kernel of a ring homom... |
pridlval 33002 | The class of prime ideals ... |
ispridl 33003 | The predicate "is a prime ... |
pridlidl 33004 | A prime ideal is an ideal.... |
pridlnr 33005 | A prime ideal is a proper ... |
pridl 33006 | The main property of a pri... |
ispridl2 33007 | A condition that shows an ... |
maxidlval 33008 | The set of maximal ideals ... |
ismaxidl 33009 | The predicate "is a maxima... |
maxidlidl 33010 | A maximal ideal is an idea... |
maxidlnr 33011 | A maximal ideal is proper.... |
maxidlmax 33012 | A maximal ideal is a maxim... |
maxidln1 33013 | One is not contained in an... |
maxidln0 33014 | A ring with a maximal idea... |
isprrngo 33019 | The predicate "is a prime ... |
prrngorngo 33020 | A prime ring is a ring. (... |
smprngopr 33021 | A simple ring (one whose o... |
divrngpr 33022 | A division ring is a prime... |
isdmn 33023 | The predicate "is a domain... |
isdmn2 33024 | The predicate "is a domain... |
dmncrng 33025 | A domain is a commutative ... |
dmnrngo 33026 | A domain is a ring. (Cont... |
flddmn 33027 | A field is a domain. (Con... |
igenval 33030 | The ideal generated by a s... |
igenss 33031 | A set is a subset of the i... |
igenidl 33032 | The ideal generated by a s... |
igenmin 33033 | The ideal generated by a s... |
igenidl2 33034 | The ideal generated by an ... |
igenval2 33035 | The ideal generated by a s... |
prnc 33036 | A principal ideal (an idea... |
isfldidl 33037 | Determine if a ring is a f... |
isfldidl2 33038 | Determine if a ring is a f... |
ispridlc 33039 | The predicate "is a prime ... |
pridlc 33040 | Property of a prime ideal ... |
pridlc2 33041 | Property of a prime ideal ... |
pridlc3 33042 | Property of a prime ideal ... |
isdmn3 33043 | The predicate "is a domain... |
dmnnzd 33044 | A domain has no zero-divis... |
dmncan1 33045 | Cancellation law for domai... |
dmncan2 33046 | Cancellation law for domai... |
efald2 33047 | A proof by contradiction. ... |
notbinot1 33048 | Simplification rule of neg... |
bicontr 33049 | Biimplication of its own n... |
impor 33050 | An equivalent formula for ... |
orfa 33051 | The falsum ` F. ` can be r... |
notbinot2 33052 | Commutation rule between n... |
biimpor 33053 | A rewriting rule for biimp... |
unitresl 33054 | A lemma for Conjunctive No... |
unitresr 33055 | A lemma for Conjunctive No... |
orfa1 33056 | Add a contradicting disjun... |
orfa2 33057 | Remove a contradicting dis... |
bifald 33058 | Infer the equivalence to a... |
orsild 33059 | A lemma for not-or-not eli... |
orsird 33060 | A lemma for not-or-not eli... |
orcomdd 33061 | Commutativity of logic dis... |
cnf1dd 33062 | A lemma for Conjunctive No... |
cnf2dd 33063 | A lemma for Conjunctive No... |
cnfn1dd 33064 | A lemma for Conjunctive No... |
cnfn2dd 33065 | A lemma for Conjunctive No... |
or32dd 33066 | A rearrangement of disjunc... |
notornotel1 33067 | A lemma for not-or-not eli... |
notornotel2 33068 | A lemma for not-or-not eli... |
contrd 33069 | A proof by contradiction, ... |
an12i 33070 | An inference from commutin... |
exmid2 33071 | An excluded middle law. (... |
selconj 33072 | An inference for selecting... |
truconj 33073 | Add true as a conjunct. (... |
orel 33074 | An inference for disjuncti... |
negel 33075 | An inference for negation ... |
botel 33076 | An inference for bottom el... |
tradd 33077 | Add top ad a conjunct. (C... |
sbtru 33078 | Substitution does not chan... |
sbfal 33079 | Substitution does not chan... |
sbcani 33080 | Distribution of class subs... |
sbcori 33081 | Distribution of class subs... |
sbcimi 33082 | Distribution of class subs... |
sbceqi 33083 | Distribution of class subs... |
sbcni 33084 | Move class substitution in... |
sbali 33085 | Discard class substitution... |
sbexi 33086 | Discard class substitution... |
sbcalf 33087 | Move universal quantifier ... |
sbcexf 33088 | Move existential quantifie... |
sbcalfi 33089 | Move universal quantifier ... |
sbcexfi 33090 | Move existential quantifie... |
csbvargi 33091 | The proper substitution of... |
csbconstgi 33092 | The proper substitution of... |
spsbcdi 33093 | A lemma for eliminating a ... |
alrimii 33094 | A lemma for introducing a ... |
spesbcdi 33095 | A lemma for introducing an... |
exlimddvf 33096 | A lemma for eliminating an... |
exlimddvfi 33097 | A lemma for eliminating an... |
sbceq1ddi 33098 | A lemma for eliminating in... |
sbccom2lem 33099 | Lemma for ~ sbccom2 . (Co... |
sbccom2 33100 | Commutative law for double... |
sbccom2f 33101 | Commutative law for double... |
sbccom2fi 33102 | Commutative law for double... |
sbcgfi 33103 | Substitution for a variabl... |
csbcom2fi 33104 | Commutative law for double... |
csbgfi 33105 | Substitution for a variabl... |
fald 33106 | Refutation of falsity, in ... |
tsim1 33107 | A Tseitin axiom for logica... |
tsim2 33108 | A Tseitin axiom for logica... |
tsim3 33109 | A Tseitin axiom for logica... |
tsbi1 33110 | A Tseitin axiom for logica... |
tsbi2 33111 | A Tseitin axiom for logica... |
tsbi3 33112 | A Tseitin axiom for logica... |
tsbi4 33113 | A Tseitin axiom for logica... |
tsxo1 33114 | A Tseitin axiom for logica... |
tsxo2 33115 | A Tseitin axiom for logica... |
tsxo3 33116 | A Tseitin axiom for logica... |
tsxo4 33117 | A Tseitin axiom for logica... |
tsan1 33118 | A Tseitin axiom for logica... |
tsan2 33119 | A Tseitin axiom for logica... |
tsan3 33120 | A Tseitin axiom for logica... |
tsna1 33121 | A Tseitin axiom for logica... |
tsna2 33122 | A Tseitin axiom for logica... |
tsna3 33123 | A Tseitin axiom for logica... |
tsor1 33124 | A Tseitin axiom for logica... |
tsor2 33125 | A Tseitin axiom for logica... |
tsor3 33126 | A Tseitin axiom for logica... |
ts3an1 33127 | A Tseitin axiom for triple... |
ts3an2 33128 | A Tseitin axiom for triple... |
ts3an3 33129 | A Tseitin axiom for triple... |
ts3or1 33130 | A Tseitin axiom for triple... |
ts3or2 33131 | A Tseitin axiom for triple... |
ts3or3 33132 | A Tseitin axiom for triple... |
iuneq2f 33133 | Equality deduction for ind... |
abeq12 33134 | Equality deduction for cla... |
rabeq12f 33135 | Equality deduction for res... |
csbeq12 33136 | Equality deduction for sub... |
nfbii2 33137 | Equality deduction for not... |
sbeqi 33138 | Equality deduction for sub... |
ralbi12f 33139 | Equality deduction for res... |
oprabbi 33140 | Equality deduction for cla... |
mpt2bi123f 33141 | Equality deduction for map... |
iuneq12f 33142 | Equality deduction for ind... |
iineq12f 33143 | Equality deduction for ind... |
opabbi 33144 | Equality deduction for cla... |
mptbi12f 33145 | Equality deduction for map... |
scottexf 33146 | A version of ~ scottex wit... |
scott0f 33147 | A version of ~ scott0 with... |
scottn0f 33148 | A version of ~ scott0f wit... |
ac6s3f 33149 | Generalization of the Axio... |
ac6s6 33150 | Generalization of the Axio... |
ac6s6f 33151 | Generalization of the Axio... |
prtlem60 33152 | Lemma for ~ prter3 . (Con... |
bicomdd 33153 | Commute two sides of a bic... |
jca2 33154 | Inference conjoining the c... |
jca2r 33155 | Inference conjoining the c... |
jca3 33156 | Inference conjoining the c... |
prtlem70 33157 | Lemma for ~ prter3 : a rea... |
ibdr 33158 | Reverse of ~ ibd . (Contr... |
pm5.31r 33159 | Variant of ~ pm5.31 . (Co... |
2r19.29 33160 | Double the quantifiers of ... |
prtlem100 33161 | Lemma for ~ prter3 . (Con... |
prtlem5 33162 | Lemma for ~ prter1 , ~ prt... |
prtlem80 33163 | Lemma for ~ prter2 . (Con... |
n0el 33164 | Negated membership of the ... |
brabsb2 33165 | A closed form of ~ brabsb ... |
eqbrrdv2 33166 | Other version of ~ eqbrrdi... |
prtlem9 33167 | Lemma for ~ prter3 . (Con... |
prtlem10 33168 | Lemma for ~ prter3 . (Con... |
prtlem11 33169 | Lemma for ~ prter2 . (Con... |
prtlem12 33170 | Lemma for ~ prtex and ~ pr... |
prtlem13 33171 | Lemma for ~ prter1 , ~ prt... |
prtlem16 33172 | Lemma for ~ prtex , ~ prte... |
prtlem400 33173 | Lemma for ~ prter2 and als... |
erprt 33176 | The quotient set of an equ... |
prtlem14 33177 | Lemma for ~ prter1 , ~ prt... |
prtlem15 33178 | Lemma for ~ prter1 and ~ p... |
prtlem17 33179 | Lemma for ~ prter2 . (Con... |
prtlem18 33180 | Lemma for ~ prter2 . (Con... |
prtlem19 33181 | Lemma for ~ prter2 . (Con... |
prter1 33182 | Every partition generates ... |
prtex 33183 | The equivalence relation g... |
prter2 33184 | The quotient set of the eq... |
prter3 33185 | For every partition there ... |
axc5 33196 | This theorem repeats ~ sp ... |
ax4fromc4 33197 | Rederivation of axiom ~ ax... |
ax10fromc7 33198 | Rederivation of axiom ~ ax... |
ax6fromc10 33199 | Rederivation of axiom ~ ax... |
hba1-o 33200 | The setvar ` x ` is not fr... |
axc4i-o 33201 | Inference version of ~ ax-... |
equid1 33202 | Proof of ~ equid from our ... |
equcomi1 33203 | Proof of ~ equcomi from ~ ... |
aecom-o 33204 | Commutation law for identi... |
aecoms-o 33205 | A commutation rule for ide... |
hbae-o 33206 | All variables are effectiv... |
dral1-o 33207 | Formula-building lemma for... |
ax12fromc15 33208 | Rederivation of axiom ~ ax... |
ax13fromc9 33209 | Derive ~ ax-13 from ~ ax-c... |
ax5ALT 33210 | Axiom to quantify a variab... |
sps-o 33211 | Generalization of antecede... |
hbequid 33212 | Bound-variable hypothesis ... |
nfequid-o 33213 | Bound-variable hypothesis ... |
axc5c7 33214 | Proof of a single axiom th... |
axc5c7toc5 33215 | Rederivation of ~ ax-c5 fr... |
axc5c7toc7 33216 | Rederivation of ~ ax-c7 fr... |
axc711 33217 | Proof of a single axiom th... |
nfa1-o 33218 | ` x ` is not free in ` A. ... |
axc711toc7 33219 | Rederivation of ~ ax-c7 fr... |
axc711to11 33220 | Rederivation of ~ ax-11 fr... |
axc5c711 33221 | Proof of a single axiom th... |
axc5c711toc5 33222 | Rederivation of ~ ax-c5 fr... |
axc5c711toc7 33223 | Rederivation of ~ ax-c7 fr... |
axc5c711to11 33224 | Rederivation of ~ ax-11 fr... |
equidqe 33225 | ~ equid with existential q... |
axc5sp1 33226 | A special case of ~ ax-c5 ... |
equidq 33227 | ~ equid with universal qua... |
equid1ALT 33228 | Alternate proof of ~ equid... |
axc11nfromc11 33229 | Rederivation of ~ ax-c11n ... |
naecoms-o 33230 | A commutation rule for dis... |
hbnae-o 33231 | All variables are effectiv... |
dvelimf-o 33232 | Proof of ~ dvelimh that us... |
dral2-o 33233 | Formula-building lemma for... |
aev-o 33234 | A "distinctor elimination"... |
ax5eq 33235 | Theorem to add distinct qu... |
dveeq2-o 33236 | Quantifier introduction wh... |
axc16g-o 33237 | A generalization of axiom ... |
dveeq1-o 33238 | Quantifier introduction wh... |
dveeq1-o16 33239 | Version of ~ dveeq1 using ... |
ax5el 33240 | Theorem to add distinct qu... |
axc11n-16 33241 | This theorem shows that, g... |
dveel2ALT 33242 | Alternate proof of ~ dveel... |
ax12f 33243 | Basis step for constructin... |
ax12eq 33244 | Basis step for constructin... |
ax12el 33245 | Basis step for constructin... |
ax12indn 33246 | Induction step for constru... |
ax12indi 33247 | Induction step for constru... |
ax12indalem 33248 | Lemma for ~ ax12inda2 and ... |
ax12inda2ALT 33249 | Alternate proof of ~ ax12i... |
ax12inda2 33250 | Induction step for constru... |
ax12inda 33251 | Induction step for constru... |
ax12v2-o 33252 | Rederivation of ~ ax-c15 f... |
ax12a2-o 33253 | Derive ~ ax-c15 from a hyp... |
axc11-o 33254 | Show that ~ ax-c11 can be ... |
fsumshftd 33255 | Index shift of a finite su... |
fsumshftdOLD 33256 | Obsolete version of ~ fsum... |
riotaclbgBAD 33258 | Closure of restricted iota... |
riotaclbBAD 33259 | Closure of restricted iota... |
riotasvd 33260 | Deduction version of ~ rio... |
riotasv2d 33261 | Value of description binde... |
riotasv2s 33262 | The value of description b... |
riotasv 33263 | Value of description binde... |
riotasv3d 33264 | A property ` ch ` holding ... |
elimhyps 33265 | A version of ~ elimhyp usi... |
dedths 33266 | A version of weak deductio... |
renegclALT 33267 | Closure law for negative o... |
elimhyps2 33268 | Generalization of ~ elimhy... |
dedths2 33269 | Generalization of ~ dedths... |
19.9alt 33270 | Version of ~ 19.9t for uni... |
nfcxfrdf 33271 | A utility lemma to transfe... |
nfded 33272 | A deduction theorem that c... |
nfded2 33273 | A deduction theorem that c... |
nfunidALT2 33274 | Deduction version of ~ nfu... |
nfunidALT 33275 | Deduction version of ~ nfu... |
nfopdALT 33276 | Deduction version of bound... |
cnaddcom 33277 | Recover the commutative la... |
toycom 33278 | Show the commutative law f... |
lshpset 33283 | The set of all hyperplanes... |
islshp 33284 | The predicate "is a hyperp... |
islshpsm 33285 | Hyperplane properties expr... |
lshplss 33286 | A hyperplane is a subspace... |
lshpne 33287 | A hyperplane is not equal ... |
lshpnel 33288 | A hyperplane's generating ... |
lshpnelb 33289 | The subspace sum of a hype... |
lshpnel2N 33290 | Condition that determines ... |
lshpne0 33291 | The member of the span in ... |
lshpdisj 33292 | A hyperplane and the span ... |
lshpcmp 33293 | If two hyperplanes are com... |
lshpinN 33294 | The intersection of two di... |
lsatset 33295 | The set of all 1-dim subsp... |
islsat 33296 | The predicate "is a 1-dim ... |
lsatlspsn2 33297 | The span of a nonzero sing... |
lsatlspsn 33298 | The span of a nonzero sing... |
islsati 33299 | A 1-dim subspace (atom) (o... |
lsateln0 33300 | A 1-dim subspace (atom) (o... |
lsatlss 33301 | The set of 1-dim subspaces... |
lsatlssel 33302 | An atom is a subspace. (C... |
lsatssv 33303 | An atom is a set of vector... |
lsatn0 33304 | A 1-dim subspace (atom) of... |
lsatspn0 33305 | The span of a vector is an... |
lsator0sp 33306 | The span of a vector is ei... |
lsatssn0 33307 | A subspace (or any class) ... |
lsatcmp 33308 | If two atoms are comparabl... |
lsatcmp2 33309 | If an atom is included in ... |
lsatel 33310 | A nonzero vector in an ato... |
lsatelbN 33311 | A nonzero vector in an ato... |
lsat2el 33312 | Two atoms sharing a nonzer... |
lsmsat 33313 | Convert comparison of atom... |
lsatfixedN 33314 | Show equality with the spa... |
lsmsatcv 33315 | Subspace sum has the cover... |
lssatomic 33316 | The lattice of subspaces i... |
lssats 33317 | The lattice of subspaces i... |
lpssat 33318 | Two subspaces in a proper ... |
lrelat 33319 | Subspaces are relatively a... |
lssatle 33320 | The ordering of two subspa... |
lssat 33321 | Two subspaces in a proper ... |
islshpat 33322 | Hyperplane properties expr... |
lcvfbr 33325 | The covers relation for a ... |
lcvbr 33326 | The covers relation for a ... |
lcvbr2 33327 | The covers relation for a ... |
lcvbr3 33328 | The covers relation for a ... |
lcvpss 33329 | The covers relation implie... |
lcvnbtwn 33330 | The covers relation implie... |
lcvntr 33331 | The covers relation is not... |
lcvnbtwn2 33332 | The covers relation implie... |
lcvnbtwn3 33333 | The covers relation implie... |
lsmcv2 33334 | Subspace sum has the cover... |
lcvat 33335 | If a subspace covers anoth... |
lsatcv0 33336 | An atom covers the zero su... |
lsatcveq0 33337 | A subspace covered by an a... |
lsat0cv 33338 | A subspace is an atom iff ... |
lcvexchlem1 33339 | Lemma for ~ lcvexch . (Co... |
lcvexchlem2 33340 | Lemma for ~ lcvexch . (Co... |
lcvexchlem3 33341 | Lemma for ~ lcvexch . (Co... |
lcvexchlem4 33342 | Lemma for ~ lcvexch . (Co... |
lcvexchlem5 33343 | Lemma for ~ lcvexch . (Co... |
lcvexch 33344 | Subspaces satisfy the exch... |
lcvp 33345 | Covering property of Defin... |
lcv1 33346 | Covering property of a sub... |
lcv2 33347 | Covering property of a sub... |
lsatexch 33348 | The atom exchange property... |
lsatnle 33349 | The meet of a subspace and... |
lsatnem0 33350 | The meet of distinct atoms... |
lsatexch1 33351 | The atom exch1ange propert... |
lsatcv0eq 33352 | If the sum of two atoms co... |
lsatcv1 33353 | Two atoms covering the zer... |
lsatcvatlem 33354 | Lemma for ~ lsatcvat . (C... |
lsatcvat 33355 | A nonzero subspace less th... |
lsatcvat2 33356 | A subspace covered by the ... |
lsatcvat3 33357 | A condition implying that ... |
islshpcv 33358 | Hyperplane properties expr... |
l1cvpat 33359 | A subspace covered by the ... |
l1cvat 33360 | Create an atom under an el... |
lshpat 33361 | Create an atom under a hyp... |
lflset 33364 | The set of linear function... |
islfl 33365 | The predicate "is a linear... |
lfli 33366 | Property of a linear funct... |
islfld 33367 | Properties that determine ... |
lflf 33368 | A linear functional is a f... |
lflcl 33369 | A linear functional value ... |
lfl0 33370 | A linear functional is zer... |
lfladd 33371 | Property of a linear funct... |
lflsub 33372 | Property of a linear funct... |
lflmul 33373 | Property of a linear funct... |
lfl0f 33374 | The zero function is a fun... |
lfl1 33375 | A nonzero functional has a... |
lfladdcl 33376 | Closure of addition of two... |
lfladdcom 33377 | Commutativity of functiona... |
lfladdass 33378 | Associativity of functiona... |
lfladd0l 33379 | Functional addition with t... |
lflnegcl 33380 | Closure of the negative of... |
lflnegl 33381 | A functional plus its nega... |
lflvscl 33382 | Closure of a scalar produc... |
lflvsdi1 33383 | Distributive law for (righ... |
lflvsdi2 33384 | Reverse distributive law f... |
lflvsdi2a 33385 | Reverse distributive law f... |
lflvsass 33386 | Associative law for (right... |
lfl0sc 33387 | The (right vector space) s... |
lflsc0N 33388 | The scalar product with th... |
lfl1sc 33389 | The (right vector space) s... |
lkrfval 33392 | The kernel of a functional... |
lkrval 33393 | Value of the kernel of a f... |
ellkr 33394 | Membership in the kernel o... |
lkrval2 33395 | Value of the kernel of a f... |
ellkr2 33396 | Membership in the kernel o... |
lkrcl 33397 | A member of the kernel of ... |
lkrf0 33398 | The value of a functional ... |
lkr0f 33399 | The kernel of the zero fun... |
lkrlss 33400 | The kernel of a linear fun... |
lkrssv 33401 | The kernel of a linear fun... |
lkrsc 33402 | The kernel of a nonzero sc... |
lkrscss 33403 | The kernel of a scalar pro... |
eqlkr 33404 | Two functionals with the s... |
eqlkr2 33405 | Two functionals with the s... |
eqlkr3 33406 | Two functionals with the s... |
lkrlsp 33407 | The subspace sum of a kern... |
lkrlsp2 33408 | The subspace sum of a kern... |
lkrlsp3 33409 | The subspace sum of a kern... |
lkrshp 33410 | The kernel of a nonzero fu... |
lkrshp3 33411 | The kernels of nonzero fun... |
lkrshpor 33412 | The kernel of a functional... |
lkrshp4 33413 | A kernel is a hyperplane i... |
lshpsmreu 33414 | Lemma for ~ lshpkrex . Sh... |
lshpkrlem1 33415 | Lemma for ~ lshpkrex . Th... |
lshpkrlem2 33416 | Lemma for ~ lshpkrex . Th... |
lshpkrlem3 33417 | Lemma for ~ lshpkrex . De... |
lshpkrlem4 33418 | Lemma for ~ lshpkrex . Pa... |
lshpkrlem5 33419 | Lemma for ~ lshpkrex . Pa... |
lshpkrlem6 33420 | Lemma for ~ lshpkrex . Sh... |
lshpkrcl 33421 | The set ` G ` defined by h... |
lshpkr 33422 | The kernel of functional `... |
lshpkrex 33423 | There exists a functional ... |
lshpset2N 33424 | The set of all hyperplanes... |
islshpkrN 33425 | The predicate "is a hyperp... |
lfl1dim 33426 | Equivalent expressions for... |
lfl1dim2N 33427 | Equivalent expressions for... |
ldualset 33430 | Define the (left) dual of ... |
ldualvbase 33431 | The vectors of a dual spac... |
ldualelvbase 33432 | Utility theorem for conver... |
ldualfvadd 33433 | Vector addition in the dua... |
ldualvadd 33434 | Vector addition in the dua... |
ldualvaddcl 33435 | The value of vector additi... |
ldualvaddval 33436 | The value of the value of ... |
ldualsca 33437 | The ring of scalars of the... |
ldualsbase 33438 | Base set of scalar ring fo... |
ldualsaddN 33439 | Scalar addition for the du... |
ldualsmul 33440 | Scalar multiplication for ... |
ldualfvs 33441 | Scalar product operation f... |
ldualvs 33442 | Scalar product operation v... |
ldualvsval 33443 | Value of scalar product op... |
ldualvscl 33444 | The scalar product operati... |
ldualvaddcom 33445 | Commutative law for vector... |
ldualvsass 33446 | Associative law for scalar... |
ldualvsass2 33447 | Associative law for scalar... |
ldualvsdi1 33448 | Distributive law for scala... |
ldualvsdi2 33449 | Reverse distributive law f... |
ldualgrplem 33450 | Lemma for ~ ldualgrp . (C... |
ldualgrp 33451 | The dual of a vector space... |
ldual0 33452 | The zero scalar of the dua... |
ldual1 33453 | The unit scalar of the dua... |
ldualneg 33454 | The negative of a scalar o... |
ldual0v 33455 | The zero vector of the dua... |
ldual0vcl 33456 | The dual zero vector is a ... |
lduallmodlem 33457 | Lemma for ~ lduallmod . (... |
lduallmod 33458 | The dual of a left module ... |
lduallvec 33459 | The dual of a left vector ... |
ldualvsub 33460 | The value of vector subtra... |
ldualvsubcl 33461 | Closure of vector subtract... |
ldualvsubval 33462 | The value of the value of ... |
ldualssvscl 33463 | Closure of scalar product ... |
ldualssvsubcl 33464 | Closure of vector subtract... |
ldual0vs 33465 | Scalar zero times a functi... |
lkr0f2 33466 | The kernel of the zero fun... |
lduallkr3 33467 | The kernels of nonzero fun... |
lkrpssN 33468 | Proper subset relation bet... |
lkrin 33469 | Intersection of the kernel... |
eqlkr4 33470 | Two functionals with the s... |
ldual1dim 33471 | Equivalent expressions for... |
ldualkrsc 33472 | The kernel of a nonzero sc... |
lkrss 33473 | The kernel of a scalar pro... |
lkrss2N 33474 | Two functionals with kerne... |
lkreqN 33475 | Proportional functionals h... |
lkrlspeqN 33476 | Condition for colinear fun... |
isopos 33485 | The predicate "is an ortho... |
opposet 33486 | Every orthoposet is a pose... |
oposlem 33487 | Lemma for orthoposet prope... |
op01dm 33488 | Conditions necessary for z... |
op0cl 33489 | An orthoposet has a zero e... |
op1cl 33490 | An orthoposet has a unit e... |
op0le 33491 | Orthoposet zero is less th... |
ople0 33492 | An element less than or eq... |
opnlen0 33493 | An element not less than a... |
lub0N 33494 | The least upper bound of t... |
opltn0 33495 | A lattice element greater ... |
ople1 33496 | Any element is less than t... |
op1le 33497 | If the orthoposet unit is ... |
glb0N 33498 | The greatest lower bound o... |
opoccl 33499 | Closure of orthocomplement... |
opococ 33500 | Double negative law for or... |
opcon3b 33501 | Contraposition law for ort... |
opcon2b 33502 | Orthocomplement contraposi... |
opcon1b 33503 | Orthocomplement contraposi... |
oplecon3 33504 | Contraposition law for ort... |
oplecon3b 33505 | Contraposition law for ort... |
oplecon1b 33506 | Contraposition law for str... |
opoc1 33507 | Orthocomplement of orthopo... |
opoc0 33508 | Orthocomplement of orthopo... |
opltcon3b 33509 | Contraposition law for str... |
opltcon1b 33510 | Contraposition law for str... |
opltcon2b 33511 | Contraposition law for str... |
opexmid 33512 | Law of excluded middle for... |
opnoncon 33513 | Law of contradiction for o... |
riotaocN 33514 | The orthocomplement of the... |
cmtfvalN 33515 | Value of commutes relation... |
cmtvalN 33516 | Equivalence for commutes r... |
isolat 33517 | The predicate "is an ortho... |
ollat 33518 | An ortholattice is a latti... |
olop 33519 | An ortholattice is an orth... |
olposN 33520 | An ortholattice is a poset... |
isolatiN 33521 | Properties that determine ... |
oldmm1 33522 | De Morgan's law for meet i... |
oldmm2 33523 | De Morgan's law for meet i... |
oldmm3N 33524 | De Morgan's law for meet i... |
oldmm4 33525 | De Morgan's law for meet i... |
oldmj1 33526 | De Morgan's law for join i... |
oldmj2 33527 | De Morgan's law for join i... |
oldmj3 33528 | De Morgan's law for join i... |
oldmj4 33529 | De Morgan's law for join i... |
olj01 33530 | An ortholattice element jo... |
olj02 33531 | An ortholattice element jo... |
olm11 33532 | The meet of an ortholattic... |
olm12 33533 | The meet of an ortholattic... |
latmassOLD 33534 | Ortholattice meet is assoc... |
latm12 33535 | A rearrangement of lattice... |
latm32 33536 | A rearrangement of lattice... |
latmrot 33537 | Rotate lattice meet of 3 c... |
latm4 33538 | Rearrangement of lattice m... |
latmmdiN 33539 | Lattice meet distributes o... |
latmmdir 33540 | Lattice meet distributes o... |
olm01 33541 | Meet with lattice zero is ... |
olm02 33542 | Meet with lattice zero is ... |
isoml 33543 | The predicate "is an ortho... |
isomliN 33544 | Properties that determine ... |
omlol 33545 | An orthomodular lattice is... |
omlop 33546 | An orthomodular lattice is... |
omllat 33547 | An orthomodular lattice is... |
omllaw 33548 | The orthomodular law. (Co... |
omllaw2N 33549 | Variation of orthomodular ... |
omllaw3 33550 | Orthomodular law equivalen... |
omllaw4 33551 | Orthomodular law equivalen... |
omllaw5N 33552 | The orthomodular law. Rem... |
cmtcomlemN 33553 | Lemma for ~ cmtcomN . ( ~... |
cmtcomN 33554 | Commutation is symmetric. ... |
cmt2N 33555 | Commutation with orthocomp... |
cmt3N 33556 | Commutation with orthocomp... |
cmt4N 33557 | Commutation with orthocomp... |
cmtbr2N 33558 | Alternate definition of th... |
cmtbr3N 33559 | Alternate definition for t... |
cmtbr4N 33560 | Alternate definition for t... |
lecmtN 33561 | Ordered elements commute. ... |
cmtidN 33562 | Any element commutes with ... |
omlfh1N 33563 | Foulis-Holland Theorem, pa... |
omlfh3N 33564 | Foulis-Holland Theorem, pa... |
omlmod1i2N 33565 | Analogue of modular law ~ ... |
omlspjN 33566 | Contraction of a Sasaki pr... |
cvrfval 33573 | Value of covers relation "... |
cvrval 33574 | Binary relation expressing... |
cvrlt 33575 | The covers relation implie... |
cvrnbtwn 33576 | There is no element betwee... |
ncvr1 33577 | No element covers the latt... |
cvrletrN 33578 | Property of an element abo... |
cvrval2 33579 | Binary relation expressing... |
cvrnbtwn2 33580 | The covers relation implie... |
cvrnbtwn3 33581 | The covers relation implie... |
cvrcon3b 33582 | Contraposition law for the... |
cvrle 33583 | The covers relation implie... |
cvrnbtwn4 33584 | The covers relation implie... |
cvrnle 33585 | The covers relation implie... |
cvrne 33586 | The covers relation implie... |
cvrnrefN 33587 | The covers relation is not... |
cvrcmp 33588 | If two lattice elements th... |
cvrcmp2 33589 | If two lattice elements co... |
pats 33590 | The set of atoms in a pose... |
isat 33591 | The predicate "is an atom"... |
isat2 33592 | The predicate "is an atom"... |
atcvr0 33593 | An atom covers zero. ( ~ ... |
atbase 33594 | An atom is a member of the... |
atssbase 33595 | The set of atoms is a subs... |
0ltat 33596 | An atom is greater than ze... |
leatb 33597 | A poset element less than ... |
leat 33598 | A poset element less than ... |
leat2 33599 | A nonzero poset element le... |
leat3 33600 | A poset element less than ... |
meetat 33601 | The meet of any element wi... |
meetat2 33602 | The meet of any element wi... |
isatl 33604 | The predicate "is an atomi... |
atllat 33605 | An atomic lattice is a lat... |
atlpos 33606 | An atomic lattice is a pos... |
atl0dm 33607 | Condition necessary for ze... |
atl0cl 33608 | An atomic lattice has a ze... |
atl0le 33609 | Orthoposet zero is less th... |
atlle0 33610 | An element less than or eq... |
atlltn0 33611 | A lattice element greater ... |
isat3 33612 | The predicate "is an atom"... |
atn0 33613 | An atom is not zero. ( ~ ... |
atnle0 33614 | An atom is not less than o... |
atlen0 33615 | A lattice element is nonze... |
atcmp 33616 | If two atoms are comparabl... |
atncmp 33617 | Frequently-used variation ... |
atnlt 33618 | Two atoms cannot satisfy t... |
atcvreq0 33619 | An element covered by an a... |
atncvrN 33620 | Two atoms cannot satisfy t... |
atlex 33621 | Every nonzero element of a... |
atnle 33622 | Two ways of expressing "an... |
atnem0 33623 | The meet of distinct atoms... |
atlatmstc 33624 | An atomic, complete, ortho... |
atlatle 33625 | The ordering of two Hilber... |
atlrelat1 33626 | An atomistic lattice with ... |
iscvlat 33628 | The predicate "is an atomi... |
iscvlat2N 33629 | The predicate "is an atomi... |
cvlatl 33630 | An atomic lattice with the... |
cvllat 33631 | An atomic lattice with the... |
cvlposN 33632 | An atomic lattice with the... |
cvlexch1 33633 | An atomic covering lattice... |
cvlexch2 33634 | An atomic covering lattice... |
cvlexchb1 33635 | An atomic covering lattice... |
cvlexchb2 33636 | An atomic covering lattice... |
cvlexch3 33637 | An atomic covering lattice... |
cvlexch4N 33638 | An atomic covering lattice... |
cvlatexchb1 33639 | A version of ~ cvlexchb1 f... |
cvlatexchb2 33640 | A version of ~ cvlexchb2 f... |
cvlatexch1 33641 | Atom exchange property. (... |
cvlatexch2 33642 | Atom exchange property. (... |
cvlatexch3 33643 | Atom exchange property. (... |
cvlcvr1 33644 | The covering property. Pr... |
cvlcvrp 33645 | A Hilbert lattice satisfie... |
cvlatcvr1 33646 | An atom is covered by its ... |
cvlatcvr2 33647 | An atom is covered by its ... |
cvlsupr2 33648 | Two equivalent ways of exp... |
cvlsupr3 33649 | Two equivalent ways of exp... |
cvlsupr4 33650 | Consequence of superpositi... |
cvlsupr5 33651 | Consequence of superpositi... |
cvlsupr6 33652 | Consequence of superpositi... |
cvlsupr7 33653 | Consequence of superpositi... |
cvlsupr8 33654 | Consequence of superpositi... |
ishlat1 33657 | The predicate "is a Hilber... |
ishlat2 33658 | The predicate "is a Hilber... |
ishlat3N 33659 | The predicate "is a Hilber... |
ishlatiN 33660 | Properties that determine ... |
hlomcmcv 33661 | A Hilbert lattice is ortho... |
hloml 33662 | A Hilbert lattice is ortho... |
hlclat 33663 | A Hilbert lattice is compl... |
hlcvl 33664 | A Hilbert lattice is an at... |
hlatl 33665 | A Hilbert lattice is atomi... |
hlol 33666 | A Hilbert lattice is an or... |
hlop 33667 | A Hilbert lattice is an or... |
hllat 33668 | A Hilbert lattice is a lat... |
hlomcmat 33669 | A Hilbert lattice is ortho... |
hlpos 33670 | A Hilbert lattice is a pos... |
hlatjcl 33671 | Closure of join operation.... |
hlatjcom 33672 | Commutatitivity of join op... |
hlatjidm 33673 | Idempotence of join operat... |
hlatjass 33674 | Lattice join is associativ... |
hlatj12 33675 | Swap 1st and 2nd members o... |
hlatj32 33676 | Swap 2nd and 3rd members o... |
hlatjrot 33677 | Rotate lattice join of 3 c... |
hlatj4 33678 | Rearrangement of lattice j... |
hlatlej1 33679 | A join's first argument is... |
hlatlej2 33680 | A join's second argument i... |
glbconN 33681 | De Morgan's law for GLB an... |
glbconxN 33682 | De Morgan's law for GLB an... |
atnlej1 33683 | If an atom is not less tha... |
atnlej2 33684 | If an atom is not less tha... |
hlsuprexch 33685 | A Hilbert lattice has the ... |
hlexch1 33686 | A Hilbert lattice has the ... |
hlexch2 33687 | A Hilbert lattice has the ... |
hlexchb1 33688 | A Hilbert lattice has the ... |
hlexchb2 33689 | A Hilbert lattice has the ... |
hlsupr 33690 | A Hilbert lattice has the ... |
hlsupr2 33691 | A Hilbert lattice has the ... |
hlhgt4 33692 | A Hilbert lattice has a he... |
hlhgt2 33693 | A Hilbert lattice has a he... |
hl0lt1N 33694 | Lattice 0 is less than lat... |
hlexch3 33695 | A Hilbert lattice has the ... |
hlexch4N 33696 | A Hilbert lattice has the ... |
hlatexchb1 33697 | A version of ~ hlexchb1 fo... |
hlatexchb2 33698 | A version of ~ hlexchb2 fo... |
hlatexch1 33699 | Atom exchange property. (... |
hlatexch2 33700 | Atom exchange property. (... |
hlatmstcOLDN 33701 | An atomic, complete, ortho... |
hlatle 33702 | The ordering of two Hilber... |
hlateq 33703 | The equality of two Hilber... |
hlrelat1 33704 | An atomistic lattice with ... |
hlrelat5N 33705 | An atomistic lattice with ... |
hlrelat 33706 | A Hilbert lattice is relat... |
hlrelat2 33707 | A consequence of relative ... |
exatleN 33708 | A condition for an atom to... |
hl2at 33709 | A Hilbert lattice has at l... |
atex 33710 | At least one atom exists. ... |
intnatN 33711 | If the intersection with a... |
2llnne2N 33712 | Condition implying that tw... |
2llnneN 33713 | Condition implying that tw... |
cvr1 33714 | A Hilbert lattice has the ... |
cvr2N 33715 | Less-than and covers equiv... |
hlrelat3 33716 | The Hilbert lattice is rel... |
cvrval3 33717 | Binary relation expressing... |
cvrval4N 33718 | Binary relation expressing... |
cvrval5 33719 | Binary relation expressing... |
cvrp 33720 | A Hilbert lattice satisfie... |
atcvr1 33721 | An atom is covered by its ... |
atcvr2 33722 | An atom is covered by its ... |
cvrexchlem 33723 | Lemma for ~ cvrexch . ( ~... |
cvrexch 33724 | A Hilbert lattice satisfie... |
cvratlem 33725 | Lemma for ~ cvrat . ( ~ a... |
cvrat 33726 | A nonzero Hilbert lattice ... |
ltltncvr 33727 | A chained strong ordering ... |
ltcvrntr 33728 | Non-transitive condition f... |
cvrntr 33729 | The covers relation is not... |
atcvr0eq 33730 | The covers relation is not... |
lnnat 33731 | A line (the join of two di... |
atcvrj0 33732 | Two atoms covering the zer... |
cvrat2 33733 | A Hilbert lattice element ... |
atcvrneN 33734 | Inequality derived from at... |
atcvrj1 33735 | Condition for an atom to b... |
atcvrj2b 33736 | Condition for an atom to b... |
atcvrj2 33737 | Condition for an atom to b... |
atleneN 33738 | Inequality derived from at... |
atltcvr 33739 | An equivalence of less-tha... |
atle 33740 | Any nonzero element has an... |
atlt 33741 | Two atoms are unequal iff ... |
atlelt 33742 | Transfer less-than relatio... |
2atlt 33743 | Given an atom less than an... |
atexchcvrN 33744 | Atom exchange property. V... |
atexchltN 33745 | Atom exchange property. V... |
cvrat3 33746 | A condition implying that ... |
cvrat4 33747 | A condition implying exist... |
cvrat42 33748 | Commuted version of ~ cvra... |
2atjm 33749 | The meet of a line (expres... |
atbtwn 33750 | Property of a 3rd atom ` R... |
atbtwnexOLDN 33751 | There exists a 3rd atom ` ... |
atbtwnex 33752 | Given atoms ` P ` in ` X `... |
3noncolr2 33753 | Two ways to express 3 non-... |
3noncolr1N 33754 | Two ways to express 3 non-... |
hlatcon3 33755 | Atom exchange combined wit... |
hlatcon2 33756 | Atom exchange combined wit... |
4noncolr3 33757 | A way to express 4 non-col... |
4noncolr2 33758 | A way to express 4 non-col... |
4noncolr1 33759 | A way to express 4 non-col... |
athgt 33760 | A Hilbert lattice, whose h... |
3dim0 33761 | There exists a 3-dimension... |
3dimlem1 33762 | Lemma for ~ 3dim1 . (Cont... |
3dimlem2 33763 | Lemma for ~ 3dim1 . (Cont... |
3dimlem3a 33764 | Lemma for ~ 3dim3 . (Cont... |
3dimlem3 33765 | Lemma for ~ 3dim1 . (Cont... |
3dimlem3OLDN 33766 | Lemma for ~ 3dim1 . (Cont... |
3dimlem4a 33767 | Lemma for ~ 3dim3 . (Cont... |
3dimlem4 33768 | Lemma for ~ 3dim1 . (Cont... |
3dimlem4OLDN 33769 | Lemma for ~ 3dim1 . (Cont... |
3dim1lem5 33770 | Lemma for ~ 3dim1 . (Cont... |
3dim1 33771 | Construct a 3-dimensional ... |
3dim2 33772 | Construct 2 new layers on ... |
3dim3 33773 | Construct a new layer on t... |
2dim 33774 | Generate a height-3 elemen... |
1dimN 33775 | An atom is covered by a he... |
1cvrco 33776 | The orthocomplement of an ... |
1cvratex 33777 | There exists an atom less ... |
1cvratlt 33778 | An atom less than or equal... |
1cvrjat 33779 | An element covered by the ... |
1cvrat 33780 | Create an atom under an el... |
ps-1 33781 | The join of two atoms ` R ... |
ps-2 33782 | Lattice analogue for the p... |
2atjlej 33783 | Two atoms are different if... |
hlatexch3N 33784 | Rearrange join of atoms in... |
hlatexch4 33785 | Exchange 2 atoms. (Contri... |
ps-2b 33786 | Variation of projective ge... |
3atlem1 33787 | Lemma for ~ 3at . (Contri... |
3atlem2 33788 | Lemma for ~ 3at . (Contri... |
3atlem3 33789 | Lemma for ~ 3at . (Contri... |
3atlem4 33790 | Lemma for ~ 3at . (Contri... |
3atlem5 33791 | Lemma for ~ 3at . (Contri... |
3atlem6 33792 | Lemma for ~ 3at . (Contri... |
3atlem7 33793 | Lemma for ~ 3at . (Contri... |
3at 33794 | Any three non-colinear ato... |
llnset 33809 | The set of lattice lines i... |
islln 33810 | The predicate "is a lattic... |
islln4 33811 | The predicate "is a lattic... |
llni 33812 | Condition implying a latti... |
llnbase 33813 | A lattice line is a lattic... |
islln3 33814 | The predicate "is a lattic... |
islln2 33815 | The predicate "is a lattic... |
llni2 33816 | The join of two different ... |
llnnleat 33817 | An atom cannot majorize a ... |
llnneat 33818 | A lattice line is not an a... |
2atneat 33819 | The join of two distinct a... |
llnn0 33820 | A lattice line is nonzero.... |
islln2a 33821 | The predicate "is a lattic... |
llnle 33822 | Any element greater than 0... |
atcvrlln2 33823 | An atom under a line is co... |
atcvrlln 33824 | An element covering an ato... |
llnexatN 33825 | Given an atom on a line, t... |
llncmp 33826 | If two lattice lines are c... |
llnnlt 33827 | Two lattice lines cannot s... |
2llnmat 33828 | Two intersecting lines int... |
2at0mat0 33829 | Special case of ~ 2atmat0 ... |
2atmat0 33830 | The meet of two unequal li... |
2atm 33831 | An atom majorized by two d... |
ps-2c 33832 | Variation of projective ge... |
lplnset 33833 | The set of lattice planes ... |
islpln 33834 | The predicate "is a lattic... |
islpln4 33835 | The predicate "is a lattic... |
lplni 33836 | Condition implying a latti... |
islpln3 33837 | The predicate "is a lattic... |
lplnbase 33838 | A lattice plane is a latti... |
islpln5 33839 | The predicate "is a lattic... |
islpln2 33840 | The predicate "is a lattic... |
lplni2 33841 | The join of 3 different at... |
lvolex3N 33842 | There is an atom outside o... |
llnmlplnN 33843 | The intersection of a line... |
lplnle 33844 | Any element greater than 0... |
lplnnle2at 33845 | A lattice line (or atom) c... |
lplnnleat 33846 | A lattice plane cannot maj... |
lplnnlelln 33847 | A lattice plane is not les... |
2atnelpln 33848 | The join of two atoms is n... |
lplnneat 33849 | No lattice plane is an ato... |
lplnnelln 33850 | No lattice plane is a latt... |
lplnn0N 33851 | A lattice plane is nonzero... |
islpln2a 33852 | The predicate "is a lattic... |
islpln2ah 33853 | The predicate "is a lattic... |
lplnriaN 33854 | Property of a lattice plan... |
lplnribN 33855 | Property of a lattice plan... |
lplnric 33856 | Property of a lattice plan... |
lplnri1 33857 | Property of a lattice plan... |
lplnri2N 33858 | Property of a lattice plan... |
lplnri3N 33859 | Property of a lattice plan... |
lplnllnneN 33860 | Two lattice lines defined ... |
llncvrlpln2 33861 | A lattice line under a lat... |
llncvrlpln 33862 | An element covering a latt... |
2lplnmN 33863 | If the join of two lattice... |
2llnmj 33864 | The meet of two lattice li... |
2atmat 33865 | The meet of two intersecti... |
lplncmp 33866 | If two lattice planes are ... |
lplnexatN 33867 | Given a lattice line on a ... |
lplnexllnN 33868 | Given an atom on a lattice... |
lplnnlt 33869 | Two lattice planes cannot ... |
2llnjaN 33870 | The join of two different ... |
2llnjN 33871 | The join of two different ... |
2llnm2N 33872 | The meet of two different ... |
2llnm3N 33873 | Two lattice lines in a lat... |
2llnm4 33874 | Two lattice lines that maj... |
2llnmeqat 33875 | An atom equals the interse... |
lvolset 33876 | The set of 3-dim lattice v... |
islvol 33877 | The predicate "is a 3-dim ... |
islvol4 33878 | The predicate "is a 3-dim ... |
lvoli 33879 | Condition implying a 3-dim... |
islvol3 33880 | The predicate "is a 3-dim ... |
lvoli3 33881 | Condition implying a 3-dim... |
lvolbase 33882 | A 3-dim lattice volume is ... |
islvol5 33883 | The predicate "is a 3-dim ... |
islvol2 33884 | The predicate "is a 3-dim ... |
lvoli2 33885 | The join of 4 different at... |
lvolnle3at 33886 | A lattice plane (or lattic... |
lvolnleat 33887 | An atom cannot majorize a ... |
lvolnlelln 33888 | A lattice line cannot majo... |
lvolnlelpln 33889 | A lattice plane cannot maj... |
3atnelvolN 33890 | The join of 3 atoms is not... |
2atnelvolN 33891 | The join of two atoms is n... |
lvolneatN 33892 | No lattice volume is an at... |
lvolnelln 33893 | No lattice volume is a lat... |
lvolnelpln 33894 | No lattice volume is a lat... |
lvoln0N 33895 | A lattice volume is nonzer... |
islvol2aN 33896 | The predicate "is a lattic... |
4atlem0a 33897 | Lemma for ~ 4at . (Contri... |
4atlem0ae 33898 | Lemma for ~ 4at . (Contri... |
4atlem0be 33899 | Lemma for ~ 4at . (Contri... |
4atlem3 33900 | Lemma for ~ 4at . Break i... |
4atlem3a 33901 | Lemma for ~ 4at . Break i... |
4atlem3b 33902 | Lemma for ~ 4at . Break i... |
4atlem4a 33903 | Lemma for ~ 4at . Frequen... |
4atlem4b 33904 | Lemma for ~ 4at . Frequen... |
4atlem4c 33905 | Lemma for ~ 4at . Frequen... |
4atlem4d 33906 | Lemma for ~ 4at . Frequen... |
4atlem9 33907 | Lemma for ~ 4at . Substit... |
4atlem10a 33908 | Lemma for ~ 4at . Substit... |
4atlem10b 33909 | Lemma for ~ 4at . Substit... |
4atlem10 33910 | Lemma for ~ 4at . Combine... |
4atlem11a 33911 | Lemma for ~ 4at . Substit... |
4atlem11b 33912 | Lemma for ~ 4at . Substit... |
4atlem11 33913 | Lemma for ~ 4at . Combine... |
4atlem12a 33914 | Lemma for ~ 4at . Substit... |
4atlem12b 33915 | Lemma for ~ 4at . Substit... |
4atlem12 33916 | Lemma for ~ 4at . Combine... |
4at 33917 | Four atoms determine a lat... |
4at2 33918 | Four atoms determine a lat... |
lplncvrlvol2 33919 | A lattice line under a lat... |
lplncvrlvol 33920 | An element covering a latt... |
lvolcmp 33921 | If two lattice planes are ... |
lvolnltN 33922 | Two lattice volumes cannot... |
2lplnja 33923 | The join of two different ... |
2lplnj 33924 | The join of two different ... |
2lplnm2N 33925 | The meet of two different ... |
2lplnmj 33926 | The meet of two lattice pl... |
dalemkehl 33927 | Lemma for ~ dath . Freque... |
dalemkelat 33928 | Lemma for ~ dath . Freque... |
dalemkeop 33929 | Lemma for ~ dath . Freque... |
dalempea 33930 | Lemma for ~ dath . Freque... |
dalemqea 33931 | Lemma for ~ dath . Freque... |
dalemrea 33932 | Lemma for ~ dath . Freque... |
dalemsea 33933 | Lemma for ~ dath . Freque... |
dalemtea 33934 | Lemma for ~ dath . Freque... |
dalemuea 33935 | Lemma for ~ dath . Freque... |
dalemyeo 33936 | Lemma for ~ dath . Freque... |
dalemzeo 33937 | Lemma for ~ dath . Freque... |
dalemclpjs 33938 | Lemma for ~ dath . Freque... |
dalemclqjt 33939 | Lemma for ~ dath . Freque... |
dalemclrju 33940 | Lemma for ~ dath . Freque... |
dalem-clpjq 33941 | Lemma for ~ dath . Freque... |
dalemceb 33942 | Lemma for ~ dath . Freque... |
dalempeb 33943 | Lemma for ~ dath . Freque... |
dalemqeb 33944 | Lemma for ~ dath . Freque... |
dalemreb 33945 | Lemma for ~ dath . Freque... |
dalemseb 33946 | Lemma for ~ dath . Freque... |
dalemteb 33947 | Lemma for ~ dath . Freque... |
dalemueb 33948 | Lemma for ~ dath . Freque... |
dalempjqeb 33949 | Lemma for ~ dath . Freque... |
dalemsjteb 33950 | Lemma for ~ dath . Freque... |
dalemtjueb 33951 | Lemma for ~ dath . Freque... |
dalemqrprot 33952 | Lemma for ~ dath . Freque... |
dalemyeb 33953 | Lemma for ~ dath . Freque... |
dalemcnes 33954 | Lemma for ~ dath . Freque... |
dalempnes 33955 | Lemma for ~ dath . Freque... |
dalemqnet 33956 | Lemma for ~ dath . Freque... |
dalempjsen 33957 | Lemma for ~ dath . Freque... |
dalemply 33958 | Lemma for ~ dath . Freque... |
dalemsly 33959 | Lemma for ~ dath . Freque... |
dalemswapyz 33960 | Lemma for ~ dath . Swap t... |
dalemrot 33961 | Lemma for ~ dath . Rotate... |
dalemrotyz 33962 | Lemma for ~ dath . Rotate... |
dalem1 33963 | Lemma for ~ dath . Show t... |
dalemcea 33964 | Lemma for ~ dath . Freque... |
dalem2 33965 | Lemma for ~ dath . Show t... |
dalemdea 33966 | Lemma for ~ dath . Freque... |
dalemeea 33967 | Lemma for ~ dath . Freque... |
dalem3 33968 | Lemma for ~ dalemdnee . (... |
dalem4 33969 | Lemma for ~ dalemdnee . (... |
dalemdnee 33970 | Lemma for ~ dath . Axis o... |
dalem5 33971 | Lemma for ~ dath . Atom `... |
dalem6 33972 | Lemma for ~ dath . Analog... |
dalem7 33973 | Lemma for ~ dath . Analog... |
dalem8 33974 | Lemma for ~ dath . Plane ... |
dalem-cly 33975 | Lemma for ~ dalem9 . Cent... |
dalem9 33976 | Lemma for ~ dath . Since ... |
dalem10 33977 | Lemma for ~ dath . Atom `... |
dalem11 33978 | Lemma for ~ dath . Analog... |
dalem12 33979 | Lemma for ~ dath . Analog... |
dalem13 33980 | Lemma for ~ dalem14 . (Co... |
dalem14 33981 | Lemma for ~ dath . Planes... |
dalem15 33982 | Lemma for ~ dath . The ax... |
dalem16 33983 | Lemma for ~ dath . The at... |
dalem17 33984 | Lemma for ~ dath . When p... |
dalem18 33985 | Lemma for ~ dath . Show t... |
dalem19 33986 | Lemma for ~ dath . Show t... |
dalemccea 33987 | Lemma for ~ dath . Freque... |
dalemddea 33988 | Lemma for ~ dath . Freque... |
dalem-ccly 33989 | Lemma for ~ dath . Freque... |
dalem-ddly 33990 | Lemma for ~ dath . Freque... |
dalemccnedd 33991 | Lemma for ~ dath . Freque... |
dalemclccjdd 33992 | Lemma for ~ dath . Freque... |
dalemcceb 33993 | Lemma for ~ dath . Freque... |
dalemswapyzps 33994 | Lemma for ~ dath . Swap t... |
dalemrotps 33995 | Lemma for ~ dath . Rotate... |
dalemcjden 33996 | Lemma for ~ dath . Show t... |
dalem20 33997 | Lemma for ~ dath . Show t... |
dalem21 33998 | Lemma for ~ dath . Show t... |
dalem22 33999 | Lemma for ~ dath . Show t... |
dalem23 34000 | Lemma for ~ dath . Show t... |
dalem24 34001 | Lemma for ~ dath . Show t... |
dalem25 34002 | Lemma for ~ dath . Show t... |
dalem27 34003 | Lemma for ~ dath . Show t... |
dalem28 34004 | Lemma for ~ dath . Lemma ... |
dalem29 34005 | Lemma for ~ dath . Analog... |
dalem30 34006 | Lemma for ~ dath . Analog... |
dalem31N 34007 | Lemma for ~ dath . Analog... |
dalem32 34008 | Lemma for ~ dath . Analog... |
dalem33 34009 | Lemma for ~ dath . Analog... |
dalem34 34010 | Lemma for ~ dath . Analog... |
dalem35 34011 | Lemma for ~ dath . Analog... |
dalem36 34012 | Lemma for ~ dath . Analog... |
dalem37 34013 | Lemma for ~ dath . Analog... |
dalem38 34014 | Lemma for ~ dath . Plane ... |
dalem39 34015 | Lemma for ~ dath . Auxili... |
dalem40 34016 | Lemma for ~ dath . Analog... |
dalem41 34017 | Lemma for ~ dath . (Contr... |
dalem42 34018 | Lemma for ~ dath . Auxili... |
dalem43 34019 | Lemma for ~ dath . Planes... |
dalem44 34020 | Lemma for ~ dath . Dummy ... |
dalem45 34021 | Lemma for ~ dath . Dummy ... |
dalem46 34022 | Lemma for ~ dath . Analog... |
dalem47 34023 | Lemma for ~ dath . Analog... |
dalem48 34024 | Lemma for ~ dath . Analog... |
dalem49 34025 | Lemma for ~ dath . Analog... |
dalem50 34026 | Lemma for ~ dath . Analog... |
dalem51 34027 | Lemma for ~ dath . Constr... |
dalem52 34028 | Lemma for ~ dath . Lines ... |
dalem53 34029 | Lemma for ~ dath . The au... |
dalem54 34030 | Lemma for ~ dath . Line `... |
dalem55 34031 | Lemma for ~ dath . Lines ... |
dalem56 34032 | Lemma for ~ dath . Analog... |
dalem57 34033 | Lemma for ~ dath . Axis o... |
dalem58 34034 | Lemma for ~ dath . Analog... |
dalem59 34035 | Lemma for ~ dath . Analog... |
dalem60 34036 | Lemma for ~ dath . ` B ` i... |
dalem61 34037 | Lemma for ~ dath . Show t... |
dalem62 34038 | Lemma for ~ dath . Elimin... |
dalem63 34039 | Lemma for ~ dath . Combin... |
dath 34040 | Desargues' Theorem of proj... |
dath2 34041 | Version of Desargues' Theo... |
lineset 34042 | The set of lines in a Hilb... |
isline 34043 | The predicate "is a line".... |
islinei 34044 | Condition implying "is a l... |
pointsetN 34045 | The set of points in a Hil... |
ispointN 34046 | The predicate "is a point"... |
atpointN 34047 | The singleton of an atom i... |
psubspset 34048 | The set of projective subs... |
ispsubsp 34049 | The predicate "is a projec... |
ispsubsp2 34050 | The predicate "is a projec... |
psubspi 34051 | Property of a projective s... |
psubspi2N 34052 | Property of a projective s... |
0psubN 34053 | The empty set is a project... |
snatpsubN 34054 | The singleton of an atom i... |
pointpsubN 34055 | A point (singleton of an a... |
linepsubN 34056 | A line is a projective sub... |
atpsubN 34057 | The set of all atoms is a ... |
psubssat 34058 | A projective subspace cons... |
psubatN 34059 | A member of a projective s... |
pmapfval 34060 | The projective map of a Hi... |
pmapval 34061 | Value of the projective ma... |
elpmap 34062 | Member of a projective map... |
pmapssat 34063 | The projective map of a Hi... |
pmapssbaN 34064 | A weakening of ~ pmapssat ... |
pmaple 34065 | The projective map of a Hi... |
pmap11 34066 | The projective map of a Hi... |
pmapat 34067 | The projective map of an a... |
elpmapat 34068 | Member of the projective m... |
pmap0 34069 | Value of the projective ma... |
pmapeq0 34070 | A projective map value is ... |
pmap1N 34071 | Value of the projective ma... |
pmapsub 34072 | The projective map of a Hi... |
pmapglbx 34073 | The projective map of the ... |
pmapglb 34074 | The projective map of the ... |
pmapglb2N 34075 | The projective map of the ... |
pmapglb2xN 34076 | The projective map of the ... |
pmapmeet 34077 | The projective map of a me... |
isline2 34078 | Definition of line in term... |
linepmap 34079 | A line described with a pr... |
isline3 34080 | Definition of line in term... |
isline4N 34081 | Definition of line in term... |
lneq2at 34082 | A line equals the join of ... |
lnatexN 34083 | There is an atom in a line... |
lnjatN 34084 | Given an atom in a line, t... |
lncvrelatN 34085 | A lattice element covered ... |
lncvrat 34086 | A line covers the atoms it... |
lncmp 34087 | If two lines are comparabl... |
2lnat 34088 | Two intersecting lines int... |
2atm2atN 34089 | Two joins with a common at... |
2llnma1b 34090 | Generalization of ~ 2llnma... |
2llnma1 34091 | Two different intersecting... |
2llnma3r 34092 | Two different intersecting... |
2llnma2 34093 | Two different intersecting... |
2llnma2rN 34094 | Two different intersecting... |
cdlema1N 34095 | A condition for required f... |
cdlema2N 34096 | A condition for required f... |
cdlemblem 34097 | Lemma for ~ cdlemb . (Con... |
cdlemb 34098 | Given two atoms not less t... |
paddfval 34101 | Projective subspace sum op... |
paddval 34102 | Projective subspace sum op... |
elpadd 34103 | Member of a projective sub... |
elpaddn0 34104 | Member of projective subsp... |
paddvaln0N 34105 | Projective subspace sum op... |
elpaddri 34106 | Condition implying members... |
elpaddatriN 34107 | Condition implying members... |
elpaddat 34108 | Membership in a projective... |
elpaddatiN 34109 | Consequence of membership ... |
elpadd2at 34110 | Membership in a projective... |
elpadd2at2 34111 | Membership in a projective... |
paddunssN 34112 | Projective subspace sum in... |
elpadd0 34113 | Member of projective subsp... |
paddval0 34114 | Projective subspace sum wi... |
padd01 34115 | Projective subspace sum wi... |
padd02 34116 | Projective subspace sum wi... |
paddcom 34117 | Projective subspace sum co... |
paddssat 34118 | A projective subspace sum ... |
sspadd1 34119 | A projective subspace sum ... |
sspadd2 34120 | A projective subspace sum ... |
paddss1 34121 | Subset law for projective ... |
paddss2 34122 | Subset law for projective ... |
paddss12 34123 | Subset law for projective ... |
paddasslem1 34124 | Lemma for ~ paddass . (Co... |
paddasslem2 34125 | Lemma for ~ paddass . (Co... |
paddasslem3 34126 | Lemma for ~ paddass . Res... |
paddasslem4 34127 | Lemma for ~ paddass . Com... |
paddasslem5 34128 | Lemma for ~ paddass . Sho... |
paddasslem6 34129 | Lemma for ~ paddass . (Co... |
paddasslem7 34130 | Lemma for ~ paddass . Com... |
paddasslem8 34131 | Lemma for ~ paddass . (Co... |
paddasslem9 34132 | Lemma for ~ paddass . Com... |
paddasslem10 34133 | Lemma for ~ paddass . Use... |
paddasslem11 34134 | Lemma for ~ paddass . The... |
paddasslem12 34135 | Lemma for ~ paddass . The... |
paddasslem13 34136 | Lemma for ~ paddass . The... |
paddasslem14 34137 | Lemma for ~ paddass . Rem... |
paddasslem15 34138 | Lemma for ~ paddass . Use... |
paddasslem16 34139 | Lemma for ~ paddass . Use... |
paddasslem17 34140 | Lemma for ~ paddass . The... |
paddasslem18 34141 | Lemma for ~ paddass . Com... |
paddass 34142 | Projective subspace sum is... |
padd12N 34143 | Commutative/associative la... |
padd4N 34144 | Rearrangement of 4 terms i... |
paddidm 34145 | Projective subspace sum is... |
paddclN 34146 | The projective sum of two ... |
paddssw1 34147 | Subset law for projective ... |
paddssw2 34148 | Subset law for projective ... |
paddss 34149 | Subset law for projective ... |
pmodlem1 34150 | Lemma for ~ pmod1i . (Con... |
pmodlem2 34151 | Lemma for ~ pmod1i . (Con... |
pmod1i 34152 | The modular law holds in a... |
pmod2iN 34153 | Dual of the modular law. ... |
pmodN 34154 | The modular law for projec... |
pmodl42N 34155 | Lemma derived from modular... |
pmapjoin 34156 | The projective map of the ... |
pmapjat1 34157 | The projective map of the ... |
pmapjat2 34158 | The projective map of the ... |
pmapjlln1 34159 | The projective map of the ... |
hlmod1i 34160 | A version of the modular l... |
atmod1i1 34161 | Version of modular law ~ p... |
atmod1i1m 34162 | Version of modular law ~ p... |
atmod1i2 34163 | Version of modular law ~ p... |
llnmod1i2 34164 | Version of modular law ~ p... |
atmod2i1 34165 | Version of modular law ~ p... |
atmod2i2 34166 | Version of modular law ~ p... |
llnmod2i2 34167 | Version of modular law ~ p... |
atmod3i1 34168 | Version of modular law tha... |
atmod3i2 34169 | Version of modular law tha... |
atmod4i1 34170 | Version of modular law tha... |
atmod4i2 34171 | Version of modular law tha... |
llnexchb2lem 34172 | Lemma for ~ llnexchb2 . (... |
llnexchb2 34173 | Line exchange property (co... |
llnexch2N 34174 | Line exchange property (co... |
dalawlem1 34175 | Lemma for ~ dalaw . Speci... |
dalawlem2 34176 | Lemma for ~ dalaw . Utili... |
dalawlem3 34177 | Lemma for ~ dalaw . First... |
dalawlem4 34178 | Lemma for ~ dalaw . Secon... |
dalawlem5 34179 | Lemma for ~ dalaw . Speci... |
dalawlem6 34180 | Lemma for ~ dalaw . First... |
dalawlem7 34181 | Lemma for ~ dalaw . Secon... |
dalawlem8 34182 | Lemma for ~ dalaw . Speci... |
dalawlem9 34183 | Lemma for ~ dalaw . Speci... |
dalawlem10 34184 | Lemma for ~ dalaw . Combi... |
dalawlem11 34185 | Lemma for ~ dalaw . First... |
dalawlem12 34186 | Lemma for ~ dalaw . Secon... |
dalawlem13 34187 | Lemma for ~ dalaw . Speci... |
dalawlem14 34188 | Lemma for ~ dalaw . Combi... |
dalawlem15 34189 | Lemma for ~ dalaw . Swap ... |
dalaw 34190 | Desargues' law, derived fr... |
pclfvalN 34193 | The projective subspace cl... |
pclvalN 34194 | Value of the projective su... |
pclclN 34195 | Closure of the projective ... |
elpclN 34196 | Membership in the projecti... |
elpcliN 34197 | Implication of membership ... |
pclssN 34198 | Ordering is preserved by s... |
pclssidN 34199 | A set of atoms is included... |
pclidN 34200 | The projective subspace cl... |
pclbtwnN 34201 | A projective subspace sand... |
pclunN 34202 | The projective subspace cl... |
pclun2N 34203 | The projective subspace cl... |
pclfinN 34204 | The projective subspace cl... |
pclcmpatN 34205 | The set of projective subs... |
polfvalN 34208 | The projective subspace po... |
polvalN 34209 | Value of the projective su... |
polval2N 34210 | Alternate expression for v... |
polsubN 34211 | The polarity of a set of a... |
polssatN 34212 | The polarity of a set of a... |
pol0N 34213 | The polarity of the empty ... |
pol1N 34214 | The polarity of the whole ... |
2pol0N 34215 | The closed subspace closur... |
polpmapN 34216 | The polarity of a projecti... |
2polpmapN 34217 | Double polarity of a proje... |
2polvalN 34218 | Value of double polarity. ... |
2polssN 34219 | A set of atoms is a subset... |
3polN 34220 | Triple polarity cancels to... |
polcon3N 34221 | Contraposition law for pol... |
2polcon4bN 34222 | Contraposition law for pol... |
polcon2N 34223 | Contraposition law for pol... |
polcon2bN 34224 | Contraposition law for pol... |
pclss2polN 34225 | The projective subspace cl... |
pcl0N 34226 | The projective subspace cl... |
pcl0bN 34227 | The projective subspace cl... |
pmaplubN 34228 | The LUB of a projective ma... |
sspmaplubN 34229 | A set of atoms is a subset... |
2pmaplubN 34230 | Double projective map of a... |
paddunN 34231 | The closure of the project... |
poldmj1N 34232 | De Morgan's law for polari... |
pmapj2N 34233 | The projective map of the ... |
pmapocjN 34234 | The projective map of the ... |
polatN 34235 | The polarity of the single... |
2polatN 34236 | Double polarity of the sin... |
pnonsingN 34237 | The intersection of a set ... |
psubclsetN 34240 | The set of closed projecti... |
ispsubclN 34241 | The predicate "is a closed... |
psubcliN 34242 | Property of a closed proje... |
psubcli2N 34243 | Property of a closed proje... |
psubclsubN 34244 | A closed projective subspa... |
psubclssatN 34245 | A closed projective subspa... |
pmapidclN 34246 | Projective map of the LUB ... |
0psubclN 34247 | The empty set is a closed ... |
1psubclN 34248 | The set of all atoms is a ... |
atpsubclN 34249 | A point (singleton of an a... |
pmapsubclN 34250 | A projective map value is ... |
ispsubcl2N 34251 | Alternate predicate for "i... |
psubclinN 34252 | The intersection of two cl... |
paddatclN 34253 | The projective sum of a cl... |
pclfinclN 34254 | The projective subspace cl... |
linepsubclN 34255 | A line is a closed project... |
polsubclN 34256 | A polarity is a closed pro... |
poml4N 34257 | Orthomodular law for proje... |
poml5N 34258 | Orthomodular law for proje... |
poml6N 34259 | Orthomodular law for proje... |
osumcllem1N 34260 | Lemma for ~ osumclN . (Co... |
osumcllem2N 34261 | Lemma for ~ osumclN . (Co... |
osumcllem3N 34262 | Lemma for ~ osumclN . (Co... |
osumcllem4N 34263 | Lemma for ~ osumclN . (Co... |
osumcllem5N 34264 | Lemma for ~ osumclN . (Co... |
osumcllem6N 34265 | Lemma for ~ osumclN . Use... |
osumcllem7N 34266 | Lemma for ~ osumclN . (Co... |
osumcllem8N 34267 | Lemma for ~ osumclN . (Co... |
osumcllem9N 34268 | Lemma for ~ osumclN . (Co... |
osumcllem10N 34269 | Lemma for ~ osumclN . Con... |
osumcllem11N 34270 | Lemma for ~ osumclN . (Co... |
osumclN 34271 | Closure of orthogonal sum.... |
pmapojoinN 34272 | For orthogonal elements, p... |
pexmidN 34273 | Excluded middle law for cl... |
pexmidlem1N 34274 | Lemma for ~ pexmidN . Hol... |
pexmidlem2N 34275 | Lemma for ~ pexmidN . (Co... |
pexmidlem3N 34276 | Lemma for ~ pexmidN . Use... |
pexmidlem4N 34277 | Lemma for ~ pexmidN . (Co... |
pexmidlem5N 34278 | Lemma for ~ pexmidN . (Co... |
pexmidlem6N 34279 | Lemma for ~ pexmidN . (Co... |
pexmidlem7N 34280 | Lemma for ~ pexmidN . Con... |
pexmidlem8N 34281 | Lemma for ~ pexmidN . The... |
pexmidALTN 34282 | Excluded middle law for cl... |
pl42lem1N 34283 | Lemma for ~ pl42N . (Cont... |
pl42lem2N 34284 | Lemma for ~ pl42N . (Cont... |
pl42lem3N 34285 | Lemma for ~ pl42N . (Cont... |
pl42lem4N 34286 | Lemma for ~ pl42N . (Cont... |
pl42N 34287 | Law holding in a Hilbert l... |
watfvalN 34296 | The W atoms function. (Co... |
watvalN 34297 | Value of the W atoms funct... |
iswatN 34298 | The predicate "is a W atom... |
lhpset 34299 | The set of co-atoms (latti... |
islhp 34300 | The predicate "is a co-ato... |
islhp2 34301 | The predicate "is a co-ato... |
lhpbase 34302 | A co-atom is a member of t... |
lhp1cvr 34303 | The lattice unit covers a ... |
lhplt 34304 | An atom under a co-atom is... |
lhp2lt 34305 | The join of two atoms unde... |
lhpexlt 34306 | There exists an atom less ... |
lhp0lt 34307 | A co-atom is greater than ... |
lhpn0 34308 | A co-atom is nonzero. TOD... |
lhpexle 34309 | There exists an atom under... |
lhpexnle 34310 | There exists an atom not u... |
lhpexle1lem 34311 | Lemma for ~ lhpexle1 and o... |
lhpexle1 34312 | There exists an atom under... |
lhpexle2lem 34313 | Lemma for ~ lhpexle2 . (C... |
lhpexle2 34314 | There exists atom under a ... |
lhpexle3lem 34315 | There exists atom under a ... |
lhpexle3 34316 | There exists atom under a ... |
lhpex2leN 34317 | There exist at least two d... |
lhpoc 34318 | The orthocomplement of a c... |
lhpoc2N 34319 | The orthocomplement of an ... |
lhpocnle 34320 | The orthocomplement of a c... |
lhpocat 34321 | The orthocomplement of a c... |
lhpocnel 34322 | The orthocomplement of a c... |
lhpocnel2 34323 | The orthocomplement of a c... |
lhpjat1 34324 | The join of a co-atom (hyp... |
lhpjat2 34325 | The join of a co-atom (hyp... |
lhpj1 34326 | The join of a co-atom (hyp... |
lhpmcvr 34327 | The meet of a lattice hype... |
lhpmcvr2 34328 | Alternate way to express t... |
lhpmcvr3 34329 | Specialization of ~ lhpmcv... |
lhpmcvr4N 34330 | Specialization of ~ lhpmcv... |
lhpmcvr5N 34331 | Specialization of ~ lhpmcv... |
lhpmcvr6N 34332 | Specialization of ~ lhpmcv... |
lhpm0atN 34333 | If the meet of a lattice h... |
lhpmat 34334 | An element covered by the ... |
lhpmatb 34335 | An element covered by the ... |
lhp2at0 34336 | Join and meet with differe... |
lhp2atnle 34337 | Inequality for 2 different... |
lhp2atne 34338 | Inequality for joins with ... |
lhp2at0nle 34339 | Inequality for 2 different... |
lhp2at0ne 34340 | Inequality for joins with ... |
lhpelim 34341 | Eliminate an atom not unde... |
lhpmod2i2 34342 | Modular law for hyperplane... |
lhpmod6i1 34343 | Modular law for hyperplane... |
lhprelat3N 34344 | The Hilbert lattice is rel... |
cdlemb2 34345 | Given two atoms not under ... |
lhple 34346 | Property of a lattice elem... |
lhpat 34347 | Create an atom under a co-... |
lhpat4N 34348 | Property of an atom under ... |
lhpat2 34349 | Create an atom under a co-... |
lhpat3 34350 | There is only one atom und... |
4atexlemk 34351 | Lemma for ~ 4atexlem7 . (... |
4atexlemw 34352 | Lemma for ~ 4atexlem7 . (... |
4atexlempw 34353 | Lemma for ~ 4atexlem7 . (... |
4atexlemp 34354 | Lemma for ~ 4atexlem7 . (... |
4atexlemq 34355 | Lemma for ~ 4atexlem7 . (... |
4atexlems 34356 | Lemma for ~ 4atexlem7 . (... |
4atexlemt 34357 | Lemma for ~ 4atexlem7 . (... |
4atexlemutvt 34358 | Lemma for ~ 4atexlem7 . (... |
4atexlempnq 34359 | Lemma for ~ 4atexlem7 . (... |
4atexlemnslpq 34360 | Lemma for ~ 4atexlem7 . (... |
4atexlemkl 34361 | Lemma for ~ 4atexlem7 . (... |
4atexlemkc 34362 | Lemma for ~ 4atexlem7 . (... |
4atexlemwb 34363 | Lemma for ~ 4atexlem7 . (... |
4atexlempsb 34364 | Lemma for ~ 4atexlem7 . (... |
4atexlemqtb 34365 | Lemma for ~ 4atexlem7 . (... |
4atexlempns 34366 | Lemma for ~ 4atexlem7 . (... |
4atexlemswapqr 34367 | Lemma for ~ 4atexlem7 . S... |
4atexlemu 34368 | Lemma for ~ 4atexlem7 . (... |
4atexlemv 34369 | Lemma for ~ 4atexlem7 . (... |
4atexlemunv 34370 | Lemma for ~ 4atexlem7 . (... |
4atexlemtlw 34371 | Lemma for ~ 4atexlem7 . (... |
4atexlemntlpq 34372 | Lemma for ~ 4atexlem7 . (... |
4atexlemc 34373 | Lemma for ~ 4atexlem7 . (... |
4atexlemnclw 34374 | Lemma for ~ 4atexlem7 . (... |
4atexlemex2 34375 | Lemma for ~ 4atexlem7 . S... |
4atexlemcnd 34376 | Lemma for ~ 4atexlem7 . (... |
4atexlemex4 34377 | Lemma for ~ 4atexlem7 . S... |
4atexlemex6 34378 | Lemma for ~ 4atexlem7 . (... |
4atexlem7 34379 | Whenever there are at leas... |
4atex 34380 | Whenever there are at leas... |
4atex2 34381 | More general version of ~ ... |
4atex2-0aOLDN 34382 | Same as ~ 4atex2 except th... |
4atex2-0bOLDN 34383 | Same as ~ 4atex2 except th... |
4atex2-0cOLDN 34384 | Same as ~ 4atex2 except th... |
4atex3 34385 | More general version of ~ ... |
lautset 34386 | The set of lattice automor... |
islaut 34387 | The predictate "is a latti... |
lautle 34388 | Less-than or equal propert... |
laut1o 34389 | A lattice automorphism is ... |
laut11 34390 | One-to-one property of a l... |
lautcl 34391 | A lattice automorphism val... |
lautcnvclN 34392 | Reverse closure of a latti... |
lautcnvle 34393 | Less-than or equal propert... |
lautcnv 34394 | The converse of a lattice ... |
lautlt 34395 | Less-than property of a la... |
lautcvr 34396 | Covering property of a lat... |
lautj 34397 | Meet property of a lattice... |
lautm 34398 | Meet property of a lattice... |
lauteq 34399 | A lattice automorphism arg... |
idlaut 34400 | The identity function is a... |
lautco 34401 | The composition of two lat... |
pautsetN 34402 | The set of projective auto... |
ispautN 34403 | The predictate "is a proje... |
ldilfset 34412 | The mapping from fiducial ... |
ldilset 34413 | The set of lattice dilatio... |
isldil 34414 | The predicate "is a lattic... |
ldillaut 34415 | A lattice dilation is an a... |
ldil1o 34416 | A lattice dilation is a on... |
ldilval 34417 | Value of a lattice dilatio... |
idldil 34418 | The identity function is a... |
ldilcnv 34419 | The converse of a lattice ... |
ldilco 34420 | The composition of two lat... |
ltrnfset 34421 | The set of all lattice tra... |
ltrnset 34422 | The set of lattice transla... |
isltrn 34423 | The predicate "is a lattic... |
isltrn2N 34424 | The predicate "is a lattic... |
ltrnu 34425 | Uniqueness property of a l... |
ltrnldil 34426 | A lattice translation is a... |
ltrnlaut 34427 | A lattice translation is a... |
ltrn1o 34428 | A lattice translation is a... |
ltrncl 34429 | Closure of a lattice trans... |
ltrn11 34430 | One-to-one property of a l... |
ltrncnvnid 34431 | If a translation is differ... |
ltrncoidN 34432 | Two translations are equal... |
ltrnle 34433 | Less-than or equal propert... |
ltrncnvleN 34434 | Less-than or equal propert... |
ltrnm 34435 | Lattice translation of a m... |
ltrnj 34436 | Lattice translation of a m... |
ltrncvr 34437 | Covering property of a lat... |
ltrnval1 34438 | Value of a lattice transla... |
ltrnid 34439 | A lattice translation is t... |
ltrnnid 34440 | If a lattice translation i... |
ltrnatb 34441 | The lattice translation of... |
ltrncnvatb 34442 | The converse of the lattic... |
ltrnel 34443 | The lattice translation of... |
ltrnat 34444 | The lattice translation of... |
ltrncnvat 34445 | The converse of the lattic... |
ltrncnvel 34446 | The converse of the lattic... |
ltrncoelN 34447 | Composition of lattice tra... |
ltrncoat 34448 | Composition of lattice tra... |
ltrncoval 34449 | Two ways to express value ... |
ltrncnv 34450 | The converse of a lattice ... |
ltrn11at 34451 | Frequently used one-to-one... |
ltrneq2 34452 | The equality of two transl... |
ltrneq 34453 | The equality of two transl... |
idltrn 34454 | The identity function is a... |
ltrnmw 34455 | Property of lattice transl... |
ltrnmwOLD 34456 | Property of lattice transl... |
dilfsetN 34457 | The mapping from fiducial ... |
dilsetN 34458 | The set of dilations for a... |
isdilN 34459 | The predicate "is a dilati... |
trnfsetN 34460 | The mapping from fiducial ... |
trnsetN 34461 | The set of translations fo... |
istrnN 34462 | The predicate "is a transl... |
trlfset 34465 | The set of all traces of l... |
trlset 34466 | The set of traces of latti... |
trlval 34467 | The value of the trace of ... |
trlval2 34468 | The value of the trace of ... |
trlcl 34469 | Closure of the trace of a ... |
trlcnv 34470 | The trace of the converse ... |
trljat1 34471 | The value of a translation... |
trljat2 34472 | The value of a translation... |
trljat3 34473 | The value of a translation... |
trlat 34474 | If an atom differs from it... |
trl0 34475 | If an atom not under the f... |
trlator0 34476 | The trace of a lattice tra... |
trlatn0 34477 | The trace of a lattice tra... |
trlnidat 34478 | The trace of a lattice tra... |
ltrnnidn 34479 | If a lattice translation i... |
ltrnideq 34480 | Property of the identity l... |
trlid0 34481 | The trace of the identity ... |
trlnidatb 34482 | A lattice translation is n... |
trlid0b 34483 | A lattice translation is t... |
trlnid 34484 | Different translations wit... |
ltrn2ateq 34485 | Property of the equality o... |
ltrnateq 34486 | If any atom (under ` W ` )... |
ltrnatneq 34487 | If any atom (under ` W ` )... |
ltrnatlw 34488 | If the value of an atom eq... |
trlle 34489 | The trace of a lattice tra... |
trlne 34490 | The trace of a lattice tra... |
trlnle 34491 | The atom not under the fid... |
trlval3 34492 | The value of the trace of ... |
trlval4 34493 | The value of the trace of ... |
trlval5 34494 | The value of the trace of ... |
arglem1N 34495 | Lemma for Desargues' law. ... |
cdlemc1 34496 | Part of proof of Lemma C i... |
cdlemc2 34497 | Part of proof of Lemma C i... |
cdlemc3 34498 | Part of proof of Lemma C i... |
cdlemc4 34499 | Part of proof of Lemma C i... |
cdlemc5 34500 | Lemma for ~ cdlemc . (Con... |
cdlemc6 34501 | Lemma for ~ cdlemc . (Con... |
cdlemc 34502 | Lemma C in [Crawley] p. 11... |
cdlemd1 34503 | Part of proof of Lemma D i... |
cdlemd2 34504 | Part of proof of Lemma D i... |
cdlemd3 34505 | Part of proof of Lemma D i... |
cdlemd4 34506 | Part of proof of Lemma D i... |
cdlemd5 34507 | Part of proof of Lemma D i... |
cdlemd6 34508 | Part of proof of Lemma D i... |
cdlemd7 34509 | Part of proof of Lemma D i... |
cdlemd8 34510 | Part of proof of Lemma D i... |
cdlemd9 34511 | Part of proof of Lemma D i... |
cdlemd 34512 | If two translations agree ... |
ltrneq3 34513 | Two translations agree at ... |
cdleme00a 34514 | Part of proof of Lemma E i... |
cdleme0aa 34515 | Part of proof of Lemma E i... |
cdleme0a 34516 | Part of proof of Lemma E i... |
cdleme0b 34517 | Part of proof of Lemma E i... |
cdleme0c 34518 | Part of proof of Lemma E i... |
cdleme0cp 34519 | Part of proof of Lemma E i... |
cdleme0cq 34520 | Part of proof of Lemma E i... |
cdleme0dN 34521 | Part of proof of Lemma E i... |
cdleme0e 34522 | Part of proof of Lemma E i... |
cdleme0fN 34523 | Part of proof of Lemma E i... |
cdleme0gN 34524 | Part of proof of Lemma E i... |
cdlemeulpq 34525 | Part of proof of Lemma E i... |
cdleme01N 34526 | Part of proof of Lemma E i... |
cdleme02N 34527 | Part of proof of Lemma E i... |
cdleme0ex1N 34528 | Part of proof of Lemma E i... |
cdleme0ex2N 34529 | Part of proof of Lemma E i... |
cdleme0moN 34530 | Part of proof of Lemma E i... |
cdleme1b 34531 | Part of proof of Lemma E i... |
cdleme1 34532 | Part of proof of Lemma E i... |
cdleme2 34533 | Part of proof of Lemma E i... |
cdleme3b 34534 | Part of proof of Lemma E i... |
cdleme3c 34535 | Part of proof of Lemma E i... |
cdleme3d 34536 | Part of proof of Lemma E i... |
cdleme3e 34537 | Part of proof of Lemma E i... |
cdleme3fN 34538 | Part of proof of Lemma E i... |
cdleme3g 34539 | Part of proof of Lemma E i... |
cdleme3h 34540 | Part of proof of Lemma E i... |
cdleme3fa 34541 | Part of proof of Lemma E i... |
cdleme3 34542 | Part of proof of Lemma E i... |
cdleme4 34543 | Part of proof of Lemma E i... |
cdleme4a 34544 | Part of proof of Lemma E i... |
cdleme5 34545 | Part of proof of Lemma E i... |
cdleme6 34546 | Part of proof of Lemma E i... |
cdleme7aa 34547 | Part of proof of Lemma E i... |
cdleme7a 34548 | Part of proof of Lemma E i... |
cdleme7b 34549 | Part of proof of Lemma E i... |
cdleme7c 34550 | Part of proof of Lemma E i... |
cdleme7d 34551 | Part of proof of Lemma E i... |
cdleme7e 34552 | Part of proof of Lemma E i... |
cdleme7ga 34553 | Part of proof of Lemma E i... |
cdleme7 34554 | Part of proof of Lemma E i... |
cdleme8 34555 | Part of proof of Lemma E i... |
cdleme9a 34556 | Part of proof of Lemma E i... |
cdleme9b 34557 | Utility lemma for Lemma E ... |
cdleme9 34558 | Part of proof of Lemma E i... |
cdleme10 34559 | Part of proof of Lemma E i... |
cdleme8tN 34560 | Part of proof of Lemma E i... |
cdleme9taN 34561 | Part of proof of Lemma E i... |
cdleme9tN 34562 | Part of proof of Lemma E i... |
cdleme10tN 34563 | Part of proof of Lemma E i... |
cdleme16aN 34564 | Part of proof of Lemma E i... |
cdleme11a 34565 | Part of proof of Lemma E i... |
cdleme11c 34566 | Part of proof of Lemma E i... |
cdleme11dN 34567 | Part of proof of Lemma E i... |
cdleme11e 34568 | Part of proof of Lemma E i... |
cdleme11fN 34569 | Part of proof of Lemma E i... |
cdleme11g 34570 | Part of proof of Lemma E i... |
cdleme11h 34571 | Part of proof of Lemma E i... |
cdleme11j 34572 | Part of proof of Lemma E i... |
cdleme11k 34573 | Part of proof of Lemma E i... |
cdleme11l 34574 | Part of proof of Lemma E i... |
cdleme11 34575 | Part of proof of Lemma E i... |
cdleme12 34576 | Part of proof of Lemma E i... |
cdleme13 34577 | Part of proof of Lemma E i... |
cdleme14 34578 | Part of proof of Lemma E i... |
cdleme15a 34579 | Part of proof of Lemma E i... |
cdleme15b 34580 | Part of proof of Lemma E i... |
cdleme15c 34581 | Part of proof of Lemma E i... |
cdleme15d 34582 | Part of proof of Lemma E i... |
cdleme15 34583 | Part of proof of Lemma E i... |
cdleme16b 34584 | Part of proof of Lemma E i... |
cdleme16c 34585 | Part of proof of Lemma E i... |
cdleme16d 34586 | Part of proof of Lemma E i... |
cdleme16e 34587 | Part of proof of Lemma E i... |
cdleme16f 34588 | Part of proof of Lemma E i... |
cdleme16g 34589 | Part of proof of Lemma E i... |
cdleme16 34590 | Part of proof of Lemma E i... |
cdleme17a 34591 | Part of proof of Lemma E i... |
cdleme17b 34592 | Lemma leading to ~ cdleme1... |
cdleme17c 34593 | Part of proof of Lemma E i... |
cdleme17d1 34594 | Part of proof of Lemma E i... |
cdleme0nex 34595 | Part of proof of Lemma E i... |
cdleme18a 34596 | Part of proof of Lemma E i... |
cdleme18b 34597 | Part of proof of Lemma E i... |
cdleme18c 34598 | Part of proof of Lemma E i... |
cdleme22gb 34599 | Utility lemma for Lemma E ... |
cdleme18d 34600 | Part of proof of Lemma E i... |
cdlemesner 34601 | Part of proof of Lemma E i... |
cdlemedb 34602 | Part of proof of Lemma E i... |
cdlemeda 34603 | Part of proof of Lemma E i... |
cdlemednpq 34604 | Part of proof of Lemma E i... |
cdlemednuN 34605 | Part of proof of Lemma E i... |
cdleme20zN 34606 | Part of proof of Lemma E i... |
cdleme20y 34607 | Part of proof of Lemma E i... |
cdleme20yOLD 34608 | Part of proof of Lemma E i... |
cdleme19a 34609 | Part of proof of Lemma E i... |
cdleme19b 34610 | Part of proof of Lemma E i... |
cdleme19c 34611 | Part of proof of Lemma E i... |
cdleme19d 34612 | Part of proof of Lemma E i... |
cdleme19e 34613 | Part of proof of Lemma E i... |
cdleme19f 34614 | Part of proof of Lemma E i... |
cdleme20aN 34615 | Part of proof of Lemma E i... |
cdleme20bN 34616 | Part of proof of Lemma E i... |
cdleme20c 34617 | Part of proof of Lemma E i... |
cdleme20d 34618 | Part of proof of Lemma E i... |
cdleme20e 34619 | Part of proof of Lemma E i... |
cdleme20f 34620 | Part of proof of Lemma E i... |
cdleme20g 34621 | Part of proof of Lemma E i... |
cdleme20h 34622 | Part of proof of Lemma E i... |
cdleme20i 34623 | Part of proof of Lemma E i... |
cdleme20j 34624 | Part of proof of Lemma E i... |
cdleme20k 34625 | Part of proof of Lemma E i... |
cdleme20l1 34626 | Part of proof of Lemma E i... |
cdleme20l2 34627 | Part of proof of Lemma E i... |
cdleme20l 34628 | Part of proof of Lemma E i... |
cdleme20m 34629 | Part of proof of Lemma E i... |
cdleme20 34630 | Combine ~ cdleme19f and ~ ... |
cdleme21a 34631 | Part of proof of Lemma E i... |
cdleme21b 34632 | Part of proof of Lemma E i... |
cdleme21c 34633 | Part of proof of Lemma E i... |
cdleme21at 34634 | Part of proof of Lemma E i... |
cdleme21ct 34635 | Part of proof of Lemma E i... |
cdleme21d 34636 | Part of proof of Lemma E i... |
cdleme21e 34637 | Part of proof of Lemma E i... |
cdleme21f 34638 | Part of proof of Lemma E i... |
cdleme21g 34639 | Part of proof of Lemma E i... |
cdleme21h 34640 | Part of proof of Lemma E i... |
cdleme21i 34641 | Part of proof of Lemma E i... |
cdleme21j 34642 | Combine ~ cdleme20 and ~ c... |
cdleme21 34643 | Part of proof of Lemma E i... |
cdleme21k 34644 | Eliminate ` S =/= T ` cond... |
cdleme22aa 34645 | Part of proof of Lemma E i... |
cdleme22a 34646 | Part of proof of Lemma E i... |
cdleme22b 34647 | Part of proof of Lemma E i... |
cdleme22cN 34648 | Part of proof of Lemma E i... |
cdleme22d 34649 | Part of proof of Lemma E i... |
cdleme22e 34650 | Part of proof of Lemma E i... |
cdleme22eALTN 34651 | Part of proof of Lemma E i... |
cdleme22f 34652 | Part of proof of Lemma E i... |
cdleme22f2 34653 | Part of proof of Lemma E i... |
cdleme22g 34654 | Part of proof of Lemma E i... |
cdleme23a 34655 | Part of proof of Lemma E i... |
cdleme23b 34656 | Part of proof of Lemma E i... |
cdleme23c 34657 | Part of proof of Lemma E i... |
cdleme24 34658 | Quantified version of ~ cd... |
cdleme25a 34659 | Lemma for ~ cdleme25b . (... |
cdleme25b 34660 | Transform ~ cdleme24 . TO... |
cdleme25c 34661 | Transform ~ cdleme25b . (... |
cdleme25dN 34662 | Transform ~ cdleme25c . (... |
cdleme25cl 34663 | Show closure of the unique... |
cdleme25cv 34664 | Change bound variables in ... |
cdleme26e 34665 | Part of proof of Lemma E i... |
cdleme26ee 34666 | Part of proof of Lemma E i... |
cdleme26eALTN 34667 | Part of proof of Lemma E i... |
cdleme26fALTN 34668 | Part of proof of Lemma E i... |
cdleme26f 34669 | Part of proof of Lemma E i... |
cdleme26f2ALTN 34670 | Part of proof of Lemma E i... |
cdleme26f2 34671 | Part of proof of Lemma E i... |
cdleme27cl 34672 | Part of proof of Lemma E i... |
cdleme27a 34673 | Part of proof of Lemma E i... |
cdleme27b 34674 | Lemma for ~ cdleme27N . (... |
cdleme27N 34675 | Part of proof of Lemma E i... |
cdleme28a 34676 | Lemma for ~ cdleme25b . T... |
cdleme28b 34677 | Lemma for ~ cdleme25b . T... |
cdleme28c 34678 | Part of proof of Lemma E i... |
cdleme28 34679 | Quantified version of ~ cd... |
cdleme29ex 34680 | Lemma for ~ cdleme29b . (... |
cdleme29b 34681 | Transform ~ cdleme28 . (C... |
cdleme29c 34682 | Transform ~ cdleme28b . (... |
cdleme29cl 34683 | Show closure of the unique... |
cdleme30a 34684 | Part of proof of Lemma E i... |
cdleme31so 34685 | Part of proof of Lemma E i... |
cdleme31sn 34686 | Part of proof of Lemma E i... |
cdleme31sn1 34687 | Part of proof of Lemma E i... |
cdleme31se 34688 | Part of proof of Lemma D i... |
cdleme31se2 34689 | Part of proof of Lemma D i... |
cdleme31sc 34690 | Part of proof of Lemma E i... |
cdleme31sde 34691 | Part of proof of Lemma D i... |
cdleme31snd 34692 | Part of proof of Lemma D i... |
cdleme31sdnN 34693 | Part of proof of Lemma E i... |
cdleme31sn1c 34694 | Part of proof of Lemma E i... |
cdleme31sn2 34695 | Part of proof of Lemma E i... |
cdleme31fv 34696 | Part of proof of Lemma E i... |
cdleme31fv1 34697 | Part of proof of Lemma E i... |
cdleme31fv1s 34698 | Part of proof of Lemma E i... |
cdleme31fv2 34699 | Part of proof of Lemma E i... |
cdleme31id 34700 | Part of proof of Lemma E i... |
cdlemefrs29pre00 34701 | ***START OF VALUE AT ATOM ... |
cdlemefrs29bpre0 34702 | TODO fix comment. (Contri... |
cdlemefrs29bpre1 34703 | TODO: FIX COMMENT. (Contr... |
cdlemefrs29cpre1 34704 | TODO: FIX COMMENT. (Contr... |
cdlemefrs29clN 34705 | TODO: NOT USED? Show clo... |
cdlemefrs32fva 34706 | Part of proof of Lemma E i... |
cdlemefrs32fva1 34707 | Part of proof of Lemma E i... |
cdlemefr29exN 34708 | Lemma for ~ cdlemefs29bpre... |
cdlemefr27cl 34709 | Part of proof of Lemma E i... |
cdlemefr32sn2aw 34710 | Show that ` [_ R / s ]_ N ... |
cdlemefr32snb 34711 | Show closure of ` [_ R / s... |
cdlemefr29bpre0N 34712 | TODO fix comment. (Contri... |
cdlemefr29clN 34713 | Show closure of the unique... |
cdleme43frv1snN 34714 | Value of ` [_ R / s ]_ N `... |
cdlemefr32fvaN 34715 | Part of proof of Lemma E i... |
cdlemefr32fva1 34716 | Part of proof of Lemma E i... |
cdlemefr31fv1 34717 | Value of ` ( F `` R ) ` wh... |
cdlemefs29pre00N 34718 | FIX COMMENT. TODO: see if ... |
cdlemefs27cl 34719 | Part of proof of Lemma E i... |
cdlemefs32sn1aw 34720 | Show that ` [_ R / s ]_ N ... |
cdlemefs32snb 34721 | Show closure of ` [_ R / s... |
cdlemefs29bpre0N 34722 | TODO: FIX COMMENT. (Contr... |
cdlemefs29bpre1N 34723 | TODO: FIX COMMENT. (Contr... |
cdlemefs29cpre1N 34724 | TODO: FIX COMMENT. (Contr... |
cdlemefs29clN 34725 | Show closure of the unique... |
cdleme43fsv1snlem 34726 | Value of ` [_ R / s ]_ N `... |
cdleme43fsv1sn 34727 | Value of ` [_ R / s ]_ N `... |
cdlemefs32fvaN 34728 | Part of proof of Lemma E i... |
cdlemefs32fva1 34729 | Part of proof of Lemma E i... |
cdlemefs31fv1 34730 | Value of ` ( F `` R ) ` wh... |
cdlemefr44 34731 | Value of f(r) when r is an... |
cdlemefs44 34732 | Value of f_s(r) when r is ... |
cdlemefr45 34733 | Value of f(r) when r is an... |
cdlemefr45e 34734 | Explicit expansion of ~ cd... |
cdlemefs45 34735 | Value of f_s(r) when r is ... |
cdlemefs45ee 34736 | Explicit expansion of ~ cd... |
cdlemefs45eN 34737 | Explicit expansion of ~ cd... |
cdleme32sn1awN 34738 | Show that ` [_ R / s ]_ N ... |
cdleme41sn3a 34739 | Show that ` [_ R / s ]_ N ... |
cdleme32sn2awN 34740 | Show that ` [_ R / s ]_ N ... |
cdleme32snaw 34741 | Show that ` [_ R / s ]_ N ... |
cdleme32snb 34742 | Show closure of ` [_ R / s... |
cdleme32fva 34743 | Part of proof of Lemma D i... |
cdleme32fva1 34744 | Part of proof of Lemma D i... |
cdleme32fvaw 34745 | Show that ` ( F `` R ) ` i... |
cdleme32fvcl 34746 | Part of proof of Lemma D i... |
cdleme32a 34747 | Part of proof of Lemma D i... |
cdleme32b 34748 | Part of proof of Lemma D i... |
cdleme32c 34749 | Part of proof of Lemma D i... |
cdleme32d 34750 | Part of proof of Lemma D i... |
cdleme32e 34751 | Part of proof of Lemma D i... |
cdleme32f 34752 | Part of proof of Lemma D i... |
cdleme32le 34753 | Part of proof of Lemma D i... |
cdleme35a 34754 | Part of proof of Lemma E i... |
cdleme35fnpq 34755 | Part of proof of Lemma E i... |
cdleme35b 34756 | Part of proof of Lemma E i... |
cdleme35c 34757 | Part of proof of Lemma E i... |
cdleme35d 34758 | Part of proof of Lemma E i... |
cdleme35e 34759 | Part of proof of Lemma E i... |
cdleme35f 34760 | Part of proof of Lemma E i... |
cdleme35g 34761 | Part of proof of Lemma E i... |
cdleme35h 34762 | Part of proof of Lemma E i... |
cdleme35h2 34763 | Part of proof of Lemma E i... |
cdleme35sn2aw 34764 | Part of proof of Lemma E i... |
cdleme35sn3a 34765 | Part of proof of Lemma E i... |
cdleme36a 34766 | Part of proof of Lemma E i... |
cdleme36m 34767 | Part of proof of Lemma E i... |
cdleme37m 34768 | Part of proof of Lemma E i... |
cdleme38m 34769 | Part of proof of Lemma E i... |
cdleme38n 34770 | Part of proof of Lemma E i... |
cdleme39a 34771 | Part of proof of Lemma E i... |
cdleme39n 34772 | Part of proof of Lemma E i... |
cdleme40m 34773 | Part of proof of Lemma E i... |
cdleme40n 34774 | Part of proof of Lemma E i... |
cdleme40v 34775 | Part of proof of Lemma E i... |
cdleme40w 34776 | Part of proof of Lemma E i... |
cdleme42a 34777 | Part of proof of Lemma E i... |
cdleme42c 34778 | Part of proof of Lemma E i... |
cdleme42d 34779 | Part of proof of Lemma E i... |
cdleme41sn3aw 34780 | Part of proof of Lemma E i... |
cdleme41sn4aw 34781 | Part of proof of Lemma E i... |
cdleme41snaw 34782 | Part of proof of Lemma E i... |
cdleme41fva11 34783 | Part of proof of Lemma E i... |
cdleme42b 34784 | Part of proof of Lemma E i... |
cdleme42e 34785 | Part of proof of Lemma E i... |
cdleme42f 34786 | Part of proof of Lemma E i... |
cdleme42g 34787 | Part of proof of Lemma E i... |
cdleme42h 34788 | Part of proof of Lemma E i... |
cdleme42i 34789 | Part of proof of Lemma E i... |
cdleme42k 34790 | Part of proof of Lemma E i... |
cdleme42ke 34791 | Part of proof of Lemma E i... |
cdleme42keg 34792 | Part of proof of Lemma E i... |
cdleme42mN 34793 | Part of proof of Lemma E i... |
cdleme42mgN 34794 | Part of proof of Lemma E i... |
cdleme43aN 34795 | Part of proof of Lemma E i... |
cdleme43bN 34796 | Lemma for Lemma E in [Craw... |
cdleme43cN 34797 | Part of proof of Lemma E i... |
cdleme43dN 34798 | Part of proof of Lemma E i... |
cdleme46f2g2 34799 | Conversion for ` G ` to re... |
cdleme46f2g1 34800 | Conversion for ` G ` to re... |
cdleme17d2 34801 | Part of proof of Lemma E i... |
cdleme17d3 34802 | TODO: FIX COMMENT. (Contr... |
cdleme17d4 34803 | TODO: FIX COMMENT. (Contr... |
cdleme17d 34804 | Part of proof of Lemma E i... |
cdleme48fv 34805 | Part of proof of Lemma D i... |
cdleme48fvg 34806 | Remove ` P =/= Q ` conditi... |
cdleme46fvaw 34807 | Show that ` ( F `` R ) ` i... |
cdleme48bw 34808 | TODO: fix comment. TODO: ... |
cdleme48b 34809 | TODO: fix comment. (Contr... |
cdleme46frvlpq 34810 | Show that ` ( F `` S ) ` i... |
cdleme46fsvlpq 34811 | Show that ` ( F `` R ) ` i... |
cdlemeg46fvcl 34812 | TODO: fix comment. (Contr... |
cdleme4gfv 34813 | Part of proof of Lemma D i... |
cdlemeg47b 34814 | TODO: FIX COMMENT. (Contr... |
cdlemeg47rv 34815 | Value of g_s(r) when r is ... |
cdlemeg47rv2 34816 | Value of g_s(r) when r is ... |
cdlemeg49le 34817 | Part of proof of Lemma D i... |
cdlemeg46bOLDN 34818 | TODO FIX COMMENT. (Contrib... |
cdlemeg46c 34819 | TODO FIX COMMENT. (Contrib... |
cdlemeg46rvOLDN 34820 | Value of g_s(r) when r is ... |
cdlemeg46rv2OLDN 34821 | Value of g_s(r) when r is ... |
cdlemeg46fvaw 34822 | Show that ` ( F `` R ) ` i... |
cdlemeg46nlpq 34823 | Show that ` ( G `` S ) ` i... |
cdlemeg46ngfr 34824 | TODO FIX COMMENT g(f(s))=s... |
cdlemeg46nfgr 34825 | TODO FIX COMMENT f(g(s))=s... |
cdlemeg46sfg 34826 | TODO FIX COMMENT f(r) ` \/... |
cdlemeg46fjgN 34827 | NOT NEEDED? TODO FIX COMM... |
cdlemeg46rjgN 34828 | NOT NEEDED? TODO FIX COMM... |
cdlemeg46fjv 34829 | TODO FIX COMMENT f(r) ` \/... |
cdlemeg46fsfv 34830 | TODO FIX COMMENT f(r) ` \/... |
cdlemeg46frv 34831 | TODO FIX COMMENT. (f(r) ` ... |
cdlemeg46v1v2 34832 | TODO FIX COMMENT v_1 = v_2... |
cdlemeg46vrg 34833 | TODO FIX COMMENT v_1 ` <_ ... |
cdlemeg46rgv 34834 | TODO FIX COMMENT r ` <_ ` ... |
cdlemeg46req 34835 | TODO FIX COMMENT r = (v_1 ... |
cdlemeg46gfv 34836 | TODO FIX COMMENT p. 115 pe... |
cdlemeg46gfr 34837 | TODO FIX COMMENT p. 116 pe... |
cdlemeg46gfre 34838 | TODO FIX COMMENT p. 116 pe... |
cdlemeg46gf 34839 | TODO FIX COMMENT Eliminate... |
cdlemeg46fgN 34840 | TODO FIX COMMENT p. 116 pe... |
cdleme48d 34841 | TODO: fix comment. (Contr... |
cdleme48gfv1 34842 | TODO: fix comment. (Contr... |
cdleme48gfv 34843 | TODO: fix comment. (Contr... |
cdleme48fgv 34844 | TODO: fix comment. (Contr... |
cdlemeg49lebilem 34845 | Part of proof of Lemma D i... |
cdleme50lebi 34846 | Part of proof of Lemma D i... |
cdleme50eq 34847 | Part of proof of Lemma D i... |
cdleme50f 34848 | Part of proof of Lemma D i... |
cdleme50f1 34849 | Part of proof of Lemma D i... |
cdleme50rnlem 34850 | Part of proof of Lemma D i... |
cdleme50rn 34851 | Part of proof of Lemma D i... |
cdleme50f1o 34852 | Part of proof of Lemma D i... |
cdleme50laut 34853 | Part of proof of Lemma D i... |
cdleme50ldil 34854 | Part of proof of Lemma D i... |
cdleme50trn1 34855 | Part of proof that ` F ` i... |
cdleme50trn2a 34856 | Part of proof that ` F ` i... |
cdleme50trn2 34857 | Part of proof that ` F ` i... |
cdleme50trn12 34858 | Part of proof that ` F ` i... |
cdleme50trn3 34859 | Part of proof that ` F ` i... |
cdleme50trn123 34860 | Part of proof that ` F ` i... |
cdleme51finvfvN 34861 | Part of proof of Lemma E i... |
cdleme51finvN 34862 | Part of proof of Lemma E i... |
cdleme50ltrn 34863 | Part of proof of Lemma E i... |
cdleme51finvtrN 34864 | Part of proof of Lemma E i... |
cdleme50ex 34865 | Part of Lemma E in [Crawle... |
cdleme 34866 | Lemma E in [Crawley] p. 11... |
cdlemf1 34867 | Part of Lemma F in [Crawle... |
cdlemf2 34868 | Part of Lemma F in [Crawle... |
cdlemf 34869 | Lemma F in [Crawley] p. 11... |
cdlemfnid 34870 | ~ cdlemf with additional c... |
cdlemftr3 34871 | Special case of ~ cdlemf s... |
cdlemftr2 34872 | Special case of ~ cdlemf s... |
cdlemftr1 34873 | Part of proof of Lemma G o... |
cdlemftr0 34874 | Special case of ~ cdlemf s... |
trlord 34875 | The ordering of two Hilber... |
cdlemg1a 34876 | Shorter expression for ` G... |
cdlemg1b2 34877 | This theorem can be used t... |
cdlemg1idlemN 34878 | Lemma for ~ cdlemg1idN . ... |
cdlemg1fvawlemN 34879 | Lemma for ~ ltrniotafvawN ... |
cdlemg1ltrnlem 34880 | Lemma for ~ ltrniotacl . ... |
cdlemg1finvtrlemN 34881 | Lemma for ~ ltrniotacnvN .... |
cdlemg1bOLDN 34882 | This theorem can be used t... |
cdlemg1idN 34883 | Version of ~ cdleme31id wi... |
ltrniotafvawN 34884 | Version of ~ cdleme46fvaw ... |
ltrniotacl 34885 | Version of ~ cdleme50ltrn ... |
ltrniotacnvN 34886 | Version of ~ cdleme51finvt... |
ltrniotaval 34887 | Value of the unique transl... |
ltrniotacnvval 34888 | Converse value of the uniq... |
ltrniotaidvalN 34889 | Value of the unique transl... |
ltrniotavalbN 34890 | Value of the unique transl... |
cdlemeiota 34891 | A translation is uniquely ... |
cdlemg1ci2 34892 | Any function of the form o... |
cdlemg1cN 34893 | Any translation belongs to... |
cdlemg1cex 34894 | Any translation is one of ... |
cdlemg2cN 34895 | Any translation belongs to... |
cdlemg2dN 34896 | This theorem can be used t... |
cdlemg2cex 34897 | Any translation is one of ... |
cdlemg2ce 34898 | Utility theorem to elimina... |
cdlemg2jlemOLDN 34899 | Part of proof of Lemma E i... |
cdlemg2fvlem 34900 | Lemma for ~ cdlemg2fv . (... |
cdlemg2klem 34901 | ~ cdleme42keg with simpler... |
cdlemg2idN 34902 | Version of ~ cdleme31id wi... |
cdlemg3a 34903 | Part of proof of Lemma G i... |
cdlemg2jOLDN 34904 | TODO: Replace this with ~... |
cdlemg2fv 34905 | Value of a translation in ... |
cdlemg2fv2 34906 | Value of a translation in ... |
cdlemg2k 34907 | ~ cdleme42keg with simpler... |
cdlemg2kq 34908 | ~ cdlemg2k with ` P ` and ... |
cdlemg2l 34909 | TODO: FIX COMMENT. (Contr... |
cdlemg2m 34910 | TODO: FIX COMMENT. (Contr... |
cdlemg5 34911 | TODO: Is there a simpler ... |
cdlemb3 34912 | Given two atoms not under ... |
cdlemg7fvbwN 34913 | Properties of a translatio... |
cdlemg4a 34914 | TODO: FIX COMMENT If fg(p... |
cdlemg4b1 34915 | TODO: FIX COMMENT. (Contr... |
cdlemg4b2 34916 | TODO: FIX COMMENT. (Contr... |
cdlemg4b12 34917 | TODO: FIX COMMENT. (Contr... |
cdlemg4c 34918 | TODO: FIX COMMENT. (Contr... |
cdlemg4d 34919 | TODO: FIX COMMENT. (Contr... |
cdlemg4e 34920 | TODO: FIX COMMENT. (Contr... |
cdlemg4f 34921 | TODO: FIX COMMENT. (Contr... |
cdlemg4g 34922 | TODO: FIX COMMENT. (Contr... |
cdlemg4 34923 | TODO: FIX COMMENT. (Contr... |
cdlemg6a 34924 | TODO: FIX COMMENT. TODO: ... |
cdlemg6b 34925 | TODO: FIX COMMENT. TODO: ... |
cdlemg6c 34926 | TODO: FIX COMMENT. (Contr... |
cdlemg6d 34927 | TODO: FIX COMMENT. (Contr... |
cdlemg6e 34928 | TODO: FIX COMMENT. (Contr... |
cdlemg6 34929 | TODO: FIX COMMENT. (Contr... |
cdlemg7fvN 34930 | Value of a translation com... |
cdlemg7aN 34931 | TODO: FIX COMMENT. (Contr... |
cdlemg7N 34932 | TODO: FIX COMMENT. (Contr... |
cdlemg8a 34933 | TODO: FIX COMMENT. (Contr... |
cdlemg8b 34934 | TODO: FIX COMMENT. (Contr... |
cdlemg8c 34935 | TODO: FIX COMMENT. (Contr... |
cdlemg8d 34936 | TODO: FIX COMMENT. (Contr... |
cdlemg8 34937 | TODO: FIX COMMENT. (Contr... |
cdlemg9a 34938 | TODO: FIX COMMENT. (Contr... |
cdlemg9b 34939 | The triples ` <. P , ( F `... |
cdlemg9 34940 | The triples ` <. P , ( F `... |
cdlemg10b 34941 | TODO: FIX COMMENT. TODO: ... |
cdlemg10bALTN 34942 | TODO: FIX COMMENT. TODO: ... |
cdlemg11a 34943 | TODO: FIX COMMENT. (Contr... |
cdlemg11aq 34944 | TODO: FIX COMMENT. TODO: ... |
cdlemg10c 34945 | TODO: FIX COMMENT. TODO: ... |
cdlemg10a 34946 | TODO: FIX COMMENT. (Contr... |
cdlemg10 34947 | TODO: FIX COMMENT. (Contr... |
cdlemg11b 34948 | TODO: FIX COMMENT. (Contr... |
cdlemg12a 34949 | TODO: FIX COMMENT. (Contr... |
cdlemg12b 34950 | The triples ` <. P , ( F `... |
cdlemg12c 34951 | The triples ` <. P , ( F `... |
cdlemg12d 34952 | TODO: FIX COMMENT. (Contr... |
cdlemg12e 34953 | TODO: FIX COMMENT. (Contr... |
cdlemg12f 34954 | TODO: FIX COMMENT. (Contr... |
cdlemg12g 34955 | TODO: FIX COMMENT. TODO: ... |
cdlemg12 34956 | TODO: FIX COMMENT. (Contr... |
cdlemg13a 34957 | TODO: FIX COMMENT. (Contr... |
cdlemg13 34958 | TODO: FIX COMMENT. (Contr... |
cdlemg14f 34959 | TODO: FIX COMMENT. (Contr... |
cdlemg14g 34960 | TODO: FIX COMMENT. (Contr... |
cdlemg15a 34961 | Eliminate the ` ( F `` P )... |
cdlemg15 34962 | Eliminate the ` ( (... |
cdlemg16 34963 | Part of proof of Lemma G o... |
cdlemg16ALTN 34964 | This version of ~ cdlemg16... |
cdlemg16z 34965 | Eliminate ` ( ( F `... |
cdlemg16zz 34966 | Eliminate ` P =/= Q ` from... |
cdlemg17a 34967 | TODO: FIX COMMENT. (Contr... |
cdlemg17b 34968 | Part of proof of Lemma G i... |
cdlemg17dN 34969 | TODO: fix comment. (Contr... |
cdlemg17dALTN 34970 | Same as ~ cdlemg17dN with ... |
cdlemg17e 34971 | TODO: fix comment. (Contr... |
cdlemg17f 34972 | TODO: fix comment. (Contr... |
cdlemg17g 34973 | TODO: fix comment. (Contr... |
cdlemg17h 34974 | TODO: fix comment. (Contr... |
cdlemg17i 34975 | TODO: fix comment. (Contr... |
cdlemg17ir 34976 | TODO: fix comment. (Contr... |
cdlemg17j 34977 | TODO: fix comment. (Contr... |
cdlemg17pq 34978 | Utility theorem for swappi... |
cdlemg17bq 34979 | ~ cdlemg17b with ` P ` and... |
cdlemg17iqN 34980 | ~ cdlemg17i with ` P ` and... |
cdlemg17irq 34981 | ~ cdlemg17ir with ` P ` an... |
cdlemg17jq 34982 | ~ cdlemg17j with ` P ` and... |
cdlemg17 34983 | Part of Lemma G of [Crawle... |
cdlemg18a 34984 | Show two lines are differe... |
cdlemg18b 34985 | Lemma for ~ cdlemg18c . T... |
cdlemg18c 34986 | Show two lines intersect a... |
cdlemg18d 34987 | Show two lines intersect a... |
cdlemg18 34988 | Show two lines intersect a... |
cdlemg19a 34989 | Show two lines intersect a... |
cdlemg19 34990 | Show two lines intersect a... |
cdlemg20 34991 | Show two lines intersect a... |
cdlemg21 34992 | Version of cdlemg19 with `... |
cdlemg22 34993 | ~ cdlemg21 with ` ( F `` P... |
cdlemg24 34994 | Combine ~ cdlemg16z and ~ ... |
cdlemg37 34995 | Use ~ cdlemg8 to eliminate... |
cdlemg25zz 34996 | ~ cdlemg16zz restated for ... |
cdlemg26zz 34997 | ~ cdlemg16zz restated for ... |
cdlemg27a 34998 | For use with case when ` (... |
cdlemg28a 34999 | Part of proof of Lemma G o... |
cdlemg31b0N 35000 | TODO: Fix comment. (Cont... |
cdlemg31b0a 35001 | TODO: Fix comment. (Cont... |
cdlemg27b 35002 | TODO: Fix comment. (Cont... |
cdlemg31a 35003 | TODO: fix comment. (Contr... |
cdlemg31b 35004 | TODO: fix comment. (Contr... |
cdlemg31c 35005 | Show that when ` N ` is an... |
cdlemg31d 35006 | Eliminate ` ( F `` P ) =/=... |
cdlemg33b0 35007 | TODO: Fix comment. (Cont... |
cdlemg33c0 35008 | TODO: Fix comment. (Cont... |
cdlemg28b 35009 | Part of proof of Lemma G o... |
cdlemg28 35010 | Part of proof of Lemma G o... |
cdlemg29 35011 | Eliminate ` ( F `` P ) =/=... |
cdlemg33a 35012 | TODO: Fix comment. (Cont... |
cdlemg33b 35013 | TODO: Fix comment. (Cont... |
cdlemg33c 35014 | TODO: Fix comment. (Cont... |
cdlemg33d 35015 | TODO: Fix comment. (Cont... |
cdlemg33e 35016 | TODO: Fix comment. (Cont... |
cdlemg33 35017 | Combine ~ cdlemg33b , ~ cd... |
cdlemg34 35018 | Use cdlemg33 to eliminate ... |
cdlemg35 35019 | TODO: Fix comment. TODO:... |
cdlemg36 35020 | Use cdlemg35 to eliminate ... |
cdlemg38 35021 | Use ~ cdlemg37 to eliminat... |
cdlemg39 35022 | Eliminate ` =/= ` conditio... |
cdlemg40 35023 | Eliminate ` P =/= Q ` cond... |
cdlemg41 35024 | Convert ~ cdlemg40 to func... |
ltrnco 35025 | The composition of two tra... |
trlcocnv 35026 | Swap the arguments of the ... |
trlcoabs 35027 | Absorption into a composit... |
trlcoabs2N 35028 | Absorption of the trace of... |
trlcoat 35029 | The trace of a composition... |
trlcocnvat 35030 | Commonly used special case... |
trlconid 35031 | The composition of two dif... |
trlcolem 35032 | Lemma for ~ trlco . (Cont... |
trlco 35033 | The trace of a composition... |
trlcone 35034 | If two translations have d... |
cdlemg42 35035 | Part of proof of Lemma G o... |
cdlemg43 35036 | Part of proof of Lemma G o... |
cdlemg44a 35037 | Part of proof of Lemma G o... |
cdlemg44b 35038 | Eliminate ` ( F `` P ) =/=... |
cdlemg44 35039 | Part of proof of Lemma G o... |
cdlemg47a 35040 | TODO: fix comment. TODO: ... |
cdlemg46 35041 | Part of proof of Lemma G o... |
cdlemg47 35042 | Part of proof of Lemma G o... |
cdlemg48 35043 | Elmininate ` h ` from ~ cd... |
ltrncom 35044 | Composition is commutative... |
ltrnco4 35045 | Rearrange a composition of... |
trljco 35046 | Trace joined with trace of... |
trljco2 35047 | Trace joined with trace of... |
tgrpfset 35050 | The translation group maps... |
tgrpset 35051 | The translation group for ... |
tgrpbase 35052 | The base set of the transl... |
tgrpopr 35053 | The group operation of the... |
tgrpov 35054 | The group operation value ... |
tgrpgrplem 35055 | Lemma for ~ tgrpgrp . (Co... |
tgrpgrp 35056 | The translation group is a... |
tgrpabl 35057 | The translation group is a... |
tendofset 35064 | The set of all trace-prese... |
tendoset 35065 | The set of trace-preservin... |
istendo 35066 | The predicate "is a trace-... |
tendotp 35067 | Trace-preserving property ... |
istendod 35068 | Deduce the predicate "is a... |
tendof 35069 | Functionality of a trace-p... |
tendoeq1 35070 | Condition determining equa... |
tendovalco 35071 | Value of composition of tr... |
tendocoval 35072 | Value of composition of en... |
tendocl 35073 | Closure of a trace-preserv... |
tendoco2 35074 | Distribution of compositio... |
tendoidcl 35075 | The identity is a trace-pr... |
tendo1mul 35076 | Multiplicative identity mu... |
tendo1mulr 35077 | Multiplicative identity mu... |
tendococl 35078 | The composition of two tra... |
tendoid 35079 | The identity value of a tr... |
tendoeq2 35080 | Condition determining equa... |
tendoplcbv 35081 | Define sum operation for t... |
tendopl 35082 | Value of endomorphism sum ... |
tendopl2 35083 | Value of result of endomor... |
tendoplcl2 35084 | Value of result of endomor... |
tendoplco2 35085 | Value of result of endomor... |
tendopltp 35086 | Trace-preserving property ... |
tendoplcl 35087 | Endomorphism sum is a trac... |
tendoplcom 35088 | The endomorphism sum opera... |
tendoplass 35089 | The endomorphism sum opera... |
tendodi1 35090 | Endomorphism composition d... |
tendodi2 35091 | Endomorphism composition d... |
tendo0cbv 35092 | Define additive identity f... |
tendo02 35093 | Value of additive identity... |
tendo0co2 35094 | The additive identity trac... |
tendo0tp 35095 | Trace-preserving property ... |
tendo0cl 35096 | The additive identity is a... |
tendo0pl 35097 | Property of the additive i... |
tendo0plr 35098 | Property of the additive i... |
tendoicbv 35099 | Define inverse function fo... |
tendoi 35100 | Value of inverse endomorph... |
tendoi2 35101 | Value of additive inverse ... |
tendoicl 35102 | Closure of the additive in... |
tendoipl 35103 | Property of the additive i... |
tendoipl2 35104 | Property of the additive i... |
erngfset 35105 | The division rings on trac... |
erngset 35106 | The division ring on trace... |
erngbase 35107 | The base set of the divisi... |
erngfplus 35108 | Ring addition operation. ... |
erngplus 35109 | Ring addition operation. ... |
erngplus2 35110 | Ring addition operation. ... |
erngfmul 35111 | Ring multiplication operat... |
erngmul 35112 | Ring addition operation. ... |
erngfset-rN 35113 | The division rings on trac... |
erngset-rN 35114 | The division ring on trace... |
erngbase-rN 35115 | The base set of the divisi... |
erngfplus-rN 35116 | Ring addition operation. ... |
erngplus-rN 35117 | Ring addition operation. ... |
erngplus2-rN 35118 | Ring addition operation. ... |
erngfmul-rN 35119 | Ring multiplication operat... |
erngmul-rN 35120 | Ring addition operation. ... |
cdlemh1 35121 | Part of proof of Lemma H o... |
cdlemh2 35122 | Part of proof of Lemma H o... |
cdlemh 35123 | Lemma H of [Crawley] p. 11... |
cdlemi1 35124 | Part of proof of Lemma I o... |
cdlemi2 35125 | Part of proof of Lemma I o... |
cdlemi 35126 | Lemma I of [Crawley] p. 11... |
cdlemj1 35127 | Part of proof of Lemma J o... |
cdlemj2 35128 | Part of proof of Lemma J o... |
cdlemj3 35129 | Part of proof of Lemma J o... |
tendocan 35130 | Cancellation law: if the v... |
tendoid0 35131 | A trace-preserving endomor... |
tendo0mul 35132 | Additive identity multipli... |
tendo0mulr 35133 | Additive identity multipli... |
tendo1ne0 35134 | The identity (unity) is no... |
tendoconid 35135 | The composition (product) ... |
tendotr 35136 | The trace of the value of ... |
cdlemk1 35137 | Part of proof of Lemma K o... |
cdlemk2 35138 | Part of proof of Lemma K o... |
cdlemk3 35139 | Part of proof of Lemma K o... |
cdlemk4 35140 | Part of proof of Lemma K o... |
cdlemk5a 35141 | Part of proof of Lemma K o... |
cdlemk5 35142 | Part of proof of Lemma K o... |
cdlemk6 35143 | Part of proof of Lemma K o... |
cdlemk8 35144 | Part of proof of Lemma K o... |
cdlemk9 35145 | Part of proof of Lemma K o... |
cdlemk9bN 35146 | Part of proof of Lemma K o... |
cdlemki 35147 | Part of proof of Lemma K o... |
cdlemkvcl 35148 | Part of proof of Lemma K o... |
cdlemk10 35149 | Part of proof of Lemma K o... |
cdlemksv 35150 | Part of proof of Lemma K o... |
cdlemksel 35151 | Part of proof of Lemma K o... |
cdlemksat 35152 | Part of proof of Lemma K o... |
cdlemksv2 35153 | Part of proof of Lemma K o... |
cdlemk7 35154 | Part of proof of Lemma K o... |
cdlemk11 35155 | Part of proof of Lemma K o... |
cdlemk12 35156 | Part of proof of Lemma K o... |
cdlemkoatnle 35157 | Utility lemma. (Contribut... |
cdlemk13 35158 | Part of proof of Lemma K o... |
cdlemkole 35159 | Utility lemma. (Contribut... |
cdlemk14 35160 | Part of proof of Lemma K o... |
cdlemk15 35161 | Part of proof of Lemma K o... |
cdlemk16a 35162 | Part of proof of Lemma K o... |
cdlemk16 35163 | Part of proof of Lemma K o... |
cdlemk17 35164 | Part of proof of Lemma K o... |
cdlemk1u 35165 | Part of proof of Lemma K o... |
cdlemk5auN 35166 | Part of proof of Lemma K o... |
cdlemk5u 35167 | Part of proof of Lemma K o... |
cdlemk6u 35168 | Part of proof of Lemma K o... |
cdlemkj 35169 | Part of proof of Lemma K o... |
cdlemkuvN 35170 | Part of proof of Lemma K o... |
cdlemkuel 35171 | Part of proof of Lemma K o... |
cdlemkuat 35172 | Part of proof of Lemma K o... |
cdlemkuv2 35173 | Part of proof of Lemma K o... |
cdlemk18 35174 | Part of proof of Lemma K o... |
cdlemk19 35175 | Part of proof of Lemma K o... |
cdlemk7u 35176 | Part of proof of Lemma K o... |
cdlemk11u 35177 | Part of proof of Lemma K o... |
cdlemk12u 35178 | Part of proof of Lemma K o... |
cdlemk21N 35179 | Part of proof of Lemma K o... |
cdlemk20 35180 | Part of proof of Lemma K o... |
cdlemkoatnle-2N 35181 | Utility lemma. (Contribut... |
cdlemk13-2N 35182 | Part of proof of Lemma K o... |
cdlemkole-2N 35183 | Utility lemma. (Contribut... |
cdlemk14-2N 35184 | Part of proof of Lemma K o... |
cdlemk15-2N 35185 | Part of proof of Lemma K o... |
cdlemk16-2N 35186 | Part of proof of Lemma K o... |
cdlemk17-2N 35187 | Part of proof of Lemma K o... |
cdlemkj-2N 35188 | Part of proof of Lemma K o... |
cdlemkuv-2N 35189 | Part of proof of Lemma K o... |
cdlemkuel-2N 35190 | Part of proof of Lemma K o... |
cdlemkuv2-2 35191 | Part of proof of Lemma K o... |
cdlemk18-2N 35192 | Part of proof of Lemma K o... |
cdlemk19-2N 35193 | Part of proof of Lemma K o... |
cdlemk7u-2N 35194 | Part of proof of Lemma K o... |
cdlemk11u-2N 35195 | Part of proof of Lemma K o... |
cdlemk12u-2N 35196 | Part of proof of Lemma K o... |
cdlemk21-2N 35197 | Part of proof of Lemma K o... |
cdlemk20-2N 35198 | Part of proof of Lemma K o... |
cdlemk22 35199 | Part of proof of Lemma K o... |
cdlemk30 35200 | Part of proof of Lemma K o... |
cdlemkuu 35201 | Convert between function a... |
cdlemk31 35202 | Part of proof of Lemma K o... |
cdlemk32 35203 | Part of proof of Lemma K o... |
cdlemkuel-3 35204 | Part of proof of Lemma K o... |
cdlemkuv2-3N 35205 | Part of proof of Lemma K o... |
cdlemk18-3N 35206 | Part of proof of Lemma K o... |
cdlemk22-3 35207 | Part of proof of Lemma K o... |
cdlemk23-3 35208 | Part of proof of Lemma K o... |
cdlemk24-3 35209 | Part of proof of Lemma K o... |
cdlemk25-3 35210 | Part of proof of Lemma K o... |
cdlemk26b-3 35211 | Part of proof of Lemma K o... |
cdlemk26-3 35212 | Part of proof of Lemma K o... |
cdlemk27-3 35213 | Part of proof of Lemma K o... |
cdlemk28-3 35214 | Part of proof of Lemma K o... |
cdlemk33N 35215 | Part of proof of Lemma K o... |
cdlemk34 35216 | Part of proof of Lemma K o... |
cdlemk29-3 35217 | Part of proof of Lemma K o... |
cdlemk35 35218 | Part of proof of Lemma K o... |
cdlemk36 35219 | Part of proof of Lemma K o... |
cdlemk37 35220 | Part of proof of Lemma K o... |
cdlemk38 35221 | Part of proof of Lemma K o... |
cdlemk39 35222 | Part of proof of Lemma K o... |
cdlemk40 35223 | TODO: fix comment. (Contr... |
cdlemk40t 35224 | TODO: fix comment. (Contr... |
cdlemk40f 35225 | TODO: fix comment. (Contr... |
cdlemk41 35226 | Part of proof of Lemma K o... |
cdlemkfid1N 35227 | Lemma for ~ cdlemkfid3N . ... |
cdlemkid1 35228 | Lemma for ~ cdlemkid . (C... |
cdlemkfid2N 35229 | Lemma for ~ cdlemkfid3N . ... |
cdlemkid2 35230 | Lemma for ~ cdlemkid . (C... |
cdlemkfid3N 35231 | TODO: is this useful or sh... |
cdlemky 35232 | Part of proof of Lemma K o... |
cdlemkyu 35233 | Convert between function a... |
cdlemkyuu 35234 | ~ cdlemkyu with some hypot... |
cdlemk11ta 35235 | Part of proof of Lemma K o... |
cdlemk19ylem 35236 | Lemma for ~ cdlemk19y . (... |
cdlemk11tb 35237 | Part of proof of Lemma K o... |
cdlemk19y 35238 | ~ cdlemk19 with simpler hy... |
cdlemkid3N 35239 | Lemma for ~ cdlemkid . (C... |
cdlemkid4 35240 | Lemma for ~ cdlemkid . (C... |
cdlemkid5 35241 | Lemma for ~ cdlemkid . (C... |
cdlemkid 35242 | The value of the tau funct... |
cdlemk35s 35243 | Substitution version of ~ ... |
cdlemk35s-id 35244 | Substitution version of ~ ... |
cdlemk39s 35245 | Substitution version of ~ ... |
cdlemk39s-id 35246 | Substitution version of ~ ... |
cdlemk42 35247 | Part of proof of Lemma K o... |
cdlemk19xlem 35248 | Lemma for ~ cdlemk19x . (... |
cdlemk19x 35249 | ~ cdlemk19 with simpler hy... |
cdlemk42yN 35250 | Part of proof of Lemma K o... |
cdlemk11tc 35251 | Part of proof of Lemma K o... |
cdlemk11t 35252 | Part of proof of Lemma K o... |
cdlemk45 35253 | Part of proof of Lemma K o... |
cdlemk46 35254 | Part of proof of Lemma K o... |
cdlemk47 35255 | Part of proof of Lemma K o... |
cdlemk48 35256 | Part of proof of Lemma K o... |
cdlemk49 35257 | Part of proof of Lemma K o... |
cdlemk50 35258 | Part of proof of Lemma K o... |
cdlemk51 35259 | Part of proof of Lemma K o... |
cdlemk52 35260 | Part of proof of Lemma K o... |
cdlemk53a 35261 | Lemma for ~ cdlemk53 . (C... |
cdlemk53b 35262 | Lemma for ~ cdlemk53 . (C... |
cdlemk53 35263 | Part of proof of Lemma K o... |
cdlemk54 35264 | Part of proof of Lemma K o... |
cdlemk55a 35265 | Lemma for ~ cdlemk55 . (C... |
cdlemk55b 35266 | Lemma for ~ cdlemk55 . (C... |
cdlemk55 35267 | Part of proof of Lemma K o... |
cdlemkyyN 35268 | Part of proof of Lemma K o... |
cdlemk43N 35269 | Part of proof of Lemma K o... |
cdlemk35u 35270 | Substitution version of ~ ... |
cdlemk55u1 35271 | Lemma for ~ cdlemk55u . (... |
cdlemk55u 35272 | Part of proof of Lemma K o... |
cdlemk39u1 35273 | Lemma for ~ cdlemk39u . (... |
cdlemk39u 35274 | Part of proof of Lemma K o... |
cdlemk19u1 35275 | ~ cdlemk19 with simpler hy... |
cdlemk19u 35276 | Part of Lemma K of [Crawle... |
cdlemk56 35277 | Part of Lemma K of [Crawle... |
cdlemk19w 35278 | Use a fixed element to eli... |
cdlemk56w 35279 | Use a fixed element to eli... |
cdlemk 35280 | Lemma K of [Crawley] p. 11... |
tendoex 35281 | Generalization of Lemma K ... |
cdleml1N 35282 | Part of proof of Lemma L o... |
cdleml2N 35283 | Part of proof of Lemma L o... |
cdleml3N 35284 | Part of proof of Lemma L o... |
cdleml4N 35285 | Part of proof of Lemma L o... |
cdleml5N 35286 | Part of proof of Lemma L o... |
cdleml6 35287 | Part of proof of Lemma L o... |
cdleml7 35288 | Part of proof of Lemma L o... |
cdleml8 35289 | Part of proof of Lemma L o... |
cdleml9 35290 | Part of proof of Lemma L o... |
dva1dim 35291 | Two expressions for the 1-... |
dvhb1dimN 35292 | Two expressions for the 1-... |
erng1lem 35293 | Value of the endomorphism ... |
erngdvlem1 35294 | Lemma for ~ eringring . (... |
erngdvlem2N 35295 | Lemma for ~ eringring . (... |
erngdvlem3 35296 | Lemma for ~ eringring . (... |
erngdvlem4 35297 | Lemma for ~ erngdv . (Con... |
eringring 35298 | An endomorphism ring is a ... |
erngdv 35299 | An endomorphism ring is a ... |
erng0g 35300 | The division ring zero of ... |
erng1r 35301 | The division ring unit of ... |
erngdvlem1-rN 35302 | Lemma for ~ eringring . (... |
erngdvlem2-rN 35303 | Lemma for ~ eringring . (... |
erngdvlem3-rN 35304 | Lemma for ~ eringring . (... |
erngdvlem4-rN 35305 | Lemma for ~ erngdv . (Con... |
erngring-rN 35306 | An endomorphism ring is a ... |
erngdv-rN 35307 | An endomorphism ring is a ... |
dvafset 35310 | The constructed partial ve... |
dvaset 35311 | The constructed partial ve... |
dvasca 35312 | The ring base set of the c... |
dvabase 35313 | The ring base set of the c... |
dvafplusg 35314 | Ring addition operation fo... |
dvaplusg 35315 | Ring addition operation fo... |
dvaplusgv 35316 | Ring addition operation fo... |
dvafmulr 35317 | Ring multiplication operat... |
dvamulr 35318 | Ring multiplication operat... |
dvavbase 35319 | The vectors (vector base s... |
dvafvadd 35320 | The vector sum operation f... |
dvavadd 35321 | Ring addition operation fo... |
dvafvsca 35322 | Ring addition operation fo... |
dvavsca 35323 | Ring addition operation fo... |
tendospid 35324 | Identity property of endom... |
tendospcl 35325 | Closure of endomorphism sc... |
tendospass 35326 | Associative law for endomo... |
tendospdi1 35327 | Forward distributive law f... |
tendocnv 35328 | Converse of a trace-preser... |
tendospdi2 35329 | Reverse distributive law f... |
tendospcanN 35330 | Cancellation law for trace... |
dvaabl 35331 | The constructed partial ve... |
dvalveclem 35332 | Lemma for ~ dvalvec . (Co... |
dvalvec 35333 | The constructed partial ve... |
dva0g 35334 | The zero vector of partial... |
diaffval 35337 | The partial isomorphism A ... |
diafval 35338 | The partial isomorphism A ... |
diaval 35339 | The partial isomorphism A ... |
diaelval 35340 | Member of the partial isom... |
diafn 35341 | Functionality and domain o... |
diadm 35342 | Domain of the partial isom... |
diaeldm 35343 | Member of domain of the pa... |
diadmclN 35344 | A member of domain of the ... |
diadmleN 35345 | A member of domain of the ... |
dian0 35346 | The value of the partial i... |
dia0eldmN 35347 | The lattice zero belongs t... |
dia1eldmN 35348 | The fiducial hyperplane (t... |
diass 35349 | The value of the partial i... |
diael 35350 | A member of the value of t... |
diatrl 35351 | Trace of a member of the p... |
diaelrnN 35352 | Any value of the partial i... |
dialss 35353 | The value of partial isomo... |
diaord 35354 | The partial isomorphism A ... |
dia11N 35355 | The partial isomorphism A ... |
diaf11N 35356 | The partial isomorphism A ... |
diaclN 35357 | Closure of partial isomorp... |
diacnvclN 35358 | Closure of partial isomorp... |
dia0 35359 | The value of the partial i... |
dia1N 35360 | The value of the partial i... |
dia1elN 35361 | The largest subspace in th... |
diaglbN 35362 | Partial isomorphism A of a... |
diameetN 35363 | Partial isomorphism A of a... |
diainN 35364 | Inverse partial isomorphis... |
diaintclN 35365 | The intersection of partia... |
diasslssN 35366 | The partial isomorphism A ... |
diassdvaN 35367 | The partial isomorphism A ... |
dia1dim 35368 | Two expressions for the 1-... |
dia1dim2 35369 | Two expressions for a 1-di... |
dia1dimid 35370 | A vector (translation) bel... |
dia2dimlem1 35371 | Lemma for ~ dia2dim . Sho... |
dia2dimlem2 35372 | Lemma for ~ dia2dim . Def... |
dia2dimlem3 35373 | Lemma for ~ dia2dim . Def... |
dia2dimlem4 35374 | Lemma for ~ dia2dim . Sho... |
dia2dimlem5 35375 | Lemma for ~ dia2dim . The... |
dia2dimlem6 35376 | Lemma for ~ dia2dim . Eli... |
dia2dimlem7 35377 | Lemma for ~ dia2dim . Eli... |
dia2dimlem8 35378 | Lemma for ~ dia2dim . Eli... |
dia2dimlem9 35379 | Lemma for ~ dia2dim . Eli... |
dia2dimlem10 35380 | Lemma for ~ dia2dim . Con... |
dia2dimlem11 35381 | Lemma for ~ dia2dim . Con... |
dia2dimlem12 35382 | Lemma for ~ dia2dim . Obt... |
dia2dimlem13 35383 | Lemma for ~ dia2dim . Eli... |
dia2dim 35384 | A two-dimensional subspace... |
dvhfset 35387 | The constructed full vecto... |
dvhset 35388 | The constructed full vecto... |
dvhsca 35389 | The ring of scalars of the... |
dvhbase 35390 | The ring base set of the c... |
dvhfplusr 35391 | Ring addition operation fo... |
dvhfmulr 35392 | Ring multiplication operat... |
dvhmulr 35393 | Ring multiplication operat... |
dvhvbase 35394 | The vectors (vector base s... |
dvhelvbasei 35395 | Vector membership in the c... |
dvhvaddcbv 35396 | Change bound variables to ... |
dvhvaddval 35397 | The vector sum operation f... |
dvhfvadd 35398 | The vector sum operation f... |
dvhvadd 35399 | The vector sum operation f... |
dvhopvadd 35400 | The vector sum operation f... |
dvhopvadd2 35401 | The vector sum operation f... |
dvhvaddcl 35402 | Closure of the vector sum ... |
dvhvaddcomN 35403 | Commutativity of vector su... |
dvhvaddass 35404 | Associativity of vector su... |
dvhvscacbv 35405 | Change bound variables to ... |
dvhvscaval 35406 | The scalar product operati... |
dvhfvsca 35407 | Scalar product operation f... |
dvhvsca 35408 | Scalar product operation f... |
dvhopvsca 35409 | Scalar product operation f... |
dvhvscacl 35410 | Closure of the scalar prod... |
tendoinvcl 35411 | Closure of multiplicative ... |
tendolinv 35412 | Left multiplicative invers... |
tendorinv 35413 | Right multiplicative inver... |
dvhgrp 35414 | The full vector space ` U ... |
dvhlveclem 35415 | Lemma for ~ dvhlvec . TOD... |
dvhlvec 35416 | The full vector space ` U ... |
dvhlmod 35417 | The full vector space ` U ... |
dvh0g 35418 | The zero vector of vector ... |
dvheveccl 35419 | Properties of a unit vecto... |
dvhopclN 35420 | Closure of a ` DVecH ` vec... |
dvhopaddN 35421 | Sum of ` DVecH ` vectors e... |
dvhopspN 35422 | Scalar product of ` DVecH ... |
dvhopN 35423 | Decompose a ` DVecH ` vect... |
dvhopellsm 35424 | Ordered pair membership in... |
cdlemm10N 35425 | The image of the map ` G `... |
docaffvalN 35428 | Subspace orthocomplement f... |
docafvalN 35429 | Subspace orthocomplement f... |
docavalN 35430 | Subspace orthocomplement f... |
docaclN 35431 | Closure of subspace orthoc... |
diaocN 35432 | Value of partial isomorphi... |
doca2N 35433 | Double orthocomplement of ... |
doca3N 35434 | Double orthocomplement of ... |
dvadiaN 35435 | Any closed subspace is a m... |
diarnN 35436 | Partial isomorphism A maps... |
diaf1oN 35437 | The partial isomorphism A ... |
djaffvalN 35440 | Subspace join for ` DVecA ... |
djafvalN 35441 | Subspace join for ` DVecA ... |
djavalN 35442 | Subspace join for ` DVecA ... |
djaclN 35443 | Closure of subspace join f... |
djajN 35444 | Transfer lattice join to `... |
dibffval 35447 | The partial isomorphism B ... |
dibfval 35448 | The partial isomorphism B ... |
dibval 35449 | The partial isomorphism B ... |
dibopelvalN 35450 | Member of the partial isom... |
dibval2 35451 | Value of the partial isomo... |
dibopelval2 35452 | Member of the partial isom... |
dibval3N 35453 | Value of the partial isomo... |
dibelval3 35454 | Member of the partial isom... |
dibopelval3 35455 | Member of the partial isom... |
dibelval1st 35456 | Membership in value of the... |
dibelval1st1 35457 | Membership in value of the... |
dibelval1st2N 35458 | Membership in value of the... |
dibelval2nd 35459 | Membership in value of the... |
dibn0 35460 | The value of the partial i... |
dibfna 35461 | Functionality and domain o... |
dibdiadm 35462 | Domain of the partial isom... |
dibfnN 35463 | Functionality and domain o... |
dibdmN 35464 | Domain of the partial isom... |
dibeldmN 35465 | Member of domain of the pa... |
dibord 35466 | The isomorphism B for a la... |
dib11N 35467 | The isomorphism B for a la... |
dibf11N 35468 | The partial isomorphism A ... |
dibclN 35469 | Closure of partial isomorp... |
dibvalrel 35470 | The value of partial isomo... |
dib0 35471 | The value of partial isomo... |
dib1dim 35472 | Two expressions for the 1-... |
dibglbN 35473 | Partial isomorphism B of a... |
dibintclN 35474 | The intersection of partia... |
dib1dim2 35475 | Two expressions for a 1-di... |
dibss 35476 | The partial isomorphism B ... |
diblss 35477 | The value of partial isomo... |
diblsmopel 35478 | Membership in subspace sum... |
dicffval 35481 | The partial isomorphism C ... |
dicfval 35482 | The partial isomorphism C ... |
dicval 35483 | The partial isomorphism C ... |
dicopelval 35484 | Membership in value of the... |
dicelvalN 35485 | Membership in value of the... |
dicval2 35486 | The partial isomorphism C ... |
dicelval3 35487 | Member of the partial isom... |
dicopelval2 35488 | Membership in value of the... |
dicelval2N 35489 | Membership in value of the... |
dicfnN 35490 | Functionality and domain o... |
dicdmN 35491 | Domain of the partial isom... |
dicvalrelN 35492 | The value of partial isomo... |
dicssdvh 35493 | The partial isomorphism C ... |
dicelval1sta 35494 | Membership in value of the... |
dicelval1stN 35495 | Membership in value of the... |
dicelval2nd 35496 | Membership in value of the... |
dicvaddcl 35497 | Membership in value of the... |
dicvscacl 35498 | Membership in value of the... |
dicn0 35499 | The value of the partial i... |
diclss 35500 | The value of partial isomo... |
diclspsn 35501 | The value of isomorphism C... |
cdlemn2 35502 | Part of proof of Lemma N o... |
cdlemn2a 35503 | Part of proof of Lemma N o... |
cdlemn3 35504 | Part of proof of Lemma N o... |
cdlemn4 35505 | Part of proof of Lemma N o... |
cdlemn4a 35506 | Part of proof of Lemma N o... |
cdlemn5pre 35507 | Part of proof of Lemma N o... |
cdlemn5 35508 | Part of proof of Lemma N o... |
cdlemn6 35509 | Part of proof of Lemma N o... |
cdlemn7 35510 | Part of proof of Lemma N o... |
cdlemn8 35511 | Part of proof of Lemma N o... |
cdlemn9 35512 | Part of proof of Lemma N o... |
cdlemn10 35513 | Part of proof of Lemma N o... |
cdlemn11a 35514 | Part of proof of Lemma N o... |
cdlemn11b 35515 | Part of proof of Lemma N o... |
cdlemn11c 35516 | Part of proof of Lemma N o... |
cdlemn11pre 35517 | Part of proof of Lemma N o... |
cdlemn11 35518 | Part of proof of Lemma N o... |
cdlemn 35519 | Lemma N of [Crawley] p. 12... |
dihordlem6 35520 | Part of proof of Lemma N o... |
dihordlem7 35521 | Part of proof of Lemma N o... |
dihordlem7b 35522 | Part of proof of Lemma N o... |
dihjustlem 35523 | Part of proof after Lemma ... |
dihjust 35524 | Part of proof after Lemma ... |
dihord1 35525 | Part of proof after Lemma ... |
dihord2a 35526 | Part of proof after Lemma ... |
dihord2b 35527 | Part of proof after Lemma ... |
dihord2cN 35528 | Part of proof after Lemma ... |
dihord11b 35529 | Part of proof after Lemma ... |
dihord10 35530 | Part of proof after Lemma ... |
dihord11c 35531 | Part of proof after Lemma ... |
dihord2pre 35532 | Part of proof after Lemma ... |
dihord2pre2 35533 | Part of proof after Lemma ... |
dihord2 35534 | Part of proof after Lemma ... |
dihffval 35537 | The isomorphism H for a la... |
dihfval 35538 | Isomorphism H for a lattic... |
dihval 35539 | Value of isomorphism H for... |
dihvalc 35540 | Value of isomorphism H for... |
dihlsscpre 35541 | Closure of isomorphism H f... |
dihvalcqpre 35542 | Value of isomorphism H for... |
dihvalcq 35543 | Value of isomorphism H for... |
dihvalb 35544 | Value of isomorphism H for... |
dihopelvalbN 35545 | Ordered pair member of the... |
dihvalcqat 35546 | Value of isomorphism H for... |
dih1dimb 35547 | Two expressions for a 1-di... |
dih1dimb2 35548 | Isomorphism H at an atom u... |
dih1dimc 35549 | Isomorphism H at an atom n... |
dib2dim 35550 | Extend ~ dia2dim to partia... |
dih2dimb 35551 | Extend ~ dib2dim to isomor... |
dih2dimbALTN 35552 | Extend ~ dia2dim to isomor... |
dihopelvalcqat 35553 | Ordered pair member of the... |
dihvalcq2 35554 | Value of isomorphism H for... |
dihopelvalcpre 35555 | Member of value of isomorp... |
dihopelvalc 35556 | Member of value of isomorp... |
dihlss 35557 | The value of isomorphism H... |
dihss 35558 | The value of isomorphism H... |
dihssxp 35559 | An isomorphism H value is ... |
dihopcl 35560 | Closure of an ordered pair... |
xihopellsmN 35561 | Ordered pair membership in... |
dihopellsm 35562 | Ordered pair membership in... |
dihord6apre 35563 | Part of proof that isomorp... |
dihord3 35564 | The isomorphism H for a la... |
dihord4 35565 | The isomorphism H for a la... |
dihord5b 35566 | Part of proof that isomorp... |
dihord6b 35567 | Part of proof that isomorp... |
dihord6a 35568 | Part of proof that isomorp... |
dihord5apre 35569 | Part of proof that isomorp... |
dihord5a 35570 | Part of proof that isomorp... |
dihord 35571 | The isomorphism H is order... |
dih11 35572 | The isomorphism H is one-t... |
dihf11lem 35573 | Functionality of the isomo... |
dihf11 35574 | The isomorphism H for a la... |
dihfn 35575 | Functionality and domain o... |
dihdm 35576 | Domain of isomorphism H. (... |
dihcl 35577 | Closure of isomorphism H. ... |
dihcnvcl 35578 | Closure of isomorphism H c... |
dihcnvid1 35579 | The converse isomorphism o... |
dihcnvid2 35580 | The isomorphism of a conve... |
dihcnvord 35581 | Ordering property for conv... |
dihcnv11 35582 | The converse of isomorphis... |
dihsslss 35583 | The isomorphism H maps to ... |
dihrnlss 35584 | The isomorphism H maps to ... |
dihrnss 35585 | The isomorphism H maps to ... |
dihvalrel 35586 | The value of isomorphism H... |
dih0 35587 | The value of isomorphism H... |
dih0bN 35588 | A lattice element is zero ... |
dih0vbN 35589 | A vector is zero iff its s... |
dih0cnv 35590 | The isomorphism H converse... |
dih0rn 35591 | The zero subspace belongs ... |
dih0sb 35592 | A subspace is zero iff the... |
dih1 35593 | The value of isomorphism H... |
dih1rn 35594 | The full vector space belo... |
dih1cnv 35595 | The isomorphism H converse... |
dihwN 35596 | Value of isomorphism H at ... |
dihmeetlem1N 35597 | Isomorphism H of a conjunc... |
dihglblem5apreN 35598 | A conjunction property of ... |
dihglblem5aN 35599 | A conjunction property of ... |
dihglblem2aN 35600 | Lemma for isomorphism H of... |
dihglblem2N 35601 | The GLB of a set of lattic... |
dihglblem3N 35602 | Isomorphism H of a lattice... |
dihglblem3aN 35603 | Isomorphism H of a lattice... |
dihglblem4 35604 | Isomorphism H of a lattice... |
dihglblem5 35605 | Isomorphism H of a lattice... |
dihmeetlem2N 35606 | Isomorphism H of a conjunc... |
dihglbcpreN 35607 | Isomorphism H of a lattice... |
dihglbcN 35608 | Isomorphism H of a lattice... |
dihmeetcN 35609 | Isomorphism H of a lattice... |
dihmeetbN 35610 | Isomorphism H of a lattice... |
dihmeetbclemN 35611 | Lemma for isomorphism H of... |
dihmeetlem3N 35612 | Lemma for isomorphism H of... |
dihmeetlem4preN 35613 | Lemma for isomorphism H of... |
dihmeetlem4N 35614 | Lemma for isomorphism H of... |
dihmeetlem5 35615 | Part of proof that isomorp... |
dihmeetlem6 35616 | Lemma for isomorphism H of... |
dihmeetlem7N 35617 | Lemma for isomorphism H of... |
dihjatc1 35618 | Lemma for isomorphism H of... |
dihjatc2N 35619 | Isomorphism H of join with... |
dihjatc3 35620 | Isomorphism H of join with... |
dihmeetlem8N 35621 | Lemma for isomorphism H of... |
dihmeetlem9N 35622 | Lemma for isomorphism H of... |
dihmeetlem10N 35623 | Lemma for isomorphism H of... |
dihmeetlem11N 35624 | Lemma for isomorphism H of... |
dihmeetlem12N 35625 | Lemma for isomorphism H of... |
dihmeetlem13N 35626 | Lemma for isomorphism H of... |
dihmeetlem14N 35627 | Lemma for isomorphism H of... |
dihmeetlem15N 35628 | Lemma for isomorphism H of... |
dihmeetlem16N 35629 | Lemma for isomorphism H of... |
dihmeetlem17N 35630 | Lemma for isomorphism H of... |
dihmeetlem18N 35631 | Lemma for isomorphism H of... |
dihmeetlem19N 35632 | Lemma for isomorphism H of... |
dihmeetlem20N 35633 | Lemma for isomorphism H of... |
dihmeetALTN 35634 | Isomorphism H of a lattice... |
dih1dimatlem0 35635 | Lemma for ~ dih1dimat . (... |
dih1dimatlem 35636 | Lemma for ~ dih1dimat . (... |
dih1dimat 35637 | Any 1-dimensional subspace... |
dihlsprn 35638 | The span of a vector belon... |
dihlspsnssN 35639 | A subspace included in a 1... |
dihlspsnat 35640 | The inverse isomorphism H ... |
dihatlat 35641 | The isomorphism H of an at... |
dihat 35642 | There exists at least one ... |
dihpN 35643 | The value of isomorphism H... |
dihlatat 35644 | The reverse isomorphism H ... |
dihatexv 35645 | There is a nonzero vector ... |
dihatexv2 35646 | There is a nonzero vector ... |
dihglblem6 35647 | Isomorphism H of a lattice... |
dihglb 35648 | Isomorphism H of a lattice... |
dihglb2 35649 | Isomorphism H of a lattice... |
dihmeet 35650 | Isomorphism H of a lattice... |
dihintcl 35651 | The intersection of closed... |
dihmeetcl 35652 | Closure of closed subspace... |
dihmeet2 35653 | Reverse isomorphism H of a... |
dochffval 35656 | Subspace orthocomplement f... |
dochfval 35657 | Subspace orthocomplement f... |
dochval 35658 | Subspace orthocomplement f... |
dochval2 35659 | Subspace orthocomplement f... |
dochcl 35660 | Closure of subspace orthoc... |
dochlss 35661 | A subspace orthocomplement... |
dochssv 35662 | A subspace orthocomplement... |
dochfN 35663 | Domain and codomain of the... |
dochvalr 35664 | Orthocomplement of a close... |
doch0 35665 | Orthocomplement of the zer... |
doch1 35666 | Orthocomplement of the uni... |
dochoc0 35667 | The zero subspace is close... |
dochoc1 35668 | The unit subspace (all vec... |
dochvalr2 35669 | Orthocomplement of a close... |
dochvalr3 35670 | Orthocomplement of a close... |
doch2val2 35671 | Double orthocomplement for... |
dochss 35672 | Subset law for orthocomple... |
dochocss 35673 | Double negative law for or... |
dochoc 35674 | Double negative law for or... |
dochsscl 35675 | If a set of vectors is inc... |
dochoccl 35676 | A set of vectors is closed... |
dochord 35677 | Ordering law for orthocomp... |
dochord2N 35678 | Ordering law for orthocomp... |
dochord3 35679 | Ordering law for orthocomp... |
doch11 35680 | Orthocomplement is one-to-... |
dochsordN 35681 | Strict ordering law for or... |
dochn0nv 35682 | An orthocomplement is nonz... |
dihoml4c 35683 | Version of ~ dihoml4 with ... |
dihoml4 35684 | Orthomodular law for const... |
dochspss 35685 | The span of a set of vecto... |
dochocsp 35686 | The span of an orthocomple... |
dochspocN 35687 | The span of an orthocomple... |
dochocsn 35688 | The double orthocomplement... |
dochsncom 35689 | Swap vectors in an orthoco... |
dochsat 35690 | The double orthocomplement... |
dochshpncl 35691 | If a hyperplane is not clo... |
dochlkr 35692 | Equivalent conditions for ... |
dochkrshp 35693 | The closure of a kernel is... |
dochkrshp2 35694 | Properties of the closure ... |
dochkrshp3 35695 | Properties of the closure ... |
dochkrshp4 35696 | Properties of the closure ... |
dochdmj1 35697 | De Morgan-like law for sub... |
dochnoncon 35698 | Law of noncontradiction. ... |
dochnel2 35699 | A nonzero member of a subs... |
dochnel 35700 | A nonzero vector doesn't b... |
djhffval 35703 | Subspace join for ` DVecH ... |
djhfval 35704 | Subspace join for ` DVecH ... |
djhval 35705 | Subspace join for ` DVecH ... |
djhval2 35706 | Value of subspace join for... |
djhcl 35707 | Closure of subspace join f... |
djhlj 35708 | Transfer lattice join to `... |
djhljjN 35709 | Lattice join in terms of `... |
djhjlj 35710 | ` DVecH ` vector space clo... |
djhj 35711 | ` DVecH ` vector space clo... |
djhcom 35712 | Subspace join commutes. (... |
djhspss 35713 | Subspace span of union is ... |
djhsumss 35714 | Subspace sum is a subset o... |
dihsumssj 35715 | The subspace sum of two is... |
djhunssN 35716 | Subspace union is a subset... |
dochdmm1 35717 | De Morgan-like law for clo... |
djhexmid 35718 | Excluded middle property o... |
djh01 35719 | Closed subspace join with ... |
djh02 35720 | Closed subspace join with ... |
djhlsmcl 35721 | A closed subspace sum equa... |
djhcvat42 35722 | A covering property. ( ~ ... |
dihjatb 35723 | Isomorphism H of lattice j... |
dihjatc 35724 | Isomorphism H of lattice j... |
dihjatcclem1 35725 | Lemma for isomorphism H of... |
dihjatcclem2 35726 | Lemma for isomorphism H of... |
dihjatcclem3 35727 | Lemma for ~ dihjatcc . (C... |
dihjatcclem4 35728 | Lemma for isomorphism H of... |
dihjatcc 35729 | Isomorphism H of lattice j... |
dihjat 35730 | Isomorphism H of lattice j... |
dihprrnlem1N 35731 | Lemma for ~ dihprrn , show... |
dihprrnlem2 35732 | Lemma for ~ dihprrn . (Co... |
dihprrn 35733 | The span of a vector pair ... |
djhlsmat 35734 | The sum of two subspace at... |
dihjat1lem 35735 | Subspace sum of a closed s... |
dihjat1 35736 | Subspace sum of a closed s... |
dihsmsprn 35737 | Subspace sum of a closed s... |
dihjat2 35738 | The subspace sum of a clos... |
dihjat3 35739 | Isomorphism H of lattice j... |
dihjat4 35740 | Transfer the subspace sum ... |
dihjat6 35741 | Transfer the subspace sum ... |
dihsmsnrn 35742 | The subspace sum of two si... |
dihsmatrn 35743 | The subspace sum of a clos... |
dihjat5N 35744 | Transfer lattice join with... |
dvh4dimat 35745 | There is an atom that is o... |
dvh3dimatN 35746 | There is an atom that is o... |
dvh2dimatN 35747 | Given an atom, there exist... |
dvh1dimat 35748 | There exists an atom. (Co... |
dvh1dim 35749 | There exists a nonzero vec... |
dvh4dimlem 35750 | Lemma for ~ dvh4dimN . (C... |
dvhdimlem 35751 | Lemma for ~ dvh2dim and ~ ... |
dvh2dim 35752 | There is a vector that is ... |
dvh3dim 35753 | There is a vector that is ... |
dvh4dimN 35754 | There is a vector that is ... |
dvh3dim2 35755 | There is a vector that is ... |
dvh3dim3N 35756 | There is a vector that is ... |
dochsnnz 35757 | The orthocomplement of a s... |
dochsatshp 35758 | The orthocomplement of a s... |
dochsatshpb 35759 | The orthocomplement of a s... |
dochsnshp 35760 | The orthocomplement of a n... |
dochshpsat 35761 | A hyperplane is closed iff... |
dochkrsat 35762 | The orthocomplement of a k... |
dochkrsat2 35763 | The orthocomplement of a k... |
dochsat0 35764 | The orthocomplement of a k... |
dochkrsm 35765 | The subspace sum of a clos... |
dochexmidat 35766 | Special case of excluded m... |
dochexmidlem1 35767 | Lemma for ~ dochexmid . H... |
dochexmidlem2 35768 | Lemma for ~ dochexmid . (... |
dochexmidlem3 35769 | Lemma for ~ dochexmid . U... |
dochexmidlem4 35770 | Lemma for ~ dochexmid . (... |
dochexmidlem5 35771 | Lemma for ~ dochexmid . (... |
dochexmidlem6 35772 | Lemma for ~ dochexmid . (... |
dochexmidlem7 35773 | Lemma for ~ dochexmid . C... |
dochexmidlem8 35774 | Lemma for ~ dochexmid . T... |
dochexmid 35775 | Excluded middle law for cl... |
dochsnkrlem1 35776 | Lemma for ~ dochsnkr . (C... |
dochsnkrlem2 35777 | Lemma for ~ dochsnkr . (C... |
dochsnkrlem3 35778 | Lemma for ~ dochsnkr . (C... |
dochsnkr 35779 | A (closed) kernel expresse... |
dochsnkr2 35780 | Kernel of the explicit fun... |
dochsnkr2cl 35781 | The ` X ` determining func... |
dochflcl 35782 | Closure of the explicit fu... |
dochfl1 35783 | The value of the explicit ... |
dochfln0 35784 | The value of a functional ... |
dochkr1 35785 | A nonzero functional has a... |
dochkr1OLDN 35786 | A nonzero functional has a... |
lpolsetN 35789 | The set of polarities of a... |
islpolN 35790 | The predicate "is a polari... |
islpoldN 35791 | Properties that determine ... |
lpolfN 35792 | Functionality of a polarit... |
lpolvN 35793 | The polarity of the whole ... |
lpolconN 35794 | Contraposition property of... |
lpolsatN 35795 | The polarity of an atomic ... |
lpolpolsatN 35796 | Property of a polarity. (... |
dochpolN 35797 | The subspace orthocompleme... |
lcfl1lem 35798 | Property of a functional w... |
lcfl1 35799 | Property of a functional w... |
lcfl2 35800 | Property of a functional w... |
lcfl3 35801 | Property of a functional w... |
lcfl4N 35802 | Property of a functional w... |
lcfl5 35803 | Property of a functional w... |
lcfl5a 35804 | Property of a functional w... |
lcfl6lem 35805 | Lemma for ~ lcfl6 . A fun... |
lcfl7lem 35806 | Lemma for ~ lcfl7N . If t... |
lcfl6 35807 | Property of a functional w... |
lcfl7N 35808 | Property of a functional w... |
lcfl8 35809 | Property of a functional w... |
lcfl8a 35810 | Property of a functional w... |
lcfl8b 35811 | Property of a nonzero func... |
lcfl9a 35812 | Property implying that a f... |
lclkrlem1 35813 | The set of functionals hav... |
lclkrlem2a 35814 | Lemma for ~ lclkr . Use ~... |
lclkrlem2b 35815 | Lemma for ~ lclkr . (Cont... |
lclkrlem2c 35816 | Lemma for ~ lclkr . (Cont... |
lclkrlem2d 35817 | Lemma for ~ lclkr . (Cont... |
lclkrlem2e 35818 | Lemma for ~ lclkr . The k... |
lclkrlem2f 35819 | Lemma for ~ lclkr . Const... |
lclkrlem2g 35820 | Lemma for ~ lclkr . Compa... |
lclkrlem2h 35821 | Lemma for ~ lclkr . Elimi... |
lclkrlem2i 35822 | Lemma for ~ lclkr . Elimi... |
lclkrlem2j 35823 | Lemma for ~ lclkr . Kerne... |
lclkrlem2k 35824 | Lemma for ~ lclkr . Kerne... |
lclkrlem2l 35825 | Lemma for ~ lclkr . Elimi... |
lclkrlem2m 35826 | Lemma for ~ lclkr . Const... |
lclkrlem2n 35827 | Lemma for ~ lclkr . (Cont... |
lclkrlem2o 35828 | Lemma for ~ lclkr . When ... |
lclkrlem2p 35829 | Lemma for ~ lclkr . When ... |
lclkrlem2q 35830 | Lemma for ~ lclkr . The s... |
lclkrlem2r 35831 | Lemma for ~ lclkr . When ... |
lclkrlem2s 35832 | Lemma for ~ lclkr . Thus,... |
lclkrlem2t 35833 | Lemma for ~ lclkr . We el... |
lclkrlem2u 35834 | Lemma for ~ lclkr . ~ lclk... |
lclkrlem2v 35835 | Lemma for ~ lclkr . When ... |
lclkrlem2w 35836 | Lemma for ~ lclkr . This ... |
lclkrlem2x 35837 | Lemma for ~ lclkr . Elimi... |
lclkrlem2y 35838 | Lemma for ~ lclkr . Resta... |
lclkrlem2 35839 | The set of functionals hav... |
lclkr 35840 | The set of functionals wit... |
lcfls1lem 35841 | Property of a functional w... |
lcfls1N 35842 | Property of a functional w... |
lcfls1c 35843 | Property of a functional w... |
lclkrslem1 35844 | The set of functionals hav... |
lclkrslem2 35845 | The set of functionals hav... |
lclkrs 35846 | The set of functionals hav... |
lclkrs2 35847 | The set of functionals wit... |
lcfrvalsnN 35848 | Reconstruction from the du... |
lcfrlem1 35849 | Lemma for ~ lcfr . Note t... |
lcfrlem2 35850 | Lemma for ~ lcfr . (Contr... |
lcfrlem3 35851 | Lemma for ~ lcfr . (Contr... |
lcfrlem4 35852 | Lemma for ~ lcfr . (Contr... |
lcfrlem5 35853 | Lemma for ~ lcfr . The se... |
lcfrlem6 35854 | Lemma for ~ lcfr . Closur... |
lcfrlem7 35855 | Lemma for ~ lcfr . Closur... |
lcfrlem8 35856 | Lemma for ~ lcf1o and ~ lc... |
lcfrlem9 35857 | Lemma for ~ lcf1o . (This... |
lcf1o 35858 | Define a function ` J ` th... |
lcfrlem10 35859 | Lemma for ~ lcfr . (Contr... |
lcfrlem11 35860 | Lemma for ~ lcfr . (Contr... |
lcfrlem12N 35861 | Lemma for ~ lcfr . (Contr... |
lcfrlem13 35862 | Lemma for ~ lcfr . (Contr... |
lcfrlem14 35863 | Lemma for ~ lcfr . (Contr... |
lcfrlem15 35864 | Lemma for ~ lcfr . (Contr... |
lcfrlem16 35865 | Lemma for ~ lcfr . (Contr... |
lcfrlem17 35866 | Lemma for ~ lcfr . Condit... |
lcfrlem18 35867 | Lemma for ~ lcfr . (Contr... |
lcfrlem19 35868 | Lemma for ~ lcfr . (Contr... |
lcfrlem20 35869 | Lemma for ~ lcfr . (Contr... |
lcfrlem21 35870 | Lemma for ~ lcfr . (Contr... |
lcfrlem22 35871 | Lemma for ~ lcfr . (Contr... |
lcfrlem23 35872 | Lemma for ~ lcfr . TODO: ... |
lcfrlem24 35873 | Lemma for ~ lcfr . (Contr... |
lcfrlem25 35874 | Lemma for ~ lcfr . Specia... |
lcfrlem26 35875 | Lemma for ~ lcfr . Specia... |
lcfrlem27 35876 | Lemma for ~ lcfr . Specia... |
lcfrlem28 35877 | Lemma for ~ lcfr . TODO: ... |
lcfrlem29 35878 | Lemma for ~ lcfr . (Contr... |
lcfrlem30 35879 | Lemma for ~ lcfr . (Contr... |
lcfrlem31 35880 | Lemma for ~ lcfr . (Contr... |
lcfrlem32 35881 | Lemma for ~ lcfr . (Contr... |
lcfrlem33 35882 | Lemma for ~ lcfr . (Contr... |
lcfrlem34 35883 | Lemma for ~ lcfr . (Contr... |
lcfrlem35 35884 | Lemma for ~ lcfr . (Contr... |
lcfrlem36 35885 | Lemma for ~ lcfr . (Contr... |
lcfrlem37 35886 | Lemma for ~ lcfr . (Contr... |
lcfrlem38 35887 | Lemma for ~ lcfr . Combin... |
lcfrlem39 35888 | Lemma for ~ lcfr . Elimin... |
lcfrlem40 35889 | Lemma for ~ lcfr . Elimin... |
lcfrlem41 35890 | Lemma for ~ lcfr . Elimin... |
lcfrlem42 35891 | Lemma for ~ lcfr . Elimin... |
lcfr 35892 | Reconstruction of a subspa... |
lcdfval 35895 | Dual vector space of funct... |
lcdval 35896 | Dual vector space of funct... |
lcdval2 35897 | Dual vector space of funct... |
lcdlvec 35898 | The dual vector space of f... |
lcdlmod 35899 | The dual vector space of f... |
lcdvbase 35900 | Vector base set of a dual ... |
lcdvbasess 35901 | The vector base set of the... |
lcdvbaselfl 35902 | A vector in the base set o... |
lcdvbasecl 35903 | Closure of the value of a ... |
lcdvadd 35904 | Vector addition for the cl... |
lcdvaddval 35905 | The value of the value of ... |
lcdsca 35906 | The ring of scalars of the... |
lcdsbase 35907 | Base set of scalar ring fo... |
lcdsadd 35908 | Scalar addition for the cl... |
lcdsmul 35909 | Scalar multiplication for ... |
lcdvs 35910 | Scalar product for the clo... |
lcdvsval 35911 | Value of scalar product op... |
lcdvscl 35912 | The scalar product operati... |
lcdlssvscl 35913 | Closure of scalar product ... |
lcdvsass 35914 | Associative law for scalar... |
lcd0 35915 | The zero scalar of the clo... |
lcd1 35916 | The unit scalar of the clo... |
lcdneg 35917 | The unit scalar of the clo... |
lcd0v 35918 | The zero functional in the... |
lcd0v2 35919 | The zero functional in the... |
lcd0vvalN 35920 | Value of the zero function... |
lcd0vcl 35921 | Closure of the zero functi... |
lcd0vs 35922 | A scalar zero times a func... |
lcdvs0N 35923 | A scalar times the zero fu... |
lcdvsub 35924 | The value of vector subtra... |
lcdvsubval 35925 | The value of the value of ... |
lcdlss 35926 | Subspaces of a dual vector... |
lcdlss2N 35927 | Subspaces of a dual vector... |
lcdlsp 35928 | Span in the set of functio... |
lcdlkreqN 35929 | Colinear functionals have ... |
lcdlkreq2N 35930 | Colinear functionals have ... |
mapdffval 35933 | Projectivity from vector s... |
mapdfval 35934 | Projectivity from vector s... |
mapdval 35935 | Value of projectivity from... |
mapdvalc 35936 | Value of projectivity from... |
mapdval2N 35937 | Value of projectivity from... |
mapdval3N 35938 | Value of projectivity from... |
mapdval4N 35939 | Value of projectivity from... |
mapdval5N 35940 | Value of projectivity from... |
mapdordlem1a 35941 | Lemma for ~ mapdord . (Co... |
mapdordlem1bN 35942 | Lemma for ~ mapdord . (Co... |
mapdordlem1 35943 | Lemma for ~ mapdord . (Co... |
mapdordlem2 35944 | Lemma for ~ mapdord . Ord... |
mapdord 35945 | Ordering property of the m... |
mapd11 35946 | The map defined by ~ df-ma... |
mapddlssN 35947 | The mapping of a subspace ... |
mapdsn 35948 | Value of the map defined b... |
mapdsn2 35949 | Value of the map defined b... |
mapdsn3 35950 | Value of the map defined b... |
mapd1dim2lem1N 35951 | Value of the map defined b... |
mapdrvallem2 35952 | Lemma for ~ mapdrval . TO... |
mapdrvallem3 35953 | Lemma for ~ mapdrval . (C... |
mapdrval 35954 | Given a dual subspace ` R ... |
mapd1o 35955 | The map defined by ~ df-ma... |
mapdrn 35956 | Range of the map defined b... |
mapdunirnN 35957 | Union of the range of the ... |
mapdrn2 35958 | Range of the map defined b... |
mapdcnvcl 35959 | Closure of the converse of... |
mapdcl 35960 | Closure the value of the m... |
mapdcnvid1N 35961 | Converse of the value of t... |
mapdsord 35962 | Strong ordering property o... |
mapdcl2 35963 | The mapping of a subspace ... |
mapdcnvid2 35964 | Value of the converse of t... |
mapdcnvordN 35965 | Ordering property of the c... |
mapdcnv11N 35966 | The converse of the map de... |
mapdcv 35967 | Covering property of the c... |
mapdincl 35968 | Closure of dual subspace i... |
mapdin 35969 | Subspace intersection is p... |
mapdlsmcl 35970 | Closure of dual subspace s... |
mapdlsm 35971 | Subspace sum is preserved ... |
mapd0 35972 | Projectivity map of the ze... |
mapdcnvatN 35973 | Atoms are preserved by the... |
mapdat 35974 | Atoms are preserved by the... |
mapdspex 35975 | The map of a span equals t... |
mapdn0 35976 | Transfer nonzero property ... |
mapdncol 35977 | Transfer non-colinearity f... |
mapdindp 35978 | Transfer (part of) vector ... |
mapdpglem1 35979 | Lemma for ~ mapdpg . Baer... |
mapdpglem2 35980 | Lemma for ~ mapdpg . Baer... |
mapdpglem2a 35981 | Lemma for ~ mapdpg . (Con... |
mapdpglem3 35982 | Lemma for ~ mapdpg . Baer... |
mapdpglem4N 35983 | Lemma for ~ mapdpg . (Con... |
mapdpglem5N 35984 | Lemma for ~ mapdpg . (Con... |
mapdpglem6 35985 | Lemma for ~ mapdpg . Baer... |
mapdpglem8 35986 | Lemma for ~ mapdpg . Baer... |
mapdpglem9 35987 | Lemma for ~ mapdpg . Baer... |
mapdpglem10 35988 | Lemma for ~ mapdpg . Baer... |
mapdpglem11 35989 | Lemma for ~ mapdpg . (Con... |
mapdpglem12 35990 | Lemma for ~ mapdpg . TODO... |
mapdpglem13 35991 | Lemma for ~ mapdpg . (Con... |
mapdpglem14 35992 | Lemma for ~ mapdpg . (Con... |
mapdpglem15 35993 | Lemma for ~ mapdpg . (Con... |
mapdpglem16 35994 | Lemma for ~ mapdpg . Baer... |
mapdpglem17N 35995 | Lemma for ~ mapdpg . Baer... |
mapdpglem18 35996 | Lemma for ~ mapdpg . Baer... |
mapdpglem19 35997 | Lemma for ~ mapdpg . Baer... |
mapdpglem20 35998 | Lemma for ~ mapdpg . Baer... |
mapdpglem21 35999 | Lemma for ~ mapdpg . (Con... |
mapdpglem22 36000 | Lemma for ~ mapdpg . Baer... |
mapdpglem23 36001 | Lemma for ~ mapdpg . Baer... |
mapdpglem30a 36002 | Lemma for ~ mapdpg . (Con... |
mapdpglem30b 36003 | Lemma for ~ mapdpg . (Con... |
mapdpglem25 36004 | Lemma for ~ mapdpg . Baer... |
mapdpglem26 36005 | Lemma for ~ mapdpg . Baer... |
mapdpglem27 36006 | Lemma for ~ mapdpg . Baer... |
mapdpglem29 36007 | Lemma for ~ mapdpg . Baer... |
mapdpglem28 36008 | Lemma for ~ mapdpg . Baer... |
mapdpglem30 36009 | Lemma for ~ mapdpg . Baer... |
mapdpglem31 36010 | Lemma for ~ mapdpg . Baer... |
mapdpglem24 36011 | Lemma for ~ mapdpg . Exis... |
mapdpglem32 36012 | Lemma for ~ mapdpg . Uniq... |
mapdpg 36013 | Part 1 of proof of the fir... |
baerlem3lem1 36014 | Lemma for ~ baerlem3 . (C... |
baerlem5alem1 36015 | Lemma for ~ baerlem5a . (... |
baerlem5blem1 36016 | Lemma for ~ baerlem5b . (... |
baerlem3lem2 36017 | Lemma for ~ baerlem3 . (C... |
baerlem5alem2 36018 | Lemma for ~ baerlem5a . (... |
baerlem5blem2 36019 | Lemma for ~ baerlem5b . (... |
baerlem3 36020 | An equality that holds whe... |
baerlem5a 36021 | An equality that holds whe... |
baerlem5b 36022 | An equality that holds whe... |
baerlem5amN 36023 | An equality that holds whe... |
baerlem5bmN 36024 | An equality that holds whe... |
baerlem5abmN 36025 | An equality that holds whe... |
mapdindp0 36026 | Vector independence lemma.... |
mapdindp1 36027 | Vector independence lemma.... |
mapdindp2 36028 | Vector independence lemma.... |
mapdindp3 36029 | Vector independence lemma.... |
mapdindp4 36030 | Vector independence lemma.... |
mapdhval 36031 | Lemmma for ~~? mapdh . (C... |
mapdhval0 36032 | Lemmma for ~~? mapdh . (C... |
mapdhval2 36033 | Lemmma for ~~? mapdh . (C... |
mapdhcl 36034 | Lemmma for ~~? mapdh . (C... |
mapdheq 36035 | Lemmma for ~~? mapdh . Th... |
mapdheq2 36036 | Lemmma for ~~? mapdh . On... |
mapdheq2biN 36037 | Lemmma for ~~? mapdh . Pa... |
mapdheq4lem 36038 | Lemma for ~ mapdheq4 . Pa... |
mapdheq4 36039 | Lemma for ~~? mapdh . Par... |
mapdh6lem1N 36040 | Lemma for ~ mapdh6N . Par... |
mapdh6lem2N 36041 | Lemma for ~ mapdh6N . Par... |
mapdh6aN 36042 | Lemma for ~ mapdh6N . Par... |
mapdh6b0N 36043 | Lemmma for ~ mapdh6N . (C... |
mapdh6bN 36044 | Lemmma for ~ mapdh6N . (C... |
mapdh6cN 36045 | Lemmma for ~ mapdh6N . (C... |
mapdh6dN 36046 | Lemmma for ~ mapdh6N . (C... |
mapdh6eN 36047 | Lemmma for ~ mapdh6N . Pa... |
mapdh6fN 36048 | Lemmma for ~ mapdh6N . Pa... |
mapdh6gN 36049 | Lemmma for ~ mapdh6N . Pa... |
mapdh6hN 36050 | Lemmma for ~ mapdh6N . Pa... |
mapdh6iN 36051 | Lemmma for ~ mapdh6N . El... |
mapdh6jN 36052 | Lemmma for ~ mapdh6N . El... |
mapdh6kN 36053 | Lemmma for ~ mapdh6N . El... |
mapdh6N 36054 | Part (6) of [Baer] p. 47 l... |
mapdh7eN 36055 | Part (7) of [Baer] p. 48 l... |
mapdh7cN 36056 | Part (7) of [Baer] p. 48 l... |
mapdh7dN 36057 | Part (7) of [Baer] p. 48 l... |
mapdh7fN 36058 | Part (7) of [Baer] p. 48 l... |
mapdh75e 36059 | Part (7) of [Baer] p. 48 l... |
mapdh75cN 36060 | Part (7) of [Baer] p. 48 l... |
mapdh75d 36061 | Part (7) of [Baer] p. 48 l... |
mapdh75fN 36062 | Part (7) of [Baer] p. 48 l... |
hvmapffval 36065 | Map from nonzero vectors t... |
hvmapfval 36066 | Map from nonzero vectors t... |
hvmapval 36067 | Value of map from nonzero ... |
hvmapvalvalN 36068 | Value of value of map (i.e... |
hvmapidN 36069 | The value of the vector to... |
hvmap1o 36070 | The vector to functional m... |
hvmapclN 36071 | Closure of the vector to f... |
hvmap1o2 36072 | The vector to functional m... |
hvmapcl2 36073 | Closure of the vector to f... |
hvmaplfl 36074 | The vector to functional m... |
hvmaplkr 36075 | Kernel of the vector to fu... |
mapdhvmap 36076 | Relationship between ` map... |
lspindp5 36077 | Obtain an independent vect... |
hdmaplem1 36078 | Lemma to convert a frequen... |
hdmaplem2N 36079 | Lemma to convert a frequen... |
hdmaplem3 36080 | Lemma to convert a frequen... |
hdmaplem4 36081 | Lemma to convert a frequen... |
mapdh8a 36082 | Part of Part (8) in [Baer]... |
mapdh8aa 36083 | Part of Part (8) in [Baer]... |
mapdh8ab 36084 | Part of Part (8) in [Baer]... |
mapdh8ac 36085 | Part of Part (8) in [Baer]... |
mapdh8ad 36086 | Part of Part (8) in [Baer]... |
mapdh8b 36087 | Part of Part (8) in [Baer]... |
mapdh8c 36088 | Part of Part (8) in [Baer]... |
mapdh8d0N 36089 | Part of Part (8) in [Baer]... |
mapdh8d 36090 | Part of Part (8) in [Baer]... |
mapdh8e 36091 | Part of Part (8) in [Baer]... |
mapdh8fN 36092 | Part of Part (8) in [Baer]... |
mapdh8g 36093 | Part of Part (8) in [Baer]... |
mapdh8i 36094 | Part of Part (8) in [Baer]... |
mapdh8j 36095 | Part of Part (8) in [Baer]... |
mapdh8 36096 | Part (8) in [Baer] p. 48. ... |
mapdh9a 36097 | Lemma for part (9) in [Bae... |
mapdh9aOLDN 36098 | Lemma for part (9) in [Bae... |
hdmap1ffval 36103 | Preliminary map from vecto... |
hdmap1fval 36104 | Preliminary map from vecto... |
hdmap1vallem 36105 | Value of preliminary map f... |
hdmap1val 36106 | Value of preliminary map f... |
hdmap1val0 36107 | Value of preliminary map f... |
hdmap1val2 36108 | Value of preliminary map f... |
hdmap1eq 36109 | The defining equation for ... |
hdmap1cbv 36110 | Frequently used lemma to c... |
hdmap1valc 36111 | Connect the value of the p... |
hdmap1cl 36112 | Convert closure theorem ~ ... |
hdmap1eq2 36113 | Convert ~ mapdheq2 to use ... |
hdmap1eq4N 36114 | Convert ~ mapdheq4 to use ... |
hdmap1l6lem1 36115 | Lemma for ~ hdmap1l6 . Pa... |
hdmap1l6lem2 36116 | Lemma for ~ hdmap1l6 . Pa... |
hdmap1l6a 36117 | Lemma for ~ hdmap1l6 . Pa... |
hdmap1l6b0N 36118 | Lemmma for ~ hdmap1l6 . (... |
hdmap1l6b 36119 | Lemmma for ~ hdmap1l6 . (... |
hdmap1l6c 36120 | Lemmma for ~ hdmap1l6 . (... |
hdmap1l6d 36121 | Lemmma for ~ hdmap1l6 . (... |
hdmap1l6e 36122 | Lemmma for ~ hdmap1l6 . P... |
hdmap1l6f 36123 | Lemmma for ~ hdmap1l6 . P... |
hdmap1l6g 36124 | Lemmma for ~ hdmap1l6 . P... |
hdmap1l6h 36125 | Lemmma for ~ hdmap1l6 . P... |
hdmap1l6i 36126 | Lemmma for ~ hdmap1l6 . E... |
hdmap1l6j 36127 | Lemmma for ~ hdmap1l6 . E... |
hdmap1l6k 36128 | Lemmma for ~ hdmap1l6 . E... |
hdmap1l6 36129 | Part (6) of [Baer] p. 47 l... |
hdmap1p6N 36130 | (Convert ~ mapdh6N to use ... |
hdmap1eulem 36131 | Lemma for ~ hdmap1eu . TO... |
hdmap1eulemOLDN 36132 | Lemma for ~ hdmap1euOLDN .... |
hdmap1eu 36133 | Convert ~ mapdh9a to use t... |
hdmap1euOLDN 36134 | Convert ~ mapdh9aOLDN to u... |
hdmap1neglem1N 36135 | Lemma for ~ hdmapneg . TO... |
hdmapffval 36136 | Map from vectors to functi... |
hdmapfval 36137 | Map from vectors to functi... |
hdmapval 36138 | Value of map from vectors ... |
hdmapfnN 36139 | Functionality of map from ... |
hdmapcl 36140 | Closure of map from vector... |
hdmapval2lem 36141 | Lemma for ~ hdmapval2 . (... |
hdmapval2 36142 | Value of map from vectors ... |
hdmapval0 36143 | Value of map from vectors ... |
hdmapeveclem 36144 | Lemma for ~ hdmapevec . T... |
hdmapevec 36145 | Value of map from vectors ... |
hdmapevec2 36146 | The inner product of the r... |
hdmapval3lemN 36147 | Value of map from vectors ... |
hdmapval3N 36148 | Value of map from vectors ... |
hdmap10lem 36149 | Lemma for ~ hdmap10 . (Co... |
hdmap10 36150 | Part 10 in [Baer] p. 48 li... |
hdmap11lem1 36151 | Lemma for ~ hdmapadd . (C... |
hdmap11lem2 36152 | Lemma for ~ hdmapadd . (C... |
hdmapadd 36153 | Part 11 in [Baer] p. 48 li... |
hdmapeq0 36154 | Part of proof of part 12 i... |
hdmapnzcl 36155 | Nonzero vector closure of ... |
hdmapneg 36156 | Part of proof of part 12 i... |
hdmapsub 36157 | Part of proof of part 12 i... |
hdmap11 36158 | Part of proof of part 12 i... |
hdmaprnlem1N 36159 | Part of proof of part 12 i... |
hdmaprnlem3N 36160 | Part of proof of part 12 i... |
hdmaprnlem3uN 36161 | Part of proof of part 12 i... |
hdmaprnlem4tN 36162 | Lemma for ~ hdmaprnN . TO... |
hdmaprnlem4N 36163 | Part of proof of part 12 i... |
hdmaprnlem6N 36164 | Part of proof of part 12 i... |
hdmaprnlem7N 36165 | Part of proof of part 12 i... |
hdmaprnlem8N 36166 | Part of proof of part 12 i... |
hdmaprnlem9N 36167 | Part of proof of part 12 i... |
hdmaprnlem3eN 36168 | Lemma for ~ hdmaprnN . (C... |
hdmaprnlem10N 36169 | Lemma for ~ hdmaprnN . Sh... |
hdmaprnlem11N 36170 | Lemma for ~ hdmaprnN . Sh... |
hdmaprnlem15N 36171 | Lemma for ~ hdmaprnN . El... |
hdmaprnlem16N 36172 | Lemma for ~ hdmaprnN . El... |
hdmaprnlem17N 36173 | Lemma for ~ hdmaprnN . In... |
hdmaprnN 36174 | Part of proof of part 12 i... |
hdmapf1oN 36175 | Part 12 in [Baer] p. 49. ... |
hdmap14lem1a 36176 | Prior to part 14 in [Baer]... |
hdmap14lem2a 36177 | Prior to part 14 in [Baer]... |
hdmap14lem1 36178 | Prior to part 14 in [Baer]... |
hdmap14lem2N 36179 | Prior to part 14 in [Baer]... |
hdmap14lem3 36180 | Prior to part 14 in [Baer]... |
hdmap14lem4a 36181 | Simplify ` ( A \ { Q } ) `... |
hdmap14lem4 36182 | Simplify ` ( A \ { Q } ) `... |
hdmap14lem6 36183 | Case where ` F ` is zero. ... |
hdmap14lem7 36184 | Combine cases of ` F ` . ... |
hdmap14lem8 36185 | Part of proof of part 14 i... |
hdmap14lem9 36186 | Part of proof of part 14 i... |
hdmap14lem10 36187 | Part of proof of part 14 i... |
hdmap14lem11 36188 | Part of proof of part 14 i... |
hdmap14lem12 36189 | Lemma for proof of part 14... |
hdmap14lem13 36190 | Lemma for proof of part 14... |
hdmap14lem14 36191 | Part of proof of part 14 i... |
hdmap14lem15 36192 | Part of proof of part 14 i... |
hgmapffval 36195 | Map from the scalar divisi... |
hgmapfval 36196 | Map from the scalar divisi... |
hgmapval 36197 | Value of map from the scal... |
hgmapfnN 36198 | Functionality of scalar si... |
hgmapcl 36199 | Closure of scalar sigma ma... |
hgmapdcl 36200 | Closure of the vector spac... |
hgmapvs 36201 | Part 15 of [Baer] p. 50 li... |
hgmapval0 36202 | Value of the scalar sigma ... |
hgmapval1 36203 | Value of the scalar sigma ... |
hgmapadd 36204 | Part 15 of [Baer] p. 50 li... |
hgmapmul 36205 | Part 15 of [Baer] p. 50 li... |
hgmaprnlem1N 36206 | Lemma for ~ hgmaprnN . (C... |
hgmaprnlem2N 36207 | Lemma for ~ hgmaprnN . Pa... |
hgmaprnlem3N 36208 | Lemma for ~ hgmaprnN . El... |
hgmaprnlem4N 36209 | Lemma for ~ hgmaprnN . El... |
hgmaprnlem5N 36210 | Lemma for ~ hgmaprnN . El... |
hgmaprnN 36211 | Part of proof of part 16 i... |
hgmap11 36212 | The scalar sigma map is on... |
hgmapf1oN 36213 | The scalar sigma map is a ... |
hgmapeq0 36214 | The scalar sigma map is ze... |
hdmapipcl 36215 | The inner product (Hermiti... |
hdmapln1 36216 | Linearity property that wi... |
hdmaplna1 36217 | Additive property of first... |
hdmaplns1 36218 | Subtraction property of fi... |
hdmaplnm1 36219 | Multiplicative property of... |
hdmaplna2 36220 | Additive property of secon... |
hdmapglnm2 36221 | g-linear property of secon... |
hdmapgln2 36222 | g-linear property that wil... |
hdmaplkr 36223 | Kernel of the vector to du... |
hdmapellkr 36224 | Membership in the kernel (... |
hdmapip0 36225 | Zero property that will be... |
hdmapip1 36226 | Construct a proportional v... |
hdmapip0com 36227 | Commutation property of Ba... |
hdmapinvlem1 36228 | Line 27 in [Baer] p. 110. ... |
hdmapinvlem2 36229 | Line 28 in [Baer] p. 110, ... |
hdmapinvlem3 36230 | Line 30 in [Baer] p. 110, ... |
hdmapinvlem4 36231 | Part 1.1 of Proposition 1 ... |
hdmapglem5 36232 | Part 1.2 in [Baer] p. 110 ... |
hgmapvvlem1 36233 | Involution property of sca... |
hgmapvvlem2 36234 | Lemma for ~ hgmapvv . Eli... |
hgmapvvlem3 36235 | Lemma for ~ hgmapvv . Eli... |
hgmapvv 36236 | Value of a double involuti... |
hdmapglem7a 36237 | Lemma for ~ hdmapg . (Con... |
hdmapglem7b 36238 | Lemma for ~ hdmapg . (Con... |
hdmapglem7 36239 | Lemma for ~ hdmapg . Line... |
hdmapg 36240 | Apply the scalar sigma fun... |
hdmapoc 36241 | Express our constructed or... |
hlhilset 36244 | The final Hilbert space co... |
hlhilsca 36245 | The scalar of the final co... |
hlhilbase 36246 | The base set of the final ... |
hlhilplus 36247 | The vector addition for th... |
hlhilslem 36248 | Lemma for ~ hlhilsbase2 . ... |
hlhilsbase 36249 | The scalar base set of the... |
hlhilsplus 36250 | Scalar addition for the fi... |
hlhilsmul 36251 | Scalar multiplication for ... |
hlhilsbase2 36252 | The scalar base set of the... |
hlhilsplus2 36253 | Scalar addition for the fi... |
hlhilsmul2 36254 | Scalar multiplication for ... |
hlhils0 36255 | The scalar ring zero for t... |
hlhils1N 36256 | The scalar ring unity for ... |
hlhilvsca 36257 | The scalar product for the... |
hlhilip 36258 | Inner product operation fo... |
hlhilipval 36259 | Value of inner product ope... |
hlhilnvl 36260 | The involution operation o... |
hlhillvec 36261 | The final constructed Hilb... |
hlhildrng 36262 | The star division ring for... |
hlhilsrnglem 36263 | Lemma for ~ hlhilsrng . (... |
hlhilsrng 36264 | The star division ring for... |
hlhil0 36265 | The zero vector for the fi... |
hlhillsm 36266 | The vector sum operation f... |
hlhilocv 36267 | The orthocomplement for th... |
hlhillcs 36268 | The closed subspaces of th... |
hlhilphllem 36269 | Lemma for ~ hlhil . (Cont... |
hlhilhillem 36270 | Lemma for ~ hlhil . (Cont... |
hlathil 36271 | Construction of a Hilbert ... |
rntrclfvOAI 36272 | The range of the transitiv... |
moxfr 36273 | Transfer at-most-one betwe... |
imaiinfv 36274 | Indexed intersection of an... |
elrfi 36275 | Elementhood in a set of re... |
elrfirn 36276 | Elementhood in a set of re... |
elrfirn2 36277 | Elementhood in a set of re... |
cmpfiiin 36278 | In a compact topology, a s... |
ismrcd1 36279 | Any function from the subs... |
ismrcd2 36280 | Second half of ~ ismrcd1 .... |
istopclsd 36281 | A closure function which s... |
ismrc 36282 | A function is a Moore clos... |
isnacs 36285 | Expand definition of Noeth... |
nacsfg 36286 | In a Noetherian-type closu... |
isnacs2 36287 | Express Noetherian-type cl... |
mrefg2 36288 | Slight variation on finite... |
mrefg3 36289 | Slight variation on finite... |
nacsacs 36290 | A closure system of Noethe... |
isnacs3 36291 | A choice-free order equiva... |
incssnn0 36292 | Transitivity induction of ... |
nacsfix 36293 | An increasing sequence of ... |
constmap 36294 | A constant (represented wi... |
mapco2g 36295 | Renaming indexes in a tupl... |
mapco2 36296 | Post-composition (renaming... |
mapfzcons 36297 | Extending a one-based mapp... |
mapfzcons1 36298 | Recover prefix mapping fro... |
mapfzcons1cl 36299 | A nonempty mapping has a p... |
mapfzcons2 36300 | Recover added element from... |
mptfcl 36301 | Interpret range of a maps-... |
mzpclval 36306 | Substitution lemma for ` m... |
elmzpcl 36307 | Double substitution lemma ... |
mzpclall 36308 | The set of all functions w... |
mzpcln0 36309 | Corrolary of ~ mzpclall : ... |
mzpcl1 36310 | Defining property 1 of a p... |
mzpcl2 36311 | Defining property 2 of a p... |
mzpcl34 36312 | Defining properties 3 and ... |
mzpval 36313 | Value of the ` mzPoly ` fu... |
dmmzp 36314 | ` mzPoly ` is defined for ... |
mzpincl 36315 | Polynomial closedness is a... |
mzpconst 36316 | Constant functions are pol... |
mzpf 36317 | A polynomial function is a... |
mzpproj 36318 | A projection function is p... |
mzpadd 36319 | The pointwise sum of two p... |
mzpmul 36320 | The pointwise product of t... |
mzpconstmpt 36321 | A constant function expres... |
mzpaddmpt 36322 | Sum of polynomial function... |
mzpmulmpt 36323 | Product of polynomial func... |
mzpsubmpt 36324 | The difference of two poly... |
mzpnegmpt 36325 | Negation of a polynomial f... |
mzpexpmpt 36326 | Raise a polynomial functio... |
mzpindd 36327 | "Structural" induction to ... |
mzpmfp 36328 | Relationship between multi... |
mzpsubst 36329 | Substituting polynomials f... |
mzprename 36330 | Simplified version of ~ mz... |
mzpresrename 36331 | A polynomial is a polynomi... |
mzpcompact2lem 36332 | Lemma for ~ mzpcompact2 . ... |
mzpcompact2 36333 | Polynomials are finitary o... |
coeq0i 36334 | ~ coeq0 but without explic... |
fzsplit1nn0 36335 | Split a finite 1-based set... |
eldiophb 36338 | Initial expression of Diop... |
eldioph 36339 | Condition for a set to be ... |
diophrw 36340 | Renaming and adding unused... |
eldioph2lem1 36341 | Lemma for ~ eldioph2 . Co... |
eldioph2lem2 36342 | Lemma for ~ eldioph2 . Co... |
eldioph2 36343 | Construct a Diophantine se... |
eldioph2b 36344 | While Diophantine sets wer... |
eldiophelnn0 36345 | Remove antecedent on ` B `... |
eldioph3b 36346 | Define Diophantine sets in... |
eldioph3 36347 | Inference version of ~ eld... |
ellz1 36348 | Membership in a lower set ... |
lzunuz 36349 | The union of a lower set o... |
fz1eqin 36350 | Express a one-based finite... |
lzenom 36351 | Lower integers are countab... |
elmapresaun 36352 | ~ fresaun transposed to ma... |
elmapresaunres2 36353 | ~ fresaunres2 transposed t... |
diophin 36354 | If two sets are Diophantin... |
diophun 36355 | If two sets are Diophantin... |
eldiophss 36356 | Diophantine sets are sets ... |
diophrex 36357 | Projecting a Diophantine s... |
eq0rabdioph 36358 | This is the first of a num... |
eqrabdioph 36359 | Diophantine set builder fo... |
0dioph 36360 | The null set is Diophantin... |
vdioph 36361 | The "universal" set (as la... |
anrabdioph 36362 | Diophantine set builder fo... |
orrabdioph 36363 | Diophantine set builder fo... |
3anrabdioph 36364 | Diophantine set builder fo... |
3orrabdioph 36365 | Diophantine set builder fo... |
2sbcrex 36366 | Exchange an existential qu... |
sbcrexgOLD 36367 | Interchange class substitu... |
2sbcrexOLD 36368 | Exchange an existential qu... |
sbc2rex 36369 | Exchange a substitution wi... |
sbc2rexgOLD 36370 | Exchange a substitution wi... |
sbc4rex 36371 | Exchange a substitution wi... |
sbc4rexgOLD 36372 | Exchange a substitution wi... |
sbcrot3 36373 | Rotate a sequence of three... |
sbcrot5 36374 | Rotate a sequence of five ... |
sbccomieg 36375 | Commute two explicit subst... |
rexrabdioph 36376 | Diophantine set builder fo... |
rexfrabdioph 36377 | Diophantine set builder fo... |
2rexfrabdioph 36378 | Diophantine set builder fo... |
3rexfrabdioph 36379 | Diophantine set builder fo... |
4rexfrabdioph 36380 | Diophantine set builder fo... |
6rexfrabdioph 36381 | Diophantine set builder fo... |
7rexfrabdioph 36382 | Diophantine set builder fo... |
rabdiophlem1 36383 | Lemma for arithmetic dioph... |
rabdiophlem2 36384 | Lemma for arithmetic dioph... |
elnn0rabdioph 36385 | Diophantine set builder fo... |
rexzrexnn0 36386 | Rewrite a quantification o... |
lerabdioph 36387 | Diophantine set builder fo... |
eluzrabdioph 36388 | Diophantine set builder fo... |
elnnrabdioph 36389 | Diophantine set builder fo... |
ltrabdioph 36390 | Diophantine set builder fo... |
nerabdioph 36391 | Diophantine set builder fo... |
dvdsrabdioph 36392 | Divisibility is a Diophant... |
eldioph4b 36393 | Membership in ` Dioph ` ex... |
eldioph4i 36394 | Forward-only version of ~ ... |
diophren 36395 | Change variables in a Diop... |
rabrenfdioph 36396 | Change variable numbers in... |
rabren3dioph 36397 | Change variable numbers in... |
fphpd 36398 | Pigeonhole principle expre... |
fphpdo 36399 | Pigeonhole principle for s... |
ctbnfien 36400 | An infinite subset of a co... |
fiphp3d 36401 | Infinite pigeonhole princi... |
rencldnfilem 36402 | Lemma for ~ rencldnfi . (... |
rencldnfi 36403 | A set of real numbers whic... |
irrapxlem1 36404 | Lemma for ~ irrapx1 . Div... |
irrapxlem2 36405 | Lemma for ~ irrapx1 . Two... |
irrapxlem3 36406 | Lemma for ~ irrapx1 . By ... |
irrapxlem4 36407 | Lemma for ~ irrapx1 . Eli... |
irrapxlem5 36408 | Lemma for ~ irrapx1 . Swi... |
irrapxlem6 36409 | Lemma for ~ irrapx1 . Exp... |
irrapx1 36410 | Dirichlet's approximation ... |
pellexlem1 36411 | Lemma for ~ pellex . Arit... |
pellexlem2 36412 | Lemma for ~ pellex . Arit... |
pellexlem3 36413 | Lemma for ~ pellex . To e... |
pellexlem4 36414 | Lemma for ~ pellex . Invo... |
pellexlem5 36415 | Lemma for ~ pellex . Invo... |
pellexlem6 36416 | Lemma for ~ pellex . Doin... |
pellex 36417 | Every Pell equation has a ... |
pell1qrval 36428 | Value of the set of first-... |
elpell1qr 36429 | Membership in a first-quad... |
pell14qrval 36430 | Value of the set of positi... |
elpell14qr 36431 | Membership in the set of p... |
pell1234qrval 36432 | Value of the set of genera... |
elpell1234qr 36433 | Membership in the set of g... |
pell1234qrre 36434 | General Pell solutions are... |
pell1234qrne0 36435 | No solution to a Pell equa... |
pell1234qrreccl 36436 | General solutions of the P... |
pell1234qrmulcl 36437 | General solutions of the P... |
pell14qrss1234 36438 | A positive Pell solution i... |
pell14qrre 36439 | A positive Pell solution i... |
pell14qrne0 36440 | A positive Pell solution i... |
pell14qrgt0 36441 | A positive Pell solution i... |
pell14qrrp 36442 | A positive Pell solution i... |
pell1234qrdich 36443 | A general Pell solution is... |
elpell14qr2 36444 | A number is a positive Pel... |
pell14qrmulcl 36445 | Positive Pell solutions ar... |
pell14qrreccl 36446 | Positive Pell solutions ar... |
pell14qrdivcl 36447 | Positive Pell solutions ar... |
pell14qrexpclnn0 36448 | Lemma for ~ pell14qrexpcl ... |
pell14qrexpcl 36449 | Positive Pell solutions ar... |
pell1qrss14 36450 | First-quadrant Pell soluti... |
pell14qrdich 36451 | A positive Pell solution i... |
pell1qrge1 36452 | A Pell solution in the fir... |
pell1qr1 36453 | 1 is a Pell solution and i... |
elpell1qr2 36454 | The first quadrant solutio... |
pell1qrgaplem 36455 | Lemma for ~ pell1qrgap . ... |
pell1qrgap 36456 | First-quadrant Pell soluti... |
pell14qrgap 36457 | Positive Pell solutions ar... |
pell14qrgapw 36458 | Positive Pell solutions ar... |
pellqrexplicit 36459 | Condition for a calculated... |
infmrgelbi 36460 | Any lower bound of a nonem... |
pellqrex 36461 | There is a nontrivial solu... |
pellfundval 36462 | Value of the fundamental s... |
pellfundre 36463 | The fundamental solution o... |
pellfundge 36464 | Lower bound on the fundame... |
pellfundgt1 36465 | Weak lower bound on the Pe... |
pellfundlb 36466 | A nontrivial first quadran... |
pellfundglb 36467 | If a real is larger than t... |
pellfundex 36468 | The fundamental solution a... |
pellfund14gap 36469 | There are no solutions bet... |
pellfundrp 36470 | The fundamental Pell solut... |
pellfundne1 36471 | The fundamental Pell solut... |
reglogcl 36472 | General logarithm is a rea... |
reglogltb 36473 | General logarithm preserve... |
reglogleb 36474 | General logarithm preserve... |
reglogmul 36475 | Multiplication law for gen... |
reglogexp 36476 | Power law for general log.... |
reglogbas 36477 | General log of the base is... |
reglog1 36478 | General log of 1 is 0. (C... |
reglogexpbas 36479 | General log of a power of ... |
pellfund14 36480 | Every positive Pell soluti... |
pellfund14b 36481 | The positive Pell solution... |
rmxfval 36486 | Value of the X sequence. ... |
rmyfval 36487 | Value of the Y sequence. ... |
rmspecsqrtnq 36488 | The discriminant used to d... |
rmspecsqrtnqOLD 36489 | Obsolete version of ~ rmsp... |
rmspecnonsq 36490 | The discriminant used to d... |
qirropth 36491 | This lemma implements the ... |
rmspecfund 36492 | The base of exponent used ... |
rmxyelqirr 36493 | The solutions used to cons... |
rmxypairf1o 36494 | The function used to extra... |
rmxyelxp 36495 | Lemma for ~ frmx and ~ frm... |
frmx 36496 | The X sequence is a nonneg... |
frmy 36497 | The Y sequence is an integ... |
rmxyval 36498 | Main definition of the X a... |
rmspecpos 36499 | The discriminant used to d... |
rmxycomplete 36500 | The X and Y sequences take... |
rmxynorm 36501 | The X and Y sequences defi... |
rmbaserp 36502 | The base of exponentiation... |
rmxyneg 36503 | Negation law for X and Y s... |
rmxyadd 36504 | Addition formula for X and... |
rmxy1 36505 | Value of the X and Y seque... |
rmxy0 36506 | Value of the X and Y seque... |
rmxneg 36507 | Negation law (even functio... |
rmx0 36508 | Value of X sequence at 0. ... |
rmx1 36509 | Value of X sequence at 1. ... |
rmxadd 36510 | Addition formula for X seq... |
rmyneg 36511 | Negation formula for Y seq... |
rmy0 36512 | Value of Y sequence at 0. ... |
rmy1 36513 | Value of Y sequence at 1. ... |
rmyadd 36514 | Addition formula for Y seq... |
rmxp1 36515 | Special addition-of-1 form... |
rmyp1 36516 | Special addition of 1 form... |
rmxm1 36517 | Subtraction of 1 formula f... |
rmym1 36518 | Subtraction of 1 formula f... |
rmxluc 36519 | The X sequence is a Lucas ... |
rmyluc 36520 | The Y sequence is a Lucas ... |
rmyluc2 36521 | Lucas sequence property of... |
rmxdbl 36522 | "Double-angle formula" for... |
rmydbl 36523 | "Double-angle formula" for... |
monotuz 36524 | A function defined on an u... |
monotoddzzfi 36525 | A function which is odd an... |
monotoddzz 36526 | A function (given implicit... |
oddcomabszz 36527 | An odd function which take... |
2nn0ind 36528 | Induction on nonnegative i... |
zindbi 36529 | Inductively transfer a pro... |
expmordi 36530 | Mantissa ordering relation... |
rpexpmord 36531 | Mantissa ordering relation... |
rmxypos 36532 | For all nonnegative indice... |
ltrmynn0 36533 | The Y-sequence is strictly... |
ltrmxnn0 36534 | The X-sequence is strictly... |
lermxnn0 36535 | The X-sequence is monotoni... |
rmxnn 36536 | The X-sequence is defined ... |
ltrmy 36537 | The Y-sequence is strictly... |
rmyeq0 36538 | Y is zero only at zero. (... |
rmyeq 36539 | Y is one-to-one. (Contrib... |
lermy 36540 | Y is monotonic (non-strict... |
rmynn 36541 | ` rmY ` is positive for po... |
rmynn0 36542 | ` rmY ` is nonnegative for... |
rmyabs 36543 | ` rmY ` commutes with ` ab... |
jm2.24nn 36544 | X(n) is strictly greater t... |
jm2.17a 36545 | First half of lemma 2.17 o... |
jm2.17b 36546 | Weak form of the second ha... |
jm2.17c 36547 | Second half of lemma 2.17 ... |
jm2.24 36548 | Lemma 2.24 of [JonesMatija... |
rmygeid 36549 | Y(n) increases faster than... |
congtr 36550 | A wff of the form ` A || (... |
congadd 36551 | If two pairs of numbers ar... |
congmul 36552 | If two pairs of numbers ar... |
congsym 36553 | Congruence mod ` A ` is a ... |
congneg 36554 | If two integers are congru... |
congsub 36555 | If two pairs of numbers ar... |
congid 36556 | Every integer is congruent... |
mzpcong 36557 | Polynomials commute with c... |
congrep 36558 | Every integer is congruent... |
congabseq 36559 | If two integers are congru... |
acongid 36560 | A wff like that in this th... |
acongsym 36561 | Symmetry of alternating co... |
acongneg2 36562 | Negate right side of alter... |
acongtr 36563 | Transitivity of alternatin... |
acongeq12d 36564 | Substitution deduction for... |
acongrep 36565 | Every integer is alternati... |
fzmaxdif 36566 | Bound on the difference be... |
fzneg 36567 | Reflection of a finite ran... |
acongeq 36568 | Two numbers in the fundame... |
dvdsacongtr 36569 | Alternating congruence pas... |
coprmdvdsb 36570 | Multiplication by a coprim... |
modabsdifz 36571 | Divisibility in terms of m... |
dvdsabsmod0 36572 | Divisibility in terms of m... |
jm2.18 36573 | Theorem 2.18 of [JonesMati... |
jm2.19lem1 36574 | Lemma for ~ jm2.19 . X an... |
jm2.19lem2 36575 | Lemma for ~ jm2.19 . (Con... |
jm2.19lem3 36576 | Lemma for ~ jm2.19 . (Con... |
jm2.19lem4 36577 | Lemma for ~ jm2.19 . Exte... |
jm2.19 36578 | Lemma 2.19 of [JonesMatija... |
jm2.21 36579 | Lemma for ~ jm2.20nn . Ex... |
jm2.22 36580 | Lemma for ~ jm2.20nn . Ap... |
jm2.23 36581 | Lemma for ~ jm2.20nn . Tr... |
jm2.20nn 36582 | Lemma 2.20 of [JonesMatija... |
jm2.25lem1 36583 | Lemma for ~ jm2.26 . (Con... |
jm2.25 36584 | Lemma for ~ jm2.26 . Rema... |
jm2.26a 36585 | Lemma for ~ jm2.26 . Reve... |
jm2.26lem3 36586 | Lemma for ~ jm2.26 . Use ... |
jm2.26 36587 | Lemma 2.26 of [JonesMatija... |
jm2.15nn0 36588 | Lemma 2.15 of [JonesMatija... |
jm2.16nn0 36589 | Lemma 2.16 of [JonesMatija... |
jm2.27a 36590 | Lemma for ~ jm2.27 . Reve... |
jm2.27b 36591 | Lemma for ~ jm2.27 . Expa... |
jm2.27c 36592 | Lemma for ~ jm2.27 . Forw... |
jm2.27 36593 | Lemma 2.27 of [JonesMatija... |
jm2.27dlem1 36594 | Lemma for ~ rmydioph . Su... |
jm2.27dlem2 36595 | Lemma for ~ rmydioph . Th... |
jm2.27dlem3 36596 | Lemma for ~ rmydioph . In... |
jm2.27dlem4 36597 | Lemma for ~ rmydioph . In... |
jm2.27dlem5 36598 | Lemma for ~ rmydioph . Us... |
rmydioph 36599 | ~ jm2.27 restated in terms... |
rmxdiophlem 36600 | X can be expressed in term... |
rmxdioph 36601 | X is a Diophantine functio... |
jm3.1lem1 36602 | Lemma for ~ jm3.1 . (Cont... |
jm3.1lem2 36603 | Lemma for ~ jm3.1 . (Cont... |
jm3.1lem3 36604 | Lemma for ~ jm3.1 . (Cont... |
jm3.1 36605 | Diophantine expression for... |
expdiophlem1 36606 | Lemma for ~ expdioph . Fu... |
expdiophlem2 36607 | Lemma for ~ expdioph . Ex... |
expdioph 36608 | The exponential function i... |
setindtr 36609 | Epsilon induction for sets... |
setindtrs 36610 | Epsilon induction scheme w... |
dford3lem1 36611 | Lemma for ~ dford3 . (Con... |
dford3lem2 36612 | Lemma for ~ dford3 . (Con... |
dford3 36613 | Ordinals are precisely the... |
dford4 36614 | ~ dford3 expressed in prim... |
wopprc 36615 | Unrelated: Wiener pairs t... |
rpnnen3lem 36616 | Lemma for ~ rpnnen3 . (Co... |
rpnnen3 36617 | Dedekind cut injection of ... |
axac10 36618 | Characterization of choice... |
harinf 36619 | The Hartogs number of an i... |
wdom2d2 36620 | Deduction for weak dominan... |
ttac 36621 | Tarski's theorem about cho... |
pw2f1ocnv 36622 | Define a bijection between... |
pw2f1o2 36623 | Define a bijection between... |
pw2f1o2val 36624 | Function value of the ~ pw... |
pw2f1o2val2 36625 | Membership in a mapped set... |
soeq12d 36626 | Equality deduction for tot... |
freq12d 36627 | Equality deduction for fou... |
weeq12d 36628 | Equality deduction for wel... |
limsuc2 36629 | Limit ordinals in the sens... |
wepwsolem 36630 | Transfer an ordering on ch... |
wepwso 36631 | A well-ordering induces a ... |
dnnumch1 36632 | Define an enumeration of a... |
dnnumch2 36633 | Define an enumeration (wea... |
dnnumch3lem 36634 | Value of the ordinal injec... |
dnnumch3 36635 | Define an injection from a... |
dnwech 36636 | Define a well-ordering fro... |
fnwe2val 36637 | Lemma for ~ fnwe2 . Subst... |
fnwe2lem1 36638 | Lemma for ~ fnwe2 . Subst... |
fnwe2lem2 36639 | Lemma for ~ fnwe2 . An el... |
fnwe2lem3 36640 | Lemma for ~ fnwe2 . Trich... |
fnwe2 36641 | A well-ordering can be con... |
aomclem1 36642 | Lemma for ~ dfac11 . This... |
aomclem2 36643 | Lemma for ~ dfac11 . Succ... |
aomclem3 36644 | Lemma for ~ dfac11 . Succ... |
aomclem4 36645 | Lemma for ~ dfac11 . Limi... |
aomclem5 36646 | Lemma for ~ dfac11 . Comb... |
aomclem6 36647 | Lemma for ~ dfac11 . Tran... |
aomclem7 36648 | Lemma for ~ dfac11 . ` ( R... |
aomclem8 36649 | Lemma for ~ dfac11 . Perf... |
dfac11 36650 | The right-hand side of thi... |
kelac1 36651 | Kelley's choice, basic for... |
kelac2lem 36652 | Lemma for ~ kelac2 and ~ d... |
kelac2 36653 | Kelley's choice, most comm... |
dfac21 36654 | Tychonoff's theorem is a c... |
islmodfg 36657 | Property of a finitely gen... |
islssfg 36658 | Property of a finitely gen... |
islssfg2 36659 | Property of a finitely gen... |
islssfgi 36660 | Finitely spanned subspaces... |
fglmod 36661 | Finitely generated left mo... |
lsmfgcl 36662 | The sum of two finitely ge... |
islnm 36665 | Property of being a Noethe... |
islnm2 36666 | Property of being a Noethe... |
lnmlmod 36667 | A Noetherian left module i... |
lnmlssfg 36668 | A submodule of Noetherian ... |
lnmlsslnm 36669 | All submodules of a Noethe... |
lnmfg 36670 | A Noetherian left module i... |
kercvrlsm 36671 | The domain of a linear fun... |
lmhmfgima 36672 | A homomorphism maps finite... |
lnmepi 36673 | Epimorphic images of Noeth... |
lmhmfgsplit 36674 | If the kernel and range of... |
lmhmlnmsplit 36675 | If the kernel and range of... |
lnmlmic 36676 | Noetherian is an invariant... |
pwssplit4 36677 | Splitting for structure po... |
filnm 36678 | Finite left modules are No... |
pwslnmlem0 36679 | Zeroeth powers are Noether... |
pwslnmlem1 36680 | First powers are Noetheria... |
pwslnmlem2 36681 | A sum of powers is Noether... |
pwslnm 36682 | Finite powers of Noetheria... |
unxpwdom3 36683 | Weaker version of ~ unxpwd... |
pwfi2f1o 36684 | The ~ pw2f1o bijection rel... |
pwfi2en 36685 | Finitely supported indicat... |
frlmpwfi 36686 | Formal linear combinations... |
gicabl 36687 | Being Abelian is a group i... |
imasgim 36688 | A relabeling of the elemen... |
basfn 36689 | Functionality of the base ... |
isnumbasgrplem1 36690 | A set which is equipollent... |
harn0 36691 | The Hartogs number of a se... |
numinfctb 36692 | A numerable infinite set c... |
isnumbasgrplem2 36693 | If the (to be thought of a... |
isnumbasgrplem3 36694 | Every nonempty numerable s... |
isnumbasabl 36695 | A set is numerable iff it ... |
isnumbasgrp 36696 | A set is numerable iff it ... |
dfacbasgrp 36697 | A choice equivalent in abs... |
islnr 36700 | Property of a left-Noether... |
lnrring 36701 | Left-Noetherian rings are ... |
lnrlnm 36702 | Left-Noetherian rings have... |
islnr2 36703 | Property of being a left-N... |
islnr3 36704 | Relate left-Noetherian rin... |
lnr2i 36705 | Given an ideal in a left-N... |
lpirlnr 36706 | Left principal ideal rings... |
lnrfrlm 36707 | Finite-dimensional free mo... |
lnrfg 36708 | Finitely-generated modules... |
lnrfgtr 36709 | A submodule of a finitely ... |
hbtlem1 36712 | Value of the leading coeff... |
hbtlem2 36713 | Leading coefficient ideals... |
hbtlem7 36714 | Functionality of leading c... |
hbtlem4 36715 | The leading ideal function... |
hbtlem3 36716 | The leading ideal function... |
hbtlem5 36717 | The leading ideal function... |
hbtlem6 36718 | There is a finite set of p... |
hbt 36719 | The Hilbert Basis Theorem ... |
dgrsub2 36724 | Subtracting two polynomial... |
elmnc 36725 | Property of a monic polyno... |
mncply 36726 | A monic polynomial is a po... |
mnccoe 36727 | A monic polynomial has lea... |
mncn0 36728 | A monic polynomial is not ... |
dgraaval 36733 | Value of the degree functi... |
dgraalem 36734 | Properties of the degree o... |
dgraacl 36735 | Closure of the degree func... |
dgraaf 36736 | Degree function on algebra... |
dgraaub 36737 | Upper bound on degree of a... |
dgraa0p 36738 | A rational polynomial of d... |
mpaaeu 36739 | An algebraic number has ex... |
mpaaval 36740 | Value of the minimal polyn... |
mpaalem 36741 | Properties of the minimal ... |
mpaacl 36742 | Minimal polynomial is a po... |
mpaadgr 36743 | Minimal polynomial has deg... |
mpaaroot 36744 | The minimal polynomial of ... |
mpaamn 36745 | Minimal polynomial is moni... |
itgoval 36750 | Value of the integral-over... |
aaitgo 36751 | The standard algebraic num... |
itgoss 36752 | An integral element is int... |
itgocn 36753 | All integral elements are ... |
cnsrexpcl 36754 | Exponentiation is closed i... |
fsumcnsrcl 36755 | Finite sums are closed in ... |
cnsrplycl 36756 | Polynomials are closed in ... |
rgspnval 36757 | Value of the ring-span of ... |
rgspncl 36758 | The ring-span of a set is ... |
rgspnssid 36759 | The ring-span of a set con... |
rgspnmin 36760 | The ring-span is contained... |
rgspnid 36761 | The span of a subring is i... |
rngunsnply 36762 | Adjoining one element to a... |
flcidc 36763 | Finite linear combinations... |
algstr 36766 | Lemma to shorten proofs of... |
algbase 36767 | The base set of a construc... |
algaddg 36768 | The additive operation of ... |
algmulr 36769 | The multiplicative operati... |
algsca 36770 | The set of scalars of a co... |
algvsca 36771 | The scalar product operati... |
mendval 36772 | Value of the module endomo... |
mendbas 36773 | Base set of the module end... |
mendplusgfval 36774 | Addition in the module end... |
mendplusg 36775 | A specific addition in the... |
mendmulrfval 36776 | Multiplication in the modu... |
mendmulr 36777 | A specific multiplication ... |
mendsca 36778 | The module endomorphism al... |
mendvscafval 36779 | Scalar multiplication in t... |
mendvsca 36780 | A specific scalar multipli... |
mendring 36781 | The module endomorphism al... |
mendlmod 36782 | The module endomorphism al... |
mendassa 36783 | The module endomorphism al... |
issdrg 36786 | Property of a division sub... |
issdrg2 36787 | Property of a division sub... |
acsfn1p 36788 | Construction of a closure ... |
subrgacs 36789 | Closure property of subrin... |
sdrgacs 36790 | Closure property of divisi... |
cntzsdrg 36791 | Centralizers in division r... |
idomrootle 36792 | No element of an integral ... |
idomodle 36793 | Limit on the number of ` N... |
fiuneneq 36794 | Two finite sets of equal s... |
idomsubgmo 36795 | The units of an integral d... |
proot1mul 36796 | Any primitive ` N ` -th ro... |
proot1hash 36797 | If an integral domain has ... |
proot1ex 36798 | The complex field has prim... |
isdomn3 36801 | Nonzero elements form a mu... |
mon1pid 36802 | Monicity and degree of the... |
mon1psubm 36803 | Monic polynomials are a mu... |
deg1mhm 36804 | Homomorphic property of th... |
cytpfn 36805 | Functionality of the cyclo... |
cytpval 36806 | Substitutions for the Nth ... |
fgraphopab 36807 | Express a function as a su... |
fgraphxp 36808 | Express a function as a su... |
hausgraph 36809 | The graph of a continuous ... |
ioounsn 36814 | The closure of the upper e... |
iocunico 36815 | Split an open interval int... |
iocinico 36816 | The intersection of two se... |
iocmbl 36817 | An open-below, closed-abov... |
cnioobibld 36818 | A bounded, continuous func... |
itgpowd 36819 | The integral of a monomial... |
arearect 36820 | The area of a rectangle wh... |
areaquad 36821 | The area of a quadrilatera... |
ifpan123g 36822 | Conjunction of conditional... |
ifpan23 36823 | Conjunction of conditional... |
ifpdfor2 36824 | Define or in terms of cond... |
ifporcor 36825 | Corollary of commutation o... |
ifpdfan2 36826 | Define and with conditiona... |
ifpancor 36827 | Corollary of commutation o... |
ifpdfor 36828 | Define or in terms of cond... |
ifpdfan 36829 | Define and with conditiona... |
ifpbi2 36830 | Equivalence theorem for co... |
ifpbi3 36831 | Equivalence theorem for co... |
ifpim1 36832 | Restate implication as con... |
ifpnot 36833 | Restate negated wff as con... |
ifpid2 36834 | Restate wff as conditional... |
ifpim2 36835 | Restate implication as con... |
ifpbi23 36836 | Equivalence theorem for co... |
ifpdfbi 36837 | Define biimplication as co... |
ifpbiidcor 36838 | Restatement of ~ biid . (... |
ifpbicor 36839 | Corollary of commutation o... |
ifpxorcor 36840 | Corollary of commutation o... |
ifpbi1 36841 | Equivalence theorem for co... |
ifpnot23 36842 | Negation of conditional lo... |
ifpnotnotb 36843 | Factor conditional logic o... |
ifpnorcor 36844 | Corollary of commutation o... |
ifpnancor 36845 | Corollary of commutation o... |
ifpnot23b 36846 | Negation of conditional lo... |
ifpbiidcor2 36847 | Restatement of ~ biid . (... |
ifpnot23c 36848 | Negation of conditional lo... |
ifpnot23d 36849 | Negation of conditional lo... |
ifpdfnan 36850 | Define nand as conditional... |
ifpdfxor 36851 | Define xor as conditional ... |
ifpbi12 36852 | Equivalence theorem for co... |
ifpbi13 36853 | Equivalence theorem for co... |
ifpbi123 36854 | Equivalence theorem for co... |
ifpidg 36855 | Restate wff as conditional... |
ifpid3g 36856 | Restate wff as conditional... |
ifpid2g 36857 | Restate wff as conditional... |
ifpid1g 36858 | Restate wff as conditional... |
ifpim23g 36859 | Restate implication as con... |
ifpim3 36860 | Restate implication as con... |
ifpnim1 36861 | Restate negated implicatio... |
ifpim4 36862 | Restate implication as con... |
ifpnim2 36863 | Restate negated implicatio... |
ifpim123g 36864 | Implication of conditional... |
ifpim1g 36865 | Implication of conditional... |
ifp1bi 36866 | Substitute the first eleme... |
ifpbi1b 36867 | When the first variable is... |
ifpimimb 36868 | Factor conditional logic o... |
ifpororb 36869 | Factor conditional logic o... |
ifpananb 36870 | Factor conditional logic o... |
ifpnannanb 36871 | Factor conditional logic o... |
ifpor123g 36872 | Disjunction of conditional... |
ifpimim 36873 | Consequnce of implication.... |
ifpbibib 36874 | Factor conditional logic o... |
ifpxorxorb 36875 | Factor conditional logic o... |
rp-fakeimass 36876 | A special case where impli... |
rp-fakeanorass 36877 | A special case where a mix... |
rp-fakeoranass 36878 | A special case where a mix... |
rp-fakenanass 36879 | A special case where nand ... |
rp-fakeinunass 36880 | A special case where a mix... |
rp-fakeuninass 36881 | A special case where a mix... |
rp-isfinite5 36882 | A set is said to be finite... |
rp-isfinite6 36883 | A set is said to be finite... |
pwelg 36884 | The powerclass is an eleme... |
pwinfig 36885 | The powerclass of an infin... |
pwinfi2 36886 | The powerclass of an infin... |
pwinfi3 36887 | The powerclass of an infin... |
pwinfi 36888 | The powerclass of an infin... |
fipjust 36889 | A definition of the finite... |
cllem0 36890 | The class of all sets with... |
superficl 36891 | The class of all supersets... |
superuncl 36892 | The class of all supersets... |
ssficl 36893 | The class of all subsets o... |
ssuncl 36894 | The class of all subsets o... |
ssdifcl 36895 | The class of all subsets o... |
sssymdifcl 36896 | The class of all subsets o... |
fiinfi 36897 | If two classes have the fi... |
rababg 36898 | Condition when restricted ... |
elintabg 36899 | Two ways of saying a set i... |
elinintab 36900 | Two ways of saying a set i... |
elmapintrab 36901 | Two ways to say a set is a... |
elinintrab 36902 | Two ways of saying a set i... |
inintabss 36903 | Upper bound on intersectio... |
inintabd 36904 | Value of the intersection ... |
xpinintabd 36905 | Value of the intersection ... |
relintabex 36906 | If the intersection of a c... |
elcnvcnvintab 36907 | Two ways of saying a set i... |
relintab 36908 | Value of the intersection ... |
nonrel 36909 | A non-relation is equal to... |
elnonrel 36910 | Only an ordered pair where... |
cnvssb 36911 | Subclass theorem for conve... |
relnonrel 36912 | The non-relation part of a... |
cnvnonrel 36913 | The converse of the non-re... |
brnonrel 36914 | A non-relation cannot rela... |
dmnonrel 36915 | The domain of the non-rela... |
rnnonrel 36916 | The range of the non-relat... |
resnonrel 36917 | A restriction of the non-r... |
imanonrel 36918 | An image under the non-rel... |
cononrel1 36919 | Composition with the non-r... |
cononrel2 36920 | Composition with the non-r... |
elmapintab 36921 | Two ways to say a set is a... |
fvnonrel 36922 | The function value of any ... |
elinlem 36923 | Two ways to say a set is a... |
elcnvcnvlem 36924 | Two ways to say a set is a... |
cnvcnvintabd 36925 | Value of the relationship ... |
elcnvlem 36926 | Two ways to say a set is a... |
elcnvintab 36927 | Two ways of saying a set i... |
cnvintabd 36928 | Value of the converse of t... |
undmrnresiss 36929 | Two ways of saying the ide... |
reflexg 36930 | Two ways of saying a relat... |
cnvssco 36931 | A condition weaker than re... |
refimssco 36932 | Reflexive relations are su... |
cleq2lem 36933 | Equality implies bijection... |
cbvcllem 36934 | Change of bound variable i... |
intabssd 36935 | When for each element ` y ... |
clublem 36936 | If a superset ` Y ` of ` X... |
clss2lem 36937 | The closure of a property ... |
dfid7 36938 | Definition of identity rel... |
mptrcllem 36939 | Show two versions of a clo... |
cotrintab 36940 | The intersection of a clas... |
rclexi 36941 | The reflexive closure of a... |
rtrclexlem 36942 | Existence of relation impl... |
rtrclex 36943 | The reflexive-transitive c... |
trclubgNEW 36944 | If a relation exists then ... |
trclubNEW 36945 | If a relation exists then ... |
trclexi 36946 | The transitive closure of ... |
rtrclexi 36947 | The reflexive-transitive c... |
clrellem 36948 | When the property ` ps ` h... |
clcnvlem 36949 | When ` A ` , an upper boun... |
cnvtrucl0 36950 | The converse of the trivia... |
cnvrcl0 36951 | The converse of the reflex... |
cnvtrcl0 36952 | The converse of the transi... |
dmtrcl 36953 | The domain of the transiti... |
rntrcl 36954 | The range of the transitiv... |
dfrtrcl5 36955 | Definition of reflexive-tr... |
trcleq2lemRP 36956 | Equality implies bijection... |
al3im 36957 | Version of ~ ax-4 for a ne... |
intima0 36958 | Two ways of expressing the... |
elimaint 36959 | Element of image of inters... |
csbcog 36960 | Distribute proper substitu... |
cnviun 36961 | Converse of indexed union.... |
imaiun1 36962 | The image of an indexed un... |
coiun1 36963 | Composition with an indexe... |
elintima 36964 | Element of intersection of... |
intimass 36965 | The image under the inters... |
intimass2 36966 | The image under the inters... |
intimag 36967 | Requirement for the image ... |
intimasn 36968 | Two ways to express the im... |
intimasn2 36969 | Two ways to express the im... |
ss2iundf 36970 | Subclass theorem for index... |
ss2iundv 36971 | Subclass theorem for index... |
cbviuneq12df 36972 | Rule used to change the bo... |
cbviuneq12dv 36973 | Rule used to change the bo... |
conrel1d 36974 | Deduction about compositio... |
conrel2d 36975 | Deduction about compositio... |
trrelind 36976 | The intersection of transi... |
xpintrreld 36977 | The intersection of a tran... |
restrreld 36978 | The restriction of a trans... |
trrelsuperreldg 36979 | Concrete construction of a... |
trficl 36980 | The class of all transitiv... |
cnvtrrel 36981 | The converse of a transiti... |
trrelsuperrel2dg 36982 | Concrete construction of a... |
dfrcl2 36985 | Reflexive closure of a rel... |
dfrcl3 36986 | Reflexive closure of a rel... |
dfrcl4 36987 | Reflexive closure of a rel... |
relexp2 36988 | A set operated on by the r... |
relexpnul 36989 | If the domain and range of... |
eliunov2 36990 | Membership in the indexed ... |
eltrclrec 36991 | Membership in the indexed ... |
elrtrclrec 36992 | Membership in the indexed ... |
briunov2 36993 | Two classes related by the... |
brmptiunrelexpd 36994 | If two elements are connec... |
fvmptiunrelexplb0d 36995 | If the indexed union range... |
fvmptiunrelexplb0da 36996 | If the indexed union range... |
fvmptiunrelexplb1d 36997 | If the indexed union range... |
brfvid 36998 | If two elements are connec... |
brfvidRP 36999 | If two elements are connec... |
fvilbd 37000 | A set is a subset of its i... |
fvilbdRP 37001 | A set is a subset of its i... |
brfvrcld 37002 | If two elements are connec... |
brfvrcld2 37003 | If two elements are connec... |
fvrcllb0d 37004 | A restriction of the ident... |
fvrcllb0da 37005 | A restriction of the ident... |
fvrcllb1d 37006 | A set is a subset of its i... |
brtrclrec 37007 | Two classes related by the... |
brrtrclrec 37008 | Two classes related by the... |
briunov2uz 37009 | Two classes related by the... |
eliunov2uz 37010 | Membership in the indexed ... |
ov2ssiunov2 37011 | Any particular operator va... |
relexp0eq 37012 | The zeroth power of relati... |
iunrelexp0 37013 | Simplification of zeroth p... |
relexpxpnnidm 37014 | Any positive power of a cr... |
relexpiidm 37015 | Any power of any restricti... |
relexpss1d 37016 | The relational power of a ... |
comptiunov2i 37017 | The composition two indexe... |
corclrcl 37018 | The reflexive closure is i... |
iunrelexpmin1 37019 | The indexed union of relat... |
relexpmulnn 37020 | With exponents limited to ... |
relexpmulg 37021 | With ordered exponents, th... |
trclrelexplem 37022 | The union of relational po... |
iunrelexpmin2 37023 | The indexed union of relat... |
relexp01min 37024 | With exponents limited to ... |
relexp1idm 37025 | Repeated raising a relatio... |
relexp0idm 37026 | Repeated raising a relatio... |
relexp0a 37027 | Absorbtion law for zeroth ... |
relexpxpmin 37028 | The composition of powers ... |
relexpaddss 37029 | The composition of two pow... |
iunrelexpuztr 37030 | The indexed union of relat... |
dftrcl3 37031 | Transitive closure of a re... |
brfvtrcld 37032 | If two elements are connec... |
fvtrcllb1d 37033 | A set is a subset of its i... |
trclfvcom 37034 | The transitive closure of ... |
cnvtrclfv 37035 | The converse of the transi... |
cotrcltrcl 37036 | The transitive closure is ... |
trclimalb2 37037 | Lower bound for image unde... |
brtrclfv2 37038 | Two ways to indicate two e... |
trclfvdecomr 37039 | The transitive closure of ... |
trclfvdecoml 37040 | The transitive closure of ... |
dmtrclfvRP 37041 | The domain of the transiti... |
rntrclfvRP 37042 | The range of the transitiv... |
rntrclfv 37043 | The range of the transitiv... |
dfrtrcl3 37044 | Reflexive-transitive closu... |
brfvrtrcld 37045 | If two elements are connec... |
fvrtrcllb0d 37046 | A restriction of the ident... |
fvrtrcllb0da 37047 | A restriction of the ident... |
fvrtrcllb1d 37048 | A set is a subset of its i... |
dfrtrcl4 37049 | Reflexive-transitive closu... |
corcltrcl 37050 | The composition of the ref... |
cortrcltrcl 37051 | Composition with the refle... |
corclrtrcl 37052 | Composition with the refle... |
cotrclrcl 37053 | The composition of the ref... |
cortrclrcl 37054 | Composition with the refle... |
cotrclrtrcl 37055 | Composition with the refle... |
cortrclrtrcl 37056 | The reflexive-transitive c... |
frege77d 37057 | If the images of both ` { ... |
frege81d 37058 | If the image of ` U ` is a... |
frege83d 37059 | If the image of the union ... |
frege96d 37060 | If ` C ` follows ` A ` in ... |
frege87d 37061 | If the images of both ` { ... |
frege91d 37062 | If ` B ` follows ` A ` in ... |
frege97d 37063 | If ` A ` contains all elem... |
frege98d 37064 | If ` C ` follows ` A ` and... |
frege102d 37065 | If either ` A ` and ` C ` ... |
frege106d 37066 | If ` B ` follows ` A ` in ... |
frege108d 37067 | If either ` A ` and ` C ` ... |
frege109d 37068 | If ` A ` contains all elem... |
frege114d 37069 | If either ` R ` relates ` ... |
frege111d 37070 | If either ` A ` and ` C ` ... |
frege122d 37071 | If ` F ` is a function, ` ... |
frege124d 37072 | If ` F ` is a function, ` ... |
frege126d 37073 | If ` F ` is a function, ` ... |
frege129d 37074 | If ` F ` is a function and... |
frege131d 37075 | If ` F ` is a function and... |
frege133d 37076 | If ` F ` is a function and... |
dfxor4 37077 | Express exclusive-or in te... |
dfxor5 37078 | Express exclusive-or in te... |
df3or2 37079 | Express triple-or in terms... |
df3an2 37080 | Express triple-and in term... |
nev 37081 | Express that not every set... |
dfss6 37082 | Another definition of subc... |
ndisj 37083 | Express that an intersecti... |
0pssin 37084 | Express that an intersecti... |
rp-imass 37085 | If the ` R ` -image of a c... |
dfhe2 37088 | The property of relation `... |
dfhe3 37089 | The property of relation `... |
heeq12 37090 | Equality law for relations... |
heeq1 37091 | Equality law for relations... |
heeq2 37092 | Equality law for relations... |
sbcheg 37093 | Distribute proper substitu... |
hess 37094 | Subclass law for relations... |
xphe 37095 | Any Cartesian product is h... |
0he 37096 | The empty relation is here... |
0heALT 37097 | The empty relation is here... |
he0 37098 | Any relation is hereditary... |
unhe1 37099 | The union of two relations... |
snhesn 37100 | Any singleton is hereditar... |
idhe 37101 | The identity relation is h... |
psshepw 37102 | The relation between sets ... |
sshepw 37103 | The relation between sets ... |
rp-simp2-frege 37106 | Simplification of triple c... |
rp-simp2 37107 | Simplification of triple c... |
rp-frege3g 37108 | Add antecedent to ~ ax-fre... |
frege3 37109 | Add antecedent to ~ ax-fre... |
rp-misc1-frege 37110 | Double-use of ~ ax-frege2 ... |
rp-frege24 37111 | Introducing an embedded an... |
rp-frege4g 37112 | Deduction related to distr... |
frege4 37113 | Special case of closed for... |
frege5 37114 | A closed form of ~ syl . ... |
rp-7frege 37115 | Distribute antecedent and ... |
rp-4frege 37116 | Elimination of a nested an... |
rp-6frege 37117 | Elimination of a nested an... |
rp-8frege 37118 | Eliminate antecedent when ... |
rp-frege25 37119 | Closed form for ~ a1dd . ... |
frege6 37120 | A closed form of ~ imim2d ... |
axfrege8 37121 | Swap antecedents. Identic... |
frege7 37122 | A closed form of ~ syl6 . ... |
frege26 37124 | Identical to ~ idd . Prop... |
frege27 37125 | We cannot (at the same tim... |
frege9 37126 | Closed form of ~ syl with ... |
frege12 37127 | A closed form of ~ com23 .... |
frege11 37128 | Elimination of a nested an... |
frege24 37129 | Closed form for ~ a1d . D... |
frege16 37130 | A closed form of ~ com34 .... |
frege25 37131 | Closed form for ~ a1dd . ... |
frege18 37132 | Closed form of a syllogism... |
frege22 37133 | A closed form of ~ com45 .... |
frege10 37134 | Result commuting anteceden... |
frege17 37135 | A closed form of ~ com3l .... |
frege13 37136 | A closed form of ~ com3r .... |
frege14 37137 | Closed form of a deduction... |
frege19 37138 | A closed form of ~ syl6 . ... |
frege23 37139 | Syllogism followed by rota... |
frege15 37140 | A closed form of ~ com4r .... |
frege21 37141 | Replace antecedent in ante... |
frege20 37142 | A closed form of ~ syl8 . ... |
axfrege28 37143 | Contraposition. Identical... |
frege29 37145 | Closed form of ~ con3d . ... |
frege30 37146 | Commuted, closed form of ~... |
axfrege31 37147 | Identical to ~ notnotr . ... |
frege32 37149 | Deduce ~ con1 from ~ con3 ... |
frege33 37150 | If ` ph ` or ` ps ` takes ... |
frege34 37151 | If as a conseqence of the ... |
frege35 37152 | Commuted, closed form of ~... |
frege36 37153 | The case in which ` ps ` i... |
frege37 37154 | If ` ch ` is a necessary c... |
frege38 37155 | Identical to ~ pm2.21 . P... |
frege39 37156 | Syllogism between ~ pm2.18... |
frege40 37157 | Anything implies ~ pm2.18 ... |
axfrege41 37158 | Identical to ~ notnot . A... |
frege42 37160 | Not not ~ id . Propositio... |
frege43 37161 | If there is a choice only ... |
frege44 37162 | Similar to a commuted ~ pm... |
frege45 37163 | Deduce ~ pm2.6 from ~ con1... |
frege46 37164 | If ` ps ` holds when ` ph ... |
frege47 37165 | Deduce consequence follows... |
frege48 37166 | Closed form of syllogism w... |
frege49 37167 | Closed form of deduction w... |
frege50 37168 | Closed form of ~ jaoi . P... |
frege51 37169 | Compare with ~ jaod . Pro... |
axfrege52a 37170 | Justification for ~ ax-fre... |
frege52aid 37172 | The case when the content ... |
frege53aid 37173 | Specialization of ~ frege5... |
frege53a 37174 | Lemma for ~ frege55a . Pr... |
axfrege54a 37175 | Justification for ~ ax-fre... |
frege54cor0a 37177 | Synonym for logical equiva... |
frege54cor1a 37178 | Reflexive equality. (Cont... |
frege55aid 37179 | Lemma for ~ frege57aid . ... |
frege55lem1a 37180 | Necessary deduction regard... |
frege55lem2a 37181 | Core proof of Proposition ... |
frege55a 37182 | Proposition 55 of [Frege18... |
frege55cor1a 37183 | Proposition 55 of [Frege18... |
frege56aid 37184 | Lemma for ~ frege57aid . ... |
frege56a 37185 | Proposition 56 of [Frege18... |
frege57aid 37186 | This is the all imporant f... |
frege57a 37187 | Analogue of ~ frege57aid .... |
axfrege58a 37188 | Identical to ~ anifp . Ju... |
frege58acor 37190 | Lemma for ~ frege59a . (C... |
frege59a 37191 | A kind of Aristotelian inf... |
frege60a 37192 | Swap antecedents of ~ ax-f... |
frege61a 37193 | Lemma for ~ frege65a . Pr... |
frege62a 37194 | A kind of Aristotelian inf... |
frege63a 37195 | Proposition 63 of [Frege18... |
frege64a 37196 | Lemma for ~ frege65a . Pr... |
frege65a 37197 | A kind of Aristotelian inf... |
frege66a 37198 | Swap antecedents of ~ freg... |
frege67a 37199 | Lemma for ~ frege68a . Pr... |
frege68a 37200 | Combination of applying a ... |
axfrege52c 37201 | Justification for ~ ax-fre... |
frege52b 37203 | The case when the content ... |
frege53b 37204 | Lemma for frege102 (via ~ ... |
axfrege54c 37205 | Reflexive equality of clas... |
frege54b 37207 | Reflexive equality of sets... |
frege54cor1b 37208 | Reflexive equality. (Cont... |
frege55lem1b 37209 | Necessary deduction regard... |
frege55lem2b 37210 | Lemma for ~ frege55b . Co... |
frege55b 37211 | Lemma for ~ frege57b . Pr... |
frege56b 37212 | Lemma for ~ frege57b . Pr... |
frege57b 37213 | Analogue of ~ frege57aid .... |
axfrege58b 37214 | If ` A. x ph ` is affirmed... |
frege58bid 37216 | If ` A. x ph ` is affirmed... |
frege58bcor 37217 | Lemma for ~ frege59b . (C... |
frege59b 37218 | A kind of Aristotelian inf... |
frege60b 37219 | Swap antecedents of ~ ax-f... |
frege61b 37220 | Lemma for ~ frege65b . Pr... |
frege62b 37221 | A kind of Aristotelian inf... |
frege63b 37222 | Lemma for ~ frege91 . Pro... |
frege64b 37223 | Lemma for ~ frege65b . Pr... |
frege65b 37224 | A kind of Aristotelian inf... |
frege66b 37225 | Swap antecedents of ~ freg... |
frege67b 37226 | Lemma for ~ frege68b . Pr... |
frege68b 37227 | Combination of applying a ... |
frege53c 37228 | Proposition 53 of [Frege18... |
frege54cor1c 37229 | Reflexive equality. (Cont... |
frege55lem1c 37230 | Necessary deduction regard... |
frege55lem2c 37231 | Core proof of Proposition ... |
frege55c 37232 | Proposition 55 of [Frege18... |
frege56c 37233 | Lemma for ~ frege57c . Pr... |
frege57c 37234 | Swap order of implication ... |
frege58c 37235 | Principle related to ~ sp ... |
frege59c 37236 | A kind of Aristotelian inf... |
frege60c 37237 | Swap antecedents of ~ freg... |
frege61c 37238 | Lemma for ~ frege65c . Pr... |
frege62c 37239 | A kind of Aristotelian inf... |
frege63c 37240 | Analogue of ~ frege63b . ... |
frege64c 37241 | Lemma for ~ frege65c . Pr... |
frege65c 37242 | A kind of Aristotelian inf... |
frege66c 37243 | Swap antecedents of ~ freg... |
frege67c 37244 | Lemma for ~ frege68c . Pr... |
frege68c 37245 | Combination of applying a ... |
dffrege69 37246 | If from the proposition th... |
frege70 37247 | Lemma for ~ frege72 . Pro... |
frege71 37248 | Lemma for ~ frege72 . Pro... |
frege72 37249 | If property ` A ` is hered... |
frege73 37250 | Lemma for ~ frege87 . Pro... |
frege74 37251 | If ` X ` has a property ` ... |
frege75 37252 | If from the proposition th... |
dffrege76 37253 | If from the two propositio... |
frege77 37254 | If ` Y ` follows ` X ` in ... |
frege78 37255 | Commuted form of of ~ freg... |
frege79 37256 | Distributed form of ~ freg... |
frege80 37257 | Add additional condition t... |
frege81 37258 | If ` X ` has a property ` ... |
frege82 37259 | Closed-form deduction base... |
frege83 37260 | Apply commuted form of ~ f... |
frege84 37261 | Commuted form of ~ frege81... |
frege85 37262 | Commuted form of ~ frege77... |
frege86 37263 | Conclusion about element o... |
frege87 37264 | If ` Z ` is a result of an... |
frege88 37265 | Commuted form of ~ frege87... |
frege89 37266 | One direction of ~ dffrege... |
frege90 37267 | Add antecedent to ~ frege8... |
frege91 37268 | Every result of an applica... |
frege92 37269 | Inference from ~ frege91 .... |
frege93 37270 | Necessary condition for tw... |
frege94 37271 | Looking one past a pair re... |
frege95 37272 | Looking one past a pair re... |
frege96 37273 | Every result of an applica... |
frege97 37274 | The property of following ... |
frege98 37275 | If ` Y ` follows ` X ` and... |
dffrege99 37276 | If ` Z ` is identical with... |
frege100 37277 | One direction of ~ dffrege... |
frege101 37278 | Lemma for ~ frege102 . Pr... |
frege102 37279 | If ` Z ` belongs to the ` ... |
frege103 37280 | Proposition 103 of [Frege1... |
frege104 37281 | Proposition 104 of [Frege1... |
frege105 37282 | Proposition 105 of [Frege1... |
frege106 37283 | Whatever follows ` X ` in ... |
frege107 37284 | Proposition 107 of [Frege1... |
frege108 37285 | If ` Y ` belongs to the ` ... |
frege109 37286 | The property of belonging ... |
frege110 37287 | Proposition 110 of [Frege1... |
frege111 37288 | If ` Y ` belongs to the ` ... |
frege112 37289 | Identity implies belonging... |
frege113 37290 | Proposition 113 of [Frege1... |
frege114 37291 | If ` X ` belongs to the ` ... |
dffrege115 37292 | If from the the circumstan... |
frege116 37293 | One direction of ~ dffrege... |
frege117 37294 | Lemma for ~ frege118 . Pr... |
frege118 37295 | Simplified application of ... |
frege119 37296 | Lemma for ~ frege120 . Pr... |
frege120 37297 | Simplified application of ... |
frege121 37298 | Lemma for ~ frege122 . Pr... |
frege122 37299 | If ` X ` is a result of an... |
frege123 37300 | Lemma for ~ frege124 . Pr... |
frege124 37301 | If ` X ` is a result of an... |
frege125 37302 | Lemma for ~ frege126 . Pr... |
frege126 37303 | If ` M ` follows ` Y ` in ... |
frege127 37304 | Communte antecedents of ~ ... |
frege128 37305 | Lemma for ~ frege129 . Pr... |
frege129 37306 | If the procedure ` R ` is ... |
frege130 37307 | Lemma for ~ frege131 . Pr... |
frege131 37308 | If the procedure ` R ` is ... |
frege132 37309 | Lemma for ~ frege133 . Pr... |
frege133 37310 | If the procedure ` R ` is ... |
enrelmap 37311 | The set of all possible re... |
enrelmapr 37312 | The set of all possible re... |
enmappw 37313 | The set of all mappings fr... |
enmappwid 37314 | The set of all mappings fr... |
rfovd 37315 | Value of the operator, ` (... |
rfovfvd 37316 | Value of the operator, ` (... |
rfovfvfvd 37317 | Value of the operator, ` (... |
rfovcnvf1od 37318 | Properties of the operator... |
rfovcnvd 37319 | Value of the converse of t... |
rfovf1od 37320 | The value of the operator,... |
rfovcnvfvd 37321 | Value of the converse of t... |
fsovd 37322 | Value of the operator, ` (... |
fsovrfovd 37323 | The operator which gives a... |
fsovfvd 37324 | Value of the operator, ` (... |
fsovfvfvd 37325 | Value of the operator, ` (... |
fsovfd 37326 | The operator, ` ( A O B ) ... |
fsovcnvlem 37327 | The ` O ` operator, which ... |
fsovcnvd 37328 | The value of the converse ... |
fsovcnvfvd 37329 | The value of the converse ... |
fsovf1od 37330 | The value of ` ( A O B ) `... |
dssmapfvd 37331 | Value of the duality opera... |
dssmapfv2d 37332 | Value of the duality opera... |
dssmapfv3d 37333 | Value of the duality opera... |
dssmapnvod 37334 | For any base set ` B ` the... |
dssmapf1od 37335 | For any base set ` B ` the... |
dssmap2d 37336 | For any base set ` B ` the... |
sscon34b 37337 | Relative complementation r... |
rcompleq 37338 | Two subclasses are equal i... |
or3or 37339 | Decompose disjunction into... |
andi3or 37340 | Distribute over triple dis... |
uneqsn 37341 | If a union of classes is e... |
df3o2 37342 | Ordinal 3 is the triplet c... |
df3o3 37343 | Ordinal 3 , fully expanded... |
brfvimex 37344 | If a binary relation holds... |
brovmptimex 37345 | If a binary relation holds... |
brovmptimex1 37346 | If a binary relation holds... |
brovmptimex2 37347 | If a binary relation holds... |
brcoffn 37348 | Conditions allowing the de... |
brcofffn 37349 | Conditions allowing the de... |
brco2f1o 37350 | Conditions allowing the de... |
brco3f1o 37351 | Conditions allowing the de... |
ntrclsbex 37352 | If (pseudo-)interior and (... |
ntrclsrcomplex 37353 | The relative complement of... |
neik0imk0p 37354 | Kuratowski's K0 axiom impl... |
ntrk2imkb 37355 | If an interior function is... |
ntrkbimka 37356 | If the interiors of disjoi... |
ntrk0kbimka 37357 | If the interiors of disjoi... |
clsk3nimkb 37358 | If the base set is not emp... |
clsk1indlem0 37359 | The ansatz closure functio... |
clsk1indlem2 37360 | The ansatz closure functio... |
clsk1indlem3 37361 | The ansatz closure functio... |
clsk1indlem4 37362 | The ansatz closure functio... |
clsk1indlem1 37363 | The ansatz closure functio... |
clsk1independent 37364 | For generalized closure fu... |
neik0pk1imk0 37365 | Kuratowski's K0' and K1 ax... |
isotone1 37366 | Two different ways to say ... |
isotone2 37367 | Two different ways to say ... |
ntrk1k3eqk13 37368 | An interior function is bo... |
ntrclsf1o 37369 | If (pseudo-)interior and (... |
ntrclsnvobr 37370 | If (pseudo-)interior and (... |
ntrclsiex 37371 | If (pseudo-)interior and (... |
ntrclskex 37372 | If (pseudo-)interior and (... |
ntrclsfv1 37373 | If (pseudo-)interior and (... |
ntrclsfv2 37374 | If (pseudo-)interior and (... |
ntrclselnel1 37375 | If (pseudo-)interior and (... |
ntrclselnel2 37376 | If (pseudo-)interior and (... |
ntrclsfv 37377 | The value of the interior ... |
ntrclsfveq1 37378 | If interior and closure fu... |
ntrclsfveq2 37379 | If interior and closure fu... |
ntrclsfveq 37380 | If interior and closure fu... |
ntrclsss 37381 | If interior and closure fu... |
ntrclsneine0lem 37382 | If (pseudo-)interior and (... |
ntrclsneine0 37383 | If (pseudo-)interior and (... |
ntrclscls00 37384 | If (pseudo-)interior and (... |
ntrclsiso 37385 | If (pseudo-)interior and (... |
ntrclsk2 37386 | An interior function is co... |
ntrclskb 37387 | The interiors of disjoint ... |
ntrclsk3 37388 | The intersection of interi... |
ntrclsk13 37389 | The interior of the inters... |
ntrclsk4 37390 | Idempotence of the interio... |
ntrneibex 37391 | If (pseudo-)interior and (... |
ntrneircomplex 37392 | The relative complement of... |
ntrneif1o 37393 | If (pseudo-)interior and (... |
ntrneiiex 37394 | If (pseudo-)interior and (... |
ntrneinex 37395 | If (pseudo-)interior and (... |
ntrneicnv 37396 | If (pseudo-)interior and (... |
ntrneifv1 37397 | If (pseudo-)interior and (... |
ntrneifv2 37398 | If (pseudo-)interior and (... |
ntrneiel 37399 | If (pseudo-)interior and (... |
ntrneifv3 37400 | The value of the neighbors... |
ntrneineine0lem 37401 | If (pseudo-)interior and (... |
ntrneineine1lem 37402 | If (pseudo-)interior and (... |
ntrneifv4 37403 | The value of the interior ... |
ntrneiel2 37404 | Membership in iterated int... |
ntrneineine0 37405 | If (pseudo-)interior and (... |
ntrneineine1 37406 | If (pseudo-)interior and (... |
ntrneicls00 37407 | If (pseudo-)interior and (... |
ntrneicls11 37408 | If (pseudo-)interior and (... |
ntrneiiso 37409 | If (pseudo-)interior and (... |
ntrneik2 37410 | An interior function is co... |
ntrneix2 37411 | An interior (closure) func... |
ntrneikb 37412 | The interiors of disjoint ... |
ntrneixb 37413 | The interiors (closures) o... |
ntrneik3 37414 | The intersection of interi... |
ntrneix3 37415 | The closure of the union o... |
ntrneik13 37416 | The interior of the inters... |
ntrneix13 37417 | The closure of the union o... |
ntrneik4w 37418 | Idempotence of the interio... |
ntrneik4 37419 | Idempotence of the interio... |
clsneibex 37420 | If (pseudo-)closure and (p... |
clsneircomplex 37421 | The relative complement of... |
clsneif1o 37422 | If a (pseudo-)closure func... |
clsneicnv 37423 | If a (pseudo-)closure func... |
clsneikex 37424 | If closure and neighborhoo... |
clsneinex 37425 | If closure and neighborhoo... |
clsneiel1 37426 | If a (pseudo-)closure func... |
clsneiel2 37427 | If a (pseudo-)closure func... |
clsneifv3 37428 | Value of the neighborhoods... |
clsneifv4 37429 | Value of the the closure (... |
neicvgbex 37430 | If (pseudo-)neighborhood a... |
neicvgrcomplex 37431 | The relative complement of... |
neicvgf1o 37432 | If neighborhood and conver... |
neicvgnvo 37433 | If neighborhood and conver... |
neicvgnvor 37434 | If neighborhood and conver... |
neicvgmex 37435 | If the neighborhoods and c... |
neicvgnex 37436 | If the neighborhoods and c... |
neicvgel1 37437 | A subset being an element ... |
neicvgel2 37438 | The complement of a subset... |
neicvgfv 37439 | The value of the neighborh... |
ntrrn 37440 | The range of the interior ... |
ntrf 37441 | The interior function of a... |
ntrf2 37442 | The interior function is a... |
ntrelmap 37443 | The interior function is a... |
clsf2 37444 | The closure function is a ... |
clselmap 37445 | The closure function is a ... |
dssmapntrcls 37446 | The interior and closure o... |
dssmapclsntr 37447 | The closure and interior o... |
gneispa 37448 | Each point ` p ` of the ne... |
gneispb 37449 | Given a neighborhood ` N `... |
gneispace2 37450 | The predicate that ` F ` i... |
gneispace3 37451 | The predicate that ` F ` i... |
gneispace 37452 | The predicate that ` F ` i... |
gneispacef 37453 | A generic neighborhood spa... |
gneispacef2 37454 | A generic neighborhood spa... |
gneispacefun 37455 | A generic neighborhood spa... |
gneispacern 37456 | A generic neighborhood spa... |
gneispacern2 37457 | A generic neighborhood spa... |
gneispace0nelrn 37458 | A generic neighborhood spa... |
gneispace0nelrn2 37459 | A generic neighborhood spa... |
gneispace0nelrn3 37460 | A generic neighborhood spa... |
gneispaceel 37461 | Every neighborhood of a po... |
gneispaceel2 37462 | Every neighborhood of a po... |
gneispacess 37463 | All supersets of a neighbo... |
gneispacess2 37464 | All supersets of a neighbo... |
k0004lem1 37465 | Application of ~ ssin to r... |
k0004lem2 37466 | A mapping with a particula... |
k0004lem3 37467 | When the value of a mappin... |
k0004val 37468 | The topological simplex of... |
k0004ss1 37469 | The topological simplex of... |
k0004ss2 37470 | The topological simplex of... |
k0004ss3 37471 | The topological simplex of... |
k0004val0 37472 | The topological simplex of... |
inductionexd 37473 | Simple induction example. ... |
wwlemuld 37474 | Natural deduction form of ... |
leeq1d 37475 | Specialization of ~ breq1d... |
leeq2d 37476 | Specialization of ~ breq2d... |
absmulrposd 37477 | Specialization of absmuld ... |
imadisjld 37478 | Natural dduction form of o... |
imadisjlnd 37479 | Natural deduction form of ... |
wnefimgd 37480 | The image of a mapping fro... |
fco2d 37481 | Natural deduction form of ... |
suprubd 37482 | Natural deduction form of ... |
suprcld 37483 | Natural deduction form of ... |
fvco3d 37484 | Natural deduction form of ... |
wfximgfd 37485 | The value of a function on... |
extoimad 37486 | If |f(x)| <= C for all x t... |
imo72b2lem0 37487 | Lemma for ~ imo72b2 . (Co... |
suprleubrd 37488 | Natural deduction form of ... |
imo72b2lem2 37489 | Lemma for ~ imo72b2 . (Co... |
syldbl2 37490 | Stacked hypotheseis implie... |
funfvima2d 37491 | A function's value in a pr... |
suprlubrd 37492 | Natural deduction form of ... |
imo72b2lem1 37493 | Lemma for ~ imo72b2 . (Co... |
lemuldiv3d 37494 | 'Less than or equal to' re... |
lemuldiv4d 37495 | 'Less than or equal to' re... |
rspcdvinvd 37496 | If something is true for a... |
imo72b2 37497 | IMO 1972 B2. (14th Intern... |
int-addcomd 37498 | AdditionCommutativity gene... |
int-addassocd 37499 | AdditionAssociativity gene... |
int-addsimpd 37500 | AdditionSimplification gen... |
int-mulcomd 37501 | MultiplicationCommutativit... |
int-mulassocd 37502 | MultiplicationAssociativit... |
int-mulsimpd 37503 | MultiplicationSimplificati... |
int-leftdistd 37504 | AdditionMultiplicationLeft... |
int-rightdistd 37505 | AdditionMultiplicationRigh... |
int-sqdefd 37506 | SquareDefinition generator... |
int-mul11d 37507 | First MultiplicationOne ge... |
int-mul12d 37508 | Second MultiplicationOne g... |
int-add01d 37509 | First AdditionZero generat... |
int-add02d 37510 | Second AdditionZero genera... |
int-sqgeq0d 37511 | SquareGEQZero generator ru... |
int-eqprincd 37512 | PrincipleOfEquality genera... |
int-eqtransd 37513 | EqualityTransitivity gener... |
int-eqmvtd 37514 | EquMoveTerm generator rule... |
int-eqineqd 37515 | EquivalenceImpliesDoubleIn... |
int-ineqmvtd 37516 | IneqMoveTerm generator rul... |
int-ineq1stprincd 37517 | FirstPrincipleOfInequality... |
int-ineq2ndprincd 37518 | SecondPrincipleOfInequalit... |
int-ineqtransd 37519 | InequalityTransitivity gen... |
unitadd 37520 | Theorem used in conjunctio... |
gsumws3 37521 | Valuation of a length 3 wo... |
gsumws4 37522 | Valuation of a length 4 wo... |
amgm2d 37523 | Arithmetic-geometric mean ... |
amgm3d 37524 | Arithmetic-geometric mean ... |
amgm4d 37525 | Arithmetic-geometric mean ... |
nanorxor 37526 | 'nand' is equivalent to th... |
undisjrab 37527 | Union of two disjoint rest... |
iso0 37528 | The empty set is an ` R , ... |
ssrecnpr 37529 | ` RR ` is a subset of both... |
seff 37530 | Let set ` S ` be the real ... |
sblpnf 37531 | The infinity ball in the a... |
prmunb2 37532 | The primes are unbounded. ... |
dvgrat 37533 | Ratio test for divergence ... |
cvgdvgrat 37534 | Ratio test for convergence... |
radcnvrat 37535 | Let ` L ` be the limit, if... |
reldvds 37536 | The divides relation is in... |
nznngen 37537 | All positive integers in t... |
nzss 37538 | The set of multiples of _m... |
nzin 37539 | The intersection of the se... |
nzprmdif 37540 | Subtract one prime's multi... |
hashnzfz 37541 | Special case of ~ hashdvds... |
hashnzfz2 37542 | Special case of ~ hashnzfz... |
hashnzfzclim 37543 | As the upper bound ` K ` o... |
caofcan 37544 | Transfer a cancellation la... |
ofsubid 37545 | Function analogue of ~ sub... |
ofmul12 37546 | Function analogue of ~ mul... |
ofdivrec 37547 | Function analogue of ~ div... |
ofdivcan4 37548 | Function analogue of ~ div... |
ofdivdiv2 37549 | Function analogue of ~ div... |
lhe4.4ex1a 37550 | Example of the Fundamental... |
dvsconst 37551 | Derivative of a constant f... |
dvsid 37552 | Derivative of the identity... |
dvsef 37553 | Derivative of the exponent... |
expgrowthi 37554 | Exponential growth and dec... |
dvconstbi 37555 | The derivative of a functi... |
expgrowth 37556 | Exponential growth and dec... |
bccval 37559 | Value of the generalized b... |
bcccl 37560 | Closure of the generalized... |
bcc0 37561 | The generalized binomial c... |
bccp1k 37562 | Generalized binomial coeff... |
bccm1k 37563 | Generalized binomial coeff... |
bccn0 37564 | Generalized binomial coeff... |
bccn1 37565 | Generalized binomial coeff... |
bccbc 37566 | The binomial coefficient a... |
uzmptshftfval 37567 | When ` F ` is a maps-to fu... |
dvradcnv2 37568 | The radius of convergence ... |
binomcxplemwb 37569 | Lemma for ~ binomcxp . Th... |
binomcxplemnn0 37570 | Lemma for ~ binomcxp . Wh... |
binomcxplemrat 37571 | Lemma for ~ binomcxp . As... |
binomcxplemfrat 37572 | Lemma for ~ binomcxp . ~ b... |
binomcxplemradcnv 37573 | Lemma for ~ binomcxp . By... |
binomcxplemdvbinom 37574 | Lemma for ~ binomcxp . By... |
binomcxplemcvg 37575 | Lemma for ~ binomcxp . Th... |
binomcxplemdvsum 37576 | Lemma for ~ binomcxp . Th... |
binomcxplemnotnn0 37577 | Lemma for ~ binomcxp . Wh... |
binomcxp 37578 | Generalize the binomial th... |
pm10.12 37579 | Theorem *10.12 in [Whitehe... |
pm10.14 37580 | Theorem *10.14 in [Whitehe... |
pm10.251 37581 | Theorem *10.251 in [Whiteh... |
pm10.252 37582 | Theorem *10.252 in [Whiteh... |
pm10.253 37583 | Theorem *10.253 in [Whiteh... |
albitr 37584 | Theorem *10.301 in [Whiteh... |
pm10.42 37585 | Theorem *10.42 in [Whitehe... |
pm10.52 37586 | Theorem *10.52 in [Whitehe... |
pm10.53 37587 | Theorem *10.53 in [Whitehe... |
pm10.541 37588 | Theorem *10.541 in [Whiteh... |
pm10.542 37589 | Theorem *10.542 in [Whiteh... |
pm10.55 37590 | Theorem *10.55 in [Whitehe... |
pm10.56 37591 | Theorem *10.56 in [Whitehe... |
pm10.57 37592 | Theorem *10.57 in [Whitehe... |
2alanimi 37593 | Removes two universal quan... |
2al2imi 37594 | Removes two universal quan... |
pm11.11 37595 | Theorem *11.11 in [Whitehe... |
pm11.12 37596 | Theorem *11.12 in [Whitehe... |
19.21vv 37597 | Compare Theorem *11.3 in [... |
2alim 37598 | Theorem *11.32 in [Whitehe... |
2albi 37599 | Theorem *11.33 in [Whitehe... |
2exim 37600 | Theorem *11.34 in [Whitehe... |
2exbi 37601 | Theorem *11.341 in [Whiteh... |
spsbce-2 37602 | Theorem *11.36 in [Whitehe... |
19.33-2 37603 | Theorem *11.421 in [Whiteh... |
19.36vv 37604 | Theorem *11.43 in [Whitehe... |
19.31vv 37605 | Theorem *11.44 in [Whitehe... |
19.37vv 37606 | Theorem *11.46 in [Whitehe... |
19.28vv 37607 | Theorem *11.47 in [Whitehe... |
pm11.52 37608 | Theorem *11.52 in [Whitehe... |
2exanali 37609 | Theorem *11.521 in [Whiteh... |
aaanv 37610 | Theorem *11.56 in [Whitehe... |
pm11.57 37611 | Theorem *11.57 in [Whitehe... |
pm11.58 37612 | Theorem *11.58 in [Whitehe... |
pm11.59 37613 | Theorem *11.59 in [Whitehe... |
pm11.6 37614 | Theorem *11.6 in [Whitehea... |
pm11.61 37615 | Theorem *11.61 in [Whitehe... |
pm11.62 37616 | Theorem *11.62 in [Whitehe... |
pm11.63 37617 | Theorem *11.63 in [Whitehe... |
pm11.7 37618 | Theorem *11.7 in [Whitehea... |
pm11.71 37619 | Theorem *11.71 in [Whitehe... |
sbeqal1 37620 | If ` x = y ` always implie... |
sbeqal1i 37621 | Suppose you know ` x = y `... |
sbeqal2i 37622 | If ` x = y ` implies ` x =... |
sbeqalbi 37623 | When both ` x ` and ` z ` ... |
axc5c4c711 37624 | Proof of a theorem that ca... |
axc5c4c711toc5 37625 | Rederivation of ~ sp from ... |
axc5c4c711toc4 37626 | Rederivation of ~ axc4 fro... |
axc5c4c711toc7 37627 | Rederivation of ~ axc7 fro... |
axc5c4c711to11 37628 | Rederivation of ~ ax-11 fr... |
axc11next 37629 | This theorem shows that, g... |
pm13.13a 37630 | One result of theorem *13.... |
pm13.13b 37631 | Theorem *13.13 in [Whitehe... |
pm13.14 37632 | Theorem *13.14 in [Whitehe... |
pm13.192 37633 | Theorem *13.192 in [Whiteh... |
pm13.193 37634 | Theorem *13.193 in [Whiteh... |
pm13.194 37635 | Theorem *13.194 in [Whiteh... |
pm13.195 37636 | Theorem *13.195 in [Whiteh... |
pm13.196a 37637 | Theorem *13.196 in [Whiteh... |
2sbc6g 37638 | Theorem *13.21 in [Whitehe... |
2sbc5g 37639 | Theorem *13.22 in [Whitehe... |
iotain 37640 | Equivalence between two di... |
iotaexeu 37641 | The iota class exists. Th... |
iotasbc 37642 | Definition *14.01 in [Whit... |
iotasbc2 37643 | Theorem *14.111 in [Whiteh... |
pm14.12 37644 | Theorem *14.12 in [Whitehe... |
pm14.122a 37645 | Theorem *14.122 in [Whiteh... |
pm14.122b 37646 | Theorem *14.122 in [Whiteh... |
pm14.122c 37647 | Theorem *14.122 in [Whiteh... |
pm14.123a 37648 | Theorem *14.123 in [Whiteh... |
pm14.123b 37649 | Theorem *14.123 in [Whiteh... |
pm14.123c 37650 | Theorem *14.123 in [Whiteh... |
pm14.18 37651 | Theorem *14.18 in [Whitehe... |
iotaequ 37652 | Theorem *14.2 in [Whitehea... |
iotavalb 37653 | Theorem *14.202 in [Whiteh... |
iotasbc5 37654 | Theorem *14.205 in [Whiteh... |
pm14.24 37655 | Theorem *14.24 in [Whitehe... |
iotavalsb 37656 | Theorem *14.242 in [Whiteh... |
sbiota1 37657 | Theorem *14.25 in [Whitehe... |
sbaniota 37658 | Theorem *14.26 in [Whitehe... |
eubi 37659 | Theorem *14.271 in [Whiteh... |
iotasbcq 37660 | Theorem *14.272 in [Whiteh... |
elnev 37661 | Any set that contains one ... |
rusbcALT 37662 | A version of Russell's par... |
compel 37663 | Equivalence between two wa... |
compeq 37664 | Equality between two ways ... |
compne 37665 | The complement of ` A ` is... |
compab 37666 | Two ways of saying "the co... |
conss34OLD 37667 | Obsolete proof of ~ compls... |
conss2 37668 | Contrapositive law for sub... |
conss1 37669 | Contrapositive law for sub... |
ralbidar 37670 | More general form of ~ ral... |
rexbidar 37671 | More general form of ~ rex... |
dropab1 37672 | Theorem to aid use of the ... |
dropab2 37673 | Theorem to aid use of the ... |
ipo0 37674 | If the identity relation p... |
ifr0 37675 | A class that is founded by... |
ordpss 37676 | ~ ordelpss with an anteced... |
fvsb 37677 | Explicit substitution of a... |
fveqsb 37678 | Implicit substitution of a... |
xpexb 37679 | A Cartesian product exists... |
trelpss 37680 | An element of a transitive... |
addcomgi 37681 | Generalization of commutat... |
addrval 37691 | Value of the operation of ... |
subrval 37692 | Value of the operation of ... |
mulvval 37693 | Value of the operation of ... |
addrfv 37694 | Vector addition at a value... |
subrfv 37695 | Vector subtraction at a va... |
mulvfv 37696 | Scalar multiplication at a... |
addrfn 37697 | Vector addition produces a... |
subrfn 37698 | Vector subtraction produce... |
mulvfn 37699 | Scalar multiplication prod... |
addrcom 37700 | Vector addition is commuta... |
idiALT 37704 | Placeholder for ~ idi . T... |
exbir 37705 | Exportation implication al... |
3impexpbicom 37706 | Version of ~ 3impexp where... |
3impexpbicomi 37707 | Inference associated with ... |
bi1imp 37708 | Importation inference simi... |
bi2imp 37709 | Importation inference simi... |
bi3impb 37710 | Similar to ~ 3impb with im... |
bi3impa 37711 | Similar to ~ 3impa with im... |
bi23impib 37712 | ~ 3impib with the inner im... |
bi13impib 37713 | ~ 3impib with the outer im... |
bi123impib 37714 | ~ 3impib with the implicat... |
bi13impia 37715 | ~ 3impia with the outer im... |
bi123impia 37716 | ~ 3impia with the implicat... |
bi33imp12 37717 | ~ 3imp with innermost impl... |
bi23imp13 37718 | ~ 3imp with middle implica... |
bi13imp23 37719 | ~ 3imp with outermost impl... |
bi13imp2 37720 | Similar to ~ 3imp except t... |
bi12imp3 37721 | Similar to ~ 3imp except a... |
bi23imp1 37722 | Similar to ~ 3imp except a... |
bi123imp0 37723 | Similar to ~ 3imp except a... |
4animp1 37724 | A single hypothesis unific... |
4an31 37725 | A rearrangement of conjunc... |
4an4132 37726 | A rearrangement of conjunc... |
expcomdg 37727 | Biconditional form of ~ ex... |
iidn3 37728 | ~ idn3 without virtual ded... |
ee222 37729 | ~ e222 without virtual ded... |
ee3bir 37730 | Right-biconditional form o... |
ee13 37731 | ~ e13 without virtual dedu... |
ee121 37732 | ~ e121 without virtual ded... |
ee122 37733 | ~ e122 without virtual ded... |
ee333 37734 | ~ e333 without virtual ded... |
ee323 37735 | ~ e323 without virtual ded... |
3ornot23 37736 | If the second and third di... |
orbi1r 37737 | ~ orbi1 with order of disj... |
3orbi123 37738 | ~ pm4.39 with a 3-conjunct... |
syl5imp 37739 | Closed form of ~ syl5 . D... |
impexpd 37740 | The following User's Proof... |
com3rgbi 37741 | The following User's Proof... |
impexpdcom 37742 | The following User's Proof... |
ee1111 37743 | Non-virtual deduction form... |
pm2.43bgbi 37744 | Logical equivalence of a 2... |
pm2.43cbi 37745 | Logical equivalence of a 3... |
ee233 37746 | Non-virtual deduction form... |
imbi13 37747 | Join three logical equival... |
ee33 37748 | Non-virtual deduction form... |
con5 37749 | Biconditional contrapositi... |
con5i 37750 | Inference form of ~ con5 .... |
exlimexi 37751 | Inference similar to Theor... |
sb5ALT 37752 | Equivalence for substituti... |
eexinst01 37753 | ~ exinst01 without virtual... |
eexinst11 37754 | ~ exinst11 without virtual... |
vk15.4j 37755 | Excercise 4j of Unit 15 of... |
notnotrALT 37756 | Converse of double negatio... |
con3ALT2 37757 | Contraposition. Alternate... |
ssralv2 37758 | Quantification restricted ... |
sbc3or 37759 | ~ sbcor with a 3-disjuncts... |
sbcangOLD 37760 | Distribution of class subs... |
sbcorgOLD 37761 | Distribution of class subs... |
sbcbiiOLD 37762 | Formula-building inference... |
sbc3orgOLD 37763 | ~ sbcorgOLD with a 3-disju... |
alrim3con13v 37764 | Closed form of ~ alrimi wi... |
rspsbc2 37765 | ~ rspsbc with two quantify... |
sbcoreleleq 37766 | Substitution of a setvar v... |
tratrb 37767 | If a class is transitive a... |
ordelordALT 37768 | An element of an ordinal c... |
sbcim2g 37769 | Distribution of class subs... |
sbcbi 37770 | Implication form of ~ sbcb... |
trsbc 37771 | Formula-building inference... |
truniALT 37772 | The union of a class of tr... |
sbcalgOLD 37773 | Move universal quantifier ... |
sbcexgOLD 37774 | Move existential quantifie... |
sbcel12gOLD 37775 | Distribute proper substitu... |
sbcel2gOLD 37776 | Move proper substitution i... |
sbcssOLD 37777 | Distribute proper substitu... |
onfrALTlem5 37778 | Lemma for ~ onfrALT . (Co... |
onfrALTlem4 37779 | Lemma for ~ onfrALT . (Co... |
onfrALTlem3 37780 | Lemma for ~ onfrALT . (Co... |
ggen31 37781 | ~ gen31 without virtual de... |
onfrALTlem2 37782 | Lemma for ~ onfrALT . (Co... |
cbvexsv 37783 | A theorem pertaining to th... |
onfrALTlem1 37784 | Lemma for ~ onfrALT . (Co... |
onfrALT 37785 | The epsilon relation is fo... |
csbeq2gOLD 37786 | Formula-building implicati... |
19.41rg 37787 | Closed form of right-to-le... |
opelopab4 37788 | Ordered pair membership in... |
2pm13.193 37789 | ~ pm13.193 for two variabl... |
hbntal 37790 | A closed form of ~ hbn . ~... |
hbimpg 37791 | A closed form of ~ hbim . ... |
hbalg 37792 | Closed form of ~ hbal . D... |
hbexg 37793 | Closed form of ~ nfex . D... |
ax6e2eq 37794 | Alternate form of ~ ax6e f... |
ax6e2nd 37795 | If at least two sets exist... |
ax6e2ndeq 37796 | "At least two sets exist" ... |
2sb5nd 37797 | Equivalence for double sub... |
2uasbanh 37798 | Distribute the unabbreviat... |
2uasban 37799 | Distribute the unabbreviat... |
e2ebind 37800 | Absorption of an existenti... |
elpwgded 37801 | ~ elpwgdedVD in convention... |
trelded 37802 | Deduction form of ~ trel .... |
jaoded 37803 | Deduction form of ~ jao . ... |
sbtT 37804 | A substitution into a theo... |
not12an2impnot1 37805 | If a double conjunction is... |
in1 37808 | Inference form of ~ df-vd1... |
iin1 37809 | ~ in1 without virtual dedu... |
dfvd1ir 37810 | Inference form of ~ df-vd1... |
idn1 37811 | Virtual deduction identity... |
dfvd1imp 37812 | Left-to-right part of defi... |
dfvd1impr 37813 | Right-to-left part of defi... |
dfvd2 37816 | Definition of a 2-hypothes... |
dfvd2an 37819 | Definition of a 2-hypothes... |
dfvd2ani 37820 | Inference form of ~ dfvd2a... |
dfvd2anir 37821 | Right-to-left inference fo... |
dfvd2i 37822 | Inference form of ~ dfvd2 ... |
dfvd2ir 37823 | Right-to-left inference fo... |
dfvd3 37828 | Definition of a 3-hypothes... |
dfvd3i 37829 | Inference form of ~ dfvd3 ... |
dfvd3ir 37830 | Right-to-left inference fo... |
dfvd3an 37831 | Definition of a 3-hypothes... |
dfvd3ani 37832 | Inference form of ~ dfvd3a... |
dfvd3anir 37833 | Right-to-left inference fo... |
vd01 37843 | A virtual hypothesis virtu... |
vd02 37844 | Two virtual hypotheses vir... |
vd03 37845 | A theorem is virtually inf... |
vd12 37846 | A virtual deduction with 1... |
vd13 37847 | A virtual deduction with 1... |
vd23 37848 | A virtual deduction with 2... |
dfvd2imp 37849 | The virtual deduction form... |
dfvd2impr 37850 | A 2-antecedent nested impl... |
in2 37851 | The virtual deduction intr... |
int2 37852 | The virtual deduction intr... |
iin2 37853 | ~ in2 without virtual dedu... |
in2an 37854 | The virtual deduction intr... |
in3 37855 | The virtual deduction intr... |
iin3 37856 | ~ in3 without virtual dedu... |
in3an 37857 | The virtual deduction intr... |
int3 37858 | The virtual deduction intr... |
idn2 37859 | Virtual deduction identity... |
iden2 37860 | Virtual deduction identity... |
idn3 37861 | Virtual deduction identity... |
gen11 37862 | Virtual deduction generali... |
gen11nv 37863 | Virtual deduction generali... |
gen12 37864 | Virtual deduction generali... |
gen21 37865 | Virtual deduction generali... |
gen21nv 37866 | Virtual deduction form of ... |
gen31 37867 | Virtual deduction generali... |
gen22 37868 | Virtual deduction generali... |
ggen22 37869 | ~ gen22 without virtual de... |
exinst 37870 | Existential Instantiation.... |
exinst01 37871 | Existential Instantiation.... |
exinst11 37872 | Existential Instantiation.... |
e1a 37873 | A Virtual deduction elimin... |
el1 37874 | A Virtual deduction elimin... |
e1bi 37875 | Biconditional form of ~ e1... |
e1bir 37876 | Right biconditional form o... |
e2 37877 | A virtual deduction elimin... |
e2bi 37878 | Biconditional form of ~ e2... |
e2bir 37879 | Right biconditional form o... |
ee223 37880 | ~ e223 without virtual ded... |
e223 37881 | A virtual deduction elimin... |
e222 37882 | A virtual deduction elimin... |
e220 37883 | A virtual deduction elimin... |
ee220 37884 | ~ e220 without virtual ded... |
e202 37885 | A virtual deduction elimin... |
ee202 37886 | ~ e202 without virtual ded... |
e022 37887 | A virtual deduction elimin... |
ee022 37888 | ~ e022 without virtual ded... |
e002 37889 | A virtual deduction elimin... |
ee002 37890 | ~ e002 without virtual ded... |
e020 37891 | A virtual deduction elimin... |
ee020 37892 | ~ e020 without virtual ded... |
e200 37893 | A virtual deduction elimin... |
ee200 37894 | ~ e200 without virtual ded... |
e221 37895 | A virtual deduction elimin... |
ee221 37896 | ~ e221 without virtual ded... |
e212 37897 | A virtual deduction elimin... |
ee212 37898 | ~ e212 without virtual ded... |
e122 37899 | A virtual deduction elimin... |
e112 37900 | A virtual deduction elimin... |
ee112 37901 | ~ e112 without virtual ded... |
e121 37902 | A virtual deduction elimin... |
e211 37903 | A virtual deduction elimin... |
ee211 37904 | ~ e211 without virtual ded... |
e210 37905 | A virtual deduction elimin... |
ee210 37906 | ~ e210 without virtual ded... |
e201 37907 | A virtual deduction elimin... |
ee201 37908 | ~ e201 without virtual ded... |
e120 37909 | A virtual deduction elimin... |
ee120 37910 | Virtual deduction rule ~ e... |
e021 37911 | A virtual deduction elimin... |
ee021 37912 | ~ e021 without virtual ded... |
e012 37913 | A virtual deduction elimin... |
ee012 37914 | ~ e012 without virtual ded... |
e102 37915 | A virtual deduction elimin... |
ee102 37916 | ~ e102 without virtual ded... |
e22 37917 | A virtual deduction elimin... |
e22an 37918 | Conjunction form of ~ e22 ... |
ee22an 37919 | ~ e22an without virtual de... |
e111 37920 | A virtual deduction elimin... |
e1111 37921 | A virtual deduction elimin... |
e110 37922 | A virtual deduction elimin... |
ee110 37923 | ~ e110 without virtual ded... |
e101 37924 | A virtual deduction elimin... |
ee101 37925 | ~ e101 without virtual ded... |
e011 37926 | A virtual deduction elimin... |
ee011 37927 | ~ e011 without virtual ded... |
e100 37928 | A virtual deduction elimin... |
ee100 37929 | ~ e100 without virtual ded... |
e010 37930 | A virtual deduction elimin... |
ee010 37931 | ~ e010 without virtual ded... |
e001 37932 | A virtual deduction elimin... |
ee001 37933 | ~ e001 without virtual ded... |
e11 37934 | A virtual deduction elimin... |
e11an 37935 | Conjunction form of ~ e11 ... |
ee11an 37936 | ~ e11an without virtual de... |
e01 37937 | A virtual deduction elimin... |
e01an 37938 | Conjunction form of ~ e01 ... |
ee01an 37939 | ~ e01an without virtual de... |
e10 37940 | A virtual deduction elimin... |
e10an 37941 | Conjunction form of ~ e10 ... |
ee10an 37942 | ~ e10an without virtual de... |
e02 37943 | A virtual deduction elimin... |
e02an 37944 | Conjunction form of ~ e02 ... |
ee02an 37945 | ~ e02an without virtual de... |
eel021old 37946 | ~ el021old without virtual... |
el021old 37947 | A virtual deduction elimin... |
eel132 37948 | ~ syl2an with antecedents ... |
eel000cT 37949 | An elimination deduction. ... |
eel0TT 37950 | An elimination deduction. ... |
eelT00 37951 | An elimination deduction. ... |
eelTTT 37952 | An elimination deduction. ... |
eelT11 37953 | An elimination deduction. ... |
eelT1 37954 | Syllogism inference combin... |
eelT12 37955 | An elimination deduction. ... |
eelTT1 37956 | An elimination deduction. ... |
eelT01 37957 | An elimination deduction. ... |
eel0T1 37958 | An elimination deduction. ... |
eel12131 37959 | An elimination deduction. ... |
eel2131 37960 | ~ syl2an with antecedents ... |
eel3132 37961 | ~ syl2an with antecedents ... |
eel0321old 37962 | ~ el0321old without virtua... |
el0321old 37963 | A virtual deduction elimin... |
eel2122old 37964 | ~ el2122old without virtua... |
el2122old 37965 | A virtual deduction elimin... |
eel0001 37966 | An elimination deduction. ... |
eel0000 37967 | Elimination rule similar t... |
eel1111 37968 | Four-hypothesis eliminatio... |
eel00001 37969 | An elimination deduction. ... |
eel00000 37970 | Elimination rule similar ~... |
eel11111 37971 | Five-hypothesis eliminatio... |
e12 37972 | A virtual deduction elimin... |
e12an 37973 | Conjunction form of ~ e12 ... |
el12 37974 | Virtual deduction form of ... |
e20 37975 | A virtual deduction elimin... |
e20an 37976 | Conjunction form of ~ e20 ... |
ee20an 37977 | ~ e20an without virtual de... |
e21 37978 | A virtual deduction elimin... |
e21an 37979 | Conjunction form of ~ e21 ... |
ee21an 37980 | ~ e21an without virtual de... |
e333 37981 | A virtual deduction elimin... |
e33 37982 | A virtual deduction elimin... |
e33an 37983 | Conjunction form of ~ e33 ... |
ee33an 37984 | ~ e33an without virtual de... |
e3 37985 | Meta-connective form of ~ ... |
e3bi 37986 | Biconditional form of ~ e3... |
e3bir 37987 | Right biconditional form o... |
e03 37988 | A virtual deduction elimin... |
ee03 37989 | ~ e03 without virtual dedu... |
e03an 37990 | Conjunction form of ~ e03 ... |
ee03an 37991 | Conjunction form of ~ ee03... |
e30 37992 | A virtual deduction elimin... |
ee30 37993 | ~ e30 without virtual dedu... |
e30an 37994 | A virtual deduction elimin... |
ee30an 37995 | Conjunction form of ~ ee30... |
e13 37996 | A virtual deduction elimin... |
e13an 37997 | A virtual deduction elimin... |
ee13an 37998 | ~ e13an without virtual de... |
e31 37999 | A virtual deduction elimin... |
ee31 38000 | ~ e31 without virtual dedu... |
e31an 38001 | A virtual deduction elimin... |
ee31an 38002 | ~ e31an without virtual de... |
e23 38003 | A virtual deduction elimin... |
e23an 38004 | A virtual deduction elimin... |
ee23an 38005 | ~ e23an without virtual de... |
e32 38006 | A virtual deduction elimin... |
ee32 38007 | ~ e32 without virtual dedu... |
e32an 38008 | A virtual deduction elimin... |
ee32an 38009 | ~ e33an without virtual de... |
e123 38010 | A virtual deduction elimin... |
ee123 38011 | ~ e123 without virtual ded... |
el123 38012 | A virtual deduction elimin... |
e233 38013 | A virtual deduction elimin... |
e323 38014 | A virtual deduction elimin... |
e000 38015 | A virtual deduction elimin... |
e00 38016 | Elimination rule identical... |
e00an 38017 | Elimination rule identical... |
eel00cT 38018 | An elimination deduction. ... |
eelTT 38019 | An elimination deduction. ... |
e0a 38020 | Elimination rule identical... |
eelT 38021 | An elimination deduction. ... |
eel0cT 38022 | An elimination deduction. ... |
eelT0 38023 | An elimination deduction. ... |
e0bi 38024 | Elimination rule identical... |
e0bir 38025 | Elimination rule identical... |
uun0.1 38026 | Convention notation form o... |
un0.1 38027 | ` T. ` is the constant tru... |
uunT1 38028 | A deduction unionizing a n... |
uunT1p1 38029 | A deduction unionizing a n... |
uunT21 38030 | A deduction unionizing a n... |
uun121 38031 | A deduction unionizing a n... |
uun121p1 38032 | A deduction unionizing a n... |
uun132 38033 | A deduction unionizing a n... |
uun132p1 38034 | A deduction unionizing a n... |
anabss7p1 38035 | A deduction unionizing a n... |
un10 38036 | A unionizing deduction. (... |
un01 38037 | A unionizing deduction. (... |
un2122 38038 | A deduction unionizing a n... |
uun2131 38039 | A deduction unionizing a n... |
uun2131p1 38040 | A deduction unionizing a n... |
uunTT1 38041 | A deduction unionizing a n... |
uunTT1p1 38042 | A deduction unionizing a n... |
uunTT1p2 38043 | A deduction unionizing a n... |
uunT11 38044 | A deduction unionizing a n... |
uunT11p1 38045 | A deduction unionizing a n... |
uunT11p2 38046 | A deduction unionizing a n... |
uunT12 38047 | A deduction unionizing a n... |
uunT12p1 38048 | A deduction unionizing a n... |
uunT12p2 38049 | A deduction unionizing a n... |
uunT12p3 38050 | A deduction unionizing a n... |
uunT12p4 38051 | A deduction unionizing a n... |
uunT12p5 38052 | A deduction unionizing a n... |
uun111 38053 | A deduction unionizing a n... |
3anidm12p1 38054 | A deduction unionizing a n... |
3anidm12p2 38055 | A deduction unionizing a n... |
uun123 38056 | A deduction unionizing a n... |
uun123p1 38057 | A deduction unionizing a n... |
uun123p2 38058 | A deduction unionizing a n... |
uun123p3 38059 | A deduction unionizing a n... |
uun123p4 38060 | A deduction unionizing a n... |
uun2221 38061 | A deduction unionizing a n... |
uun2221p1 38062 | A deduction unionizing a n... |
uun2221p2 38063 | A deduction unionizing a n... |
3impdirp1 38064 | A deduction unionizing a n... |
3impcombi 38065 | A 1-hypothesis proposition... |
3imp231 38066 | Importation inference. (C... |
trsspwALT 38067 | Virtual deduction proof of... |
trsspwALT2 38068 | Virtual deduction proof of... |
trsspwALT3 38069 | Short predicate calculus p... |
sspwtr 38070 | Virtual deduction proof of... |
sspwtrALT 38071 | Virtual deduction proof of... |
csbabgOLD 38072 | Move substitution into a c... |
csbunigOLD 38073 | Distribute proper substitu... |
csbfv12gALTOLD 38074 | Move class substitution in... |
csbxpgOLD 38075 | Distribute proper substitu... |
csbingOLD 38076 | Distribute proper substitu... |
csbresgOLD 38077 | Distribute proper substitu... |
csbrngOLD 38078 | Distribute proper substitu... |
csbima12gALTOLD 38079 | Move class substitution in... |
sspwtrALT2 38080 | Short predicate calculus p... |
pwtrVD 38081 | Virtual deduction proof of... |
pwtrrVD 38082 | Virtual deduction proof of... |
suctrALT 38083 | The successor of a transit... |
snssiALTVD 38084 | Virtual deduction proof of... |
snssiALT 38085 | If a class is an element o... |
snsslVD 38086 | Virtual deduction proof of... |
snssl 38087 | If a singleton is a subcla... |
snelpwrVD 38088 | Virtual deduction proof of... |
unipwrVD 38089 | Virtual deduction proof of... |
unipwr 38090 | A class is a subclass of t... |
sstrALT2VD 38091 | Virtual deduction proof of... |
sstrALT2 38092 | Virtual deduction proof of... |
suctrALT2VD 38093 | Virtual deduction proof of... |
suctrALT2 38094 | Virtual deduction proof of... |
elex2VD 38095 | Virtual deduction proof of... |
elex22VD 38096 | Virtual deduction proof of... |
eqsbc3rVD 38097 | Virtual deduction proof of... |
zfregs2VD 38098 | Virtual deduction proof of... |
tpid3gVD 38099 | Virtual deduction proof of... |
en3lplem1VD 38100 | Virtual deduction proof of... |
en3lplem2VD 38101 | Virtual deduction proof of... |
en3lpVD 38102 | Virtual deduction proof of... |
simplbi2VD 38103 | Virtual deduction proof of... |
3ornot23VD 38104 | Virtual deduction proof of... |
orbi1rVD 38105 | Virtual deduction proof of... |
bitr3VD 38106 | Virtual deduction proof of... |
3orbi123VD 38107 | Virtual deduction proof of... |
sbc3orgVD 38108 | Virtual deduction proof of... |
19.21a3con13vVD 38109 | Virtual deduction proof of... |
exbirVD 38110 | Virtual deduction proof of... |
exbiriVD 38111 | Virtual deduction proof of... |
rspsbc2VD 38112 | Virtual deduction proof of... |
3impexpVD 38113 | Virtual deduction proof of... |
3impexpbicomVD 38114 | Virtual deduction proof of... |
3impexpbicomiVD 38115 | Virtual deduction proof of... |
sbcel1gvOLD 38116 | Class substitution into a ... |
sbcoreleleqVD 38117 | Virtual deduction proof of... |
hbra2VD 38118 | Virtual deduction proof of... |
tratrbVD 38119 | Virtual deduction proof of... |
al2imVD 38120 | Virtual deduction proof of... |
syl5impVD 38121 | Virtual deduction proof of... |
idiVD 38122 | Virtual deduction proof of... |
ancomstVD 38123 | Closed form of ~ ancoms . ... |
ssralv2VD 38124 | Quantification restricted ... |
ordelordALTVD 38125 | An element of an ordinal c... |
equncomVD 38126 | If a class equals the unio... |
equncomiVD 38127 | Inference form of ~ equnco... |
sucidALTVD 38128 | A set belongs to its succe... |
sucidALT 38129 | A set belongs to its succe... |
sucidVD 38130 | A set belongs to its succe... |
imbi12VD 38131 | Implication form of ~ imbi... |
imbi13VD 38132 | Join three logical equival... |
sbcim2gVD 38133 | Distribution of class subs... |
sbcbiVD 38134 | Implication form of ~ sbcb... |
trsbcVD 38135 | Formula-building inference... |
truniALTVD 38136 | The union of a class of tr... |
ee33VD 38137 | Non-virtual deduction form... |
trintALTVD 38138 | The intersection of a clas... |
trintALT 38139 | The intersection of a clas... |
undif3VD 38140 | The first equality of Exer... |
sbcssgVD 38141 | Virtual deduction proof of... |
csbingVD 38142 | Virtual deduction proof of... |
onfrALTlem5VD 38143 | Virtual deduction proof of... |
onfrALTlem4VD 38144 | Virtual deduction proof of... |
onfrALTlem3VD 38145 | Virtual deduction proof of... |
simplbi2comtVD 38146 | Virtual deduction proof of... |
onfrALTlem2VD 38147 | Virtual deduction proof of... |
onfrALTlem1VD 38148 | Virtual deduction proof of... |
onfrALTVD 38149 | Virtual deduction proof of... |
csbeq2gVD 38150 | Virtual deduction proof of... |
csbsngVD 38151 | Virtual deduction proof of... |
csbxpgVD 38152 | Virtual deduction proof of... |
csbresgVD 38153 | Virtual deduction proof of... |
csbrngVD 38154 | Virtual deduction proof of... |
csbima12gALTVD 38155 | Virtual deduction proof of... |
csbunigVD 38156 | Virtual deduction proof of... |
csbfv12gALTVD 38157 | Virtual deduction proof of... |
con5VD 38158 | Virtual deduction proof of... |
relopabVD 38159 | Virtual deduction proof of... |
19.41rgVD 38160 | Virtual deduction proof of... |
2pm13.193VD 38161 | Virtual deduction proof of... |
hbimpgVD 38162 | Virtual deduction proof of... |
hbalgVD 38163 | Virtual deduction proof of... |
hbexgVD 38164 | Virtual deduction proof of... |
ax6e2eqVD 38165 | The following User's Proof... |
ax6e2ndVD 38166 | The following User's Proof... |
ax6e2ndeqVD 38167 | The following User's Proof... |
2sb5ndVD 38168 | The following User's Proof... |
2uasbanhVD 38169 | The following User's Proof... |
e2ebindVD 38170 | The following User's Proof... |
sb5ALTVD 38171 | The following User's Proof... |
vk15.4jVD 38172 | The following User's Proof... |
notnotrALTVD 38173 | The following User's Proof... |
con3ALTVD 38174 | The following User's Proof... |
elpwgdedVD 38175 | Membership in a power clas... |
sspwimp 38176 | If a class is a subclass o... |
sspwimpVD 38177 | The following User's Proof... |
sspwimpcf 38178 | If a class is a subclass o... |
sspwimpcfVD 38179 | The following User's Proof... |
suctrALTcf 38180 | The sucessor of a transiti... |
suctrALTcfVD 38181 | The following User's Proof... |
suctrALT3 38182 | The successor of a transit... |
sspwimpALT 38183 | If a class is a subclass o... |
unisnALT 38184 | A set equals the union of ... |
notnotrALT2 38185 | Converse of double negatio... |
sspwimpALT2 38186 | If a class is a subclass o... |
e2ebindALT 38187 | Absorption of an existenti... |
ax6e2ndALT 38188 | If at least two sets exist... |
ax6e2ndeqALT 38189 | "At least two sets exist" ... |
2sb5ndALT 38190 | Equivalence for double sub... |
chordthmALT 38191 | The intersecting chords th... |
isosctrlem1ALT 38192 | Lemma for ~ isosctr . Thi... |
iunconlem2 38193 | The indexed union of conne... |
iunconALT 38194 | The indexed union of conne... |
sineq0ALT 38195 | A complex number whose sin... |
fnvinran 38196 | the function value belongs... |
evth2f 38197 | A version of ~ evth2 using... |
elunif 38198 | A version of ~ eluni using... |
rzalf 38199 | A version of ~ rzal using ... |
fvelrnbf 38200 | A version of ~ fvelrnb usi... |
rfcnpre1 38201 | If F is a continuous funct... |
ubelsupr 38202 | If U belongs to A and U is... |
fsumcnf 38203 | A finite sum of functions ... |
mulltgt0 38204 | The product of a negative ... |
rspcegf 38205 | A version of ~ rspcev usin... |
rabexgf 38206 | A version of ~ rabexg usin... |
fcnre 38207 | A function continuous with... |
sumsnd 38208 | A sum of a singleton is th... |
evthf 38209 | A version of ~ evth using ... |
cnfex 38210 | The class of continuous fu... |
fnchoice 38211 | For a finite set, a choice... |
refsumcn 38212 | A finite sum of continuous... |
rfcnpre2 38213 | If ` F ` is a continuous f... |
cncmpmax 38214 | When the hypothesis for th... |
rfcnpre3 38215 | If F is a continuous funct... |
rfcnpre4 38216 | If F is a continuous funct... |
sumpair 38217 | Sum of two distinct comple... |
rfcnnnub 38218 | Given a real continuous fu... |
refsum2cnlem1 38219 | This is the core Lemma for... |
refsum2cn 38220 | The sum of two continuus r... |
elunnel2 38221 | A member of a union that i... |
adantlllr 38222 | Deduction adding a conjunc... |
3adantlr3 38223 | Deduction adding a conjunc... |
nnxrd 38224 | A natural number is an ext... |
3adantll2 38225 | Deduction adding a conjunc... |
3adantll3 38226 | Deduction adding a conjunc... |
ssnel 38227 | If not element of a set, t... |
jcn 38228 | Inference joining the cons... |
elabrexg 38229 | Elementhood in an image se... |
unicntop 38230 | The underlying set of the ... |
ifeq123d 38231 | Equality deduction for con... |
sncldre 38232 | A singleton is closed w.r.... |
cnopn 38233 | The set of complex numbers... |
n0p 38234 | A polynomial with a nonzer... |
pm2.65ni 38235 | Inference rule for proof b... |
pwssfi 38236 | Every element of the power... |
iuneq2df 38237 | Equality deduction for ind... |
nnfoctb 38238 | There exists a mapping fro... |
ssinss1d 38239 | Intersection preserves sub... |
0un 38240 | The union of the empty set... |
elpwinss 38241 | An element of the powerset... |
unidmex 38242 | If ` F ` is a set, then ` ... |
ndisj2 38243 | A non disjointness conditi... |
zenom 38244 | The set of integer numbers... |
rexsngf 38245 | Restricted existential qua... |
uzwo4 38246 | Well-ordering principle: a... |
unisn0 38247 | The union of the singleton... |
ssin0 38248 | If two classes are disjoin... |
inabs3 38249 | Absorption law for interse... |
pwpwuni 38250 | Relationship between power... |
disjiun2 38251 | In a disjoint collection, ... |
0pwfi 38252 | The empty set is in any po... |
ssinss2d 38253 | Intersection preserves sub... |
zct 38254 | The set of integer numbers... |
iunxsngf2 38255 | A singleton index picks ou... |
pwfin0 38256 | A finite set always belong... |
uzct 38257 | An upper integer set is co... |
iunxsnf 38258 | A singleton index picks ou... |
fiiuncl 38259 | If a set is closed under t... |
iunp1 38260 | The addition of the next s... |
fiunicl 38261 | If a set is closed under t... |
ixpeq2d 38262 | Equality theorem for infin... |
disjxp1 38263 | The sets of a cartesian pr... |
elpwd 38264 | Membership in a power clas... |
disjsnxp 38265 | The sets in the cartesian ... |
eliind 38266 | Membership in indexed inte... |
rspcef 38267 | Restricted existential spe... |
inn0f 38268 | A non-empty intersection. ... |
ixpssmapc 38269 | An infinite Cartesian prod... |
inn0 38270 | A non-empty intersection. ... |
elintd 38271 | Membership in class inters... |
eqneltri 38272 | If a class is not an eleme... |
ssdf 38273 | A sufficient condition for... |
brneqtrd 38274 | Substitution of equal clas... |
ssnct 38275 | A set containing an uncoun... |
ssuniint 38276 | Sufficient condition for b... |
elintdv 38277 | Membership in class inters... |
ssd 38278 | A sufficient condition for... |
ralimralim 38279 | Introducing any antecedent... |
snelmap 38280 | Membership of the element ... |
dfcleqf 38281 | Equality connective betwee... |
xrnmnfpnf 38282 | An extended real that is n... |
nelrnmpt 38283 | Non-membership in the rang... |
snn0d 38284 | The singleton of a set is ... |
rabid3 38285 | Membership in a restricted... |
iuneq1i 38286 | Equality theorem for index... |
nssrex 38287 | Negation of subclass relat... |
nelpr2 38288 | If a class is not an eleme... |
nelpr1 38289 | If a class is not an eleme... |
iunssf 38290 | Subset theorem for an inde... |
elpwi2 38291 | Membership in a power clas... |
ssinc 38292 | Inclusion relation for a m... |
ssdec 38293 | Inclusion relation for a m... |
elixpconstg 38294 | Membership in an infinite ... |
iineq1d 38295 | Equality theorem for index... |
metpsmet 38296 | A metric is a pseudometric... |
ixpssixp 38297 | Subclass theorem for infin... |
ballss3 38298 | A sufficient condition for... |
iunssd 38299 | Subset theorem for an inde... |
iunincfi 38300 | Given a sequence of increa... |
nsstr 38301 | If it's not a subclass, it... |
rabbida 38302 | Equivalent wff's yield equ... |
rexanuz3 38303 | Combine two different uppe... |
rabeqd 38304 | Equality theorem for restr... |
cbvmpt22 38305 | Rule to change the second ... |
cbvmpt21 38306 | Rule to change the first b... |
eliuniin 38307 | Indexed union of indexed i... |
ssabf 38308 | Subclass of a class abstra... |
rabbia2 38309 | Equivalent wff's yield equ... |
uniexd 38310 | Deduction version of the Z... |
pwexd 38311 | Deduction version of the p... |
pssnssi 38312 | A proper subclass does not... |
rabidim2 38313 | Membership in a restricted... |
xpexd 38314 | The Cartesian product of t... |
eluni2f 38315 | Membership in class union.... |
eliin2f 38316 | Membership in indexed inte... |
nssd 38317 | Negation of subclass relat... |
rabidim1 38318 | Membership in a restricted... |
iineq12dv 38319 | Equality deduction for ind... |
rabeqif 38320 | Equality theorem for restr... |
supxrcld 38321 | The supremum of an arbitra... |
elrestd 38322 | A sufficient condition for... |
eliuniincex 38323 | Counterexample to show tha... |
eliincex 38324 | Counterexample to show tha... |
eliinid 38325 | Membership in an indexed i... |
abssf 38326 | Class abstraction in a sub... |
fexd 38327 | If the domain of a mapping... |
supxrubd 38328 | A member of a set of exten... |
ssrabf 38329 | Subclass of a restricted c... |
eliin2 38330 | Membership in indexed inte... |
ssrab2f 38331 | Subclass relation for a re... |
rabeqi 38332 | Equality theorem for restr... |
restuni3 38333 | The underlying set of a su... |
rabssf 38334 | Restricted class abstracti... |
eliuniin2 38335 | Indexed union of indexed i... |
restuni4 38336 | The underlying set of a su... |
restuni6 38337 | The underlying set of a su... |
restuni5 38338 | The underlying set of a su... |
unirestss 38339 | The union of an elementwis... |
unima 38340 | Image of a union. (Contri... |
feq1dd 38341 | Equality deduction for fun... |
fnresdmss 38342 | A function does not change... |
fmptsnxp 38343 | Maps-to notation and cross... |
mptex2 38344 | If a class given as a map-... |
fvmpt2bd 38345 | Value of a function given ... |
rnmptfi 38346 | The range of a function wi... |
fresin2 38347 | Restriction of a function ... |
rnmptc 38348 | Range of a constant functi... |
ffi 38349 | A function with finite dom... |
suprnmpt 38350 | An explicit bound for the ... |
rnffi 38351 | The range of a function wi... |
mptelpm 38352 | A function in maps-to nota... |
rnmptpr 38353 | Range of a function define... |
resmpti 38354 | Restriction of the mapping... |
founiiun 38355 | Union expressed as an inde... |
f1oeq2d 38356 | Equality deduction for one... |
rnresun 38357 | Distribution law for range... |
f1oeq1d 38358 | Equality deduction for one... |
dffo3f 38359 | An onto mapping expressed ... |
rnresss 38360 | The range of a restriction... |
elrnmptd 38361 | The range of a function in... |
elrnmptf 38362 | The range of a function in... |
rnmptssrn 38363 | Inclusion relation for two... |
disjf1 38364 | A 1 to 1 mapping built fro... |
rnsnf 38365 | The range of a function wh... |
wessf1ornlem 38366 | Given a function ` F ` on ... |
wessf1orn 38367 | Given a function ` F ` on ... |
foelrnf 38368 | Property of a surjective f... |
nelrnres 38369 | If ` A ` is not in the ran... |
disjrnmpt2 38370 | Disjointness of the range ... |
elrnmpt1sf 38371 | Elementhood in an image se... |
founiiun0 38372 | Union expressed as an inde... |
disjf1o 38373 | A bijection built from dis... |
fompt 38374 | Express being onto for a m... |
disjinfi 38375 | Only a finite number of di... |
fvovco 38376 | Value of the composition o... |
ssnnf1octb 38377 | There exists a bijection b... |
mapdm0 38378 | The empty set is the only ... |
nnf1oxpnn 38379 | There is a bijection betwe... |
rnmptssd 38380 | The range of an operation ... |
projf1o 38381 | A biijection from a set to... |
fvmap 38382 | Function value for a membe... |
mapsnd 38383 | The value of set exponenti... |
fvixp2 38384 | Projection of a factor of ... |
fidmfisupp 38385 | A function with a finite d... |
mapsnend 38386 | Set exponentiation to a si... |
choicefi 38387 | For a finite set, a choice... |
mpct 38388 | The exponentiation of a co... |
cnmetcoval 38389 | Value of the distance func... |
fcomptss 38390 | Express composition of two... |
elmapsnd 38391 | Membership in a set expone... |
mapss2 38392 | Subset inheritance for set... |
fsneq 38393 | Equality condition for two... |
difmap 38394 | Difference of two sets exp... |
unirnmap 38395 | Given a subset of a set ex... |
inmap 38396 | Intersection of two sets e... |
fcoss 38397 | Composition of two mapping... |
fsneqrn 38398 | Equality condition for two... |
difmapsn 38399 | Difference of two sets exp... |
mapssbi 38400 | Subset inheritance for set... |
unirnmapsn 38401 | Equality theorem for a sub... |
iunmapss 38402 | The indexed union of set e... |
ssmapsn 38403 | A subset ` C ` of a set ex... |
iunmapsn 38404 | The indexed union of set e... |
absfico 38405 | Mapping domain and codomai... |
icof 38406 | The set of left-closed rig... |
rnmpt0 38407 | The range of a function in... |
rnmptn0 38408 | The range of a function in... |
elpmrn 38409 | The range of a partial fun... |
imaexi 38410 | The image of a set is a se... |
axccdom 38411 | Relax the constraint on ax... |
dmmptdf 38412 | The domain of the mapping ... |
elpmi2 38413 | The domain of a partial fu... |
dmrelrnrel 38414 | A relation preserving func... |
fdmd 38415 | The domain of a mapping. ... |
fco3 38416 | Functionality of a composi... |
dmexd 38417 | The domain of a set is a s... |
fvcod 38418 | Value of a function compos... |
fcod 38419 | Composition of two mapping... |
freld 38420 | A mapping is a relation. ... |
frnd 38421 | The range of a mapping. (... |
elrnmpt2id 38422 | Membership in the range of... |
fvmptelrn 38423 | A function's value belongs... |
axccd 38424 | An alternative version of ... |
axccd2 38425 | An alternative version of ... |
sub2times 38426 | Subtracting from a number,... |
xrltled 38427 | 'Less than' implies 'less ... |
abssubrp 38428 | The distance of two distin... |
elfzfzo 38429 | Relationship between membe... |
oddfl 38430 | Odd number representation ... |
abscosbd 38431 | Bound for the absolute val... |
mul13d 38432 | Commutative/associative la... |
negpilt0 38433 | Negative ` _pi ` is negati... |
dstregt0 38434 | A complex number ` A ` tha... |
subadd4b 38435 | Rearrangement of 4 terms i... |
xrlttri5d 38436 | Not equal and not larger i... |
neglt 38437 | The negative of a positive... |
zltlesub 38438 | If an integer ` N ` is sma... |
divlt0gt0d 38439 | The ratio of a negative nu... |
subsub23d 38440 | Swap subtrahend and result... |
2timesgt 38441 | Double of a positive real ... |
reopn 38442 | The reals are open with re... |
elfzop1le2 38443 | A member in a half-open in... |
sub31 38444 | Swap the first and third t... |
nnne1ge2 38445 | A positive integer which i... |
lefldiveq 38446 | A closed enough, smaller r... |
negsubdi3d 38447 | Distribution of negative o... |
ltdiv2dd 38448 | Division of a positive num... |
absnpncand 38449 | Triangular inequality, com... |
abssinbd 38450 | Bound for the absolute val... |
halffl 38451 | Floor of ` ( 1 / 2 ) ` . ... |
monoords 38452 | Ordering relation for a st... |
hashssle 38453 | The size of a subset of a ... |
lttri5d 38454 | Not equal and not larger i... |
fzisoeu 38455 | A finite ordered set has a... |
lt3addmuld 38456 | If three real numbers are ... |
absnpncan2d 38457 | Triangular inequality, com... |
fperiodmullem 38458 | A function with period T i... |
fperiodmul 38459 | A function with period T i... |
upbdrech 38460 | Choice of an upper bound f... |
lt4addmuld 38461 | If four real numbers are l... |
absnpncan3d 38462 | Triangular inequality, com... |
upbdrech2 38463 | Choice of an upper bound f... |
ssfiunibd 38464 | A finite union of bounded ... |
fz1ssfz0 38465 | Subset relationship for fi... |
fzdifsuc2 38466 | Remove a successor from th... |
fzsscn 38467 | A finite sequence of integ... |
divcan8d 38468 | A cancellation law for div... |
dmmcand 38469 | Cancellation law for divis... |
fzssre 38470 | A finite sequence of integ... |
elfzelzd 38471 | A member of a finite set o... |
bccld 38472 | A binomial coefficient, in... |
leadd12dd 38473 | Addition to both sides of ... |
fzssnn0 38474 | A finite set of sequential... |
xreqle 38475 | Equality implies 'less tha... |
xaddid2d 38476 | ` 0 ` is a left identity f... |
xadd0ge 38477 | A number is less than or e... |
elfzolem1 38478 | A member in a half-open in... |
xrgtned 38479 | 'Greater than' implies not... |
xrleneltd 38480 | 'Less than or equal to' an... |
xaddcomd 38481 | The extended real addition... |
supxrre3 38482 | The supremum of a nonempty... |
uzfissfz 38483 | For any finite subset of t... |
xleadd2d 38484 | Addition of extended reals... |
suprltrp 38485 | The supremum of a nonempty... |
xleadd1d 38486 | Addition of extended reals... |
xreqled 38487 | Equality implies 'less tha... |
xrgepnfd 38488 | An extended real greater o... |
xrge0nemnfd 38489 | A nonnegative extended rea... |
supxrgere 38490 | If a real number can be ap... |
iuneqfzuzlem 38491 | Lemma for ~ iuneqfzuz : he... |
iuneqfzuz 38492 | If two unions indexed by u... |
xle2addd 38493 | Adding both side of two in... |
supxrgelem 38494 | If an extended real number... |
supxrge 38495 | If an extended real number... |
suplesup 38496 | If any element of ` A ` ca... |
infxrglb 38497 | The infimum of a set of ex... |
xadd0ge2 38498 | A number is less than or e... |
nepnfltpnf 38499 | An extended real that is n... |
ltadd12dd 38500 | Addition to both sides of ... |
nemnftgtmnft 38501 | An extended real that is n... |
xrgtso 38502 | 'Greater than' is a strict... |
rpex 38503 | The positive reals form a ... |
xrge0ge0 38504 | A nonnegative extended rea... |
xrssre 38505 | A subset of extended reals... |
ssuzfz 38506 | A finite subset of the upp... |
absfun 38507 | The absolute value is a fu... |
infrpge 38508 | The infimum of a non empty... |
xrlexaddrp 38509 | If an extended real number... |
supsubc 38510 | The supremum function dist... |
xralrple2 38511 | Show that ` A ` is less th... |
nnuzdisj 38512 | The first ` N ` elements o... |
ltdivgt1 38513 | Divsion by a number greate... |
xrltned 38514 | 'Less than' implies not eq... |
nnsplit 38515 | Express the set of positiv... |
divdiv3d 38516 | Division into a fraction. ... |
abslt2sqd 38517 | Comparison of the square o... |
qenom 38518 | The set of rational number... |
qct 38519 | The set of rational number... |
xrltnled 38520 | 'Less than' in terms of 'l... |
lenlteq 38521 | 'less than or equal to' bu... |
xrred 38522 | An extended real that is n... |
rr2sscn2 38523 | ` RR^ 2 ` is a subset of C... |
infxr 38524 | The infimum of a set of ex... |
infxrunb2 38525 | The infimum of an unbounde... |
infxrbnd2 38526 | The infimum of a bounded-b... |
infleinflem1 38527 | Lemma for ~ infleinf , cas... |
infleinflem2 38528 | Lemma for ~ infleinf , whe... |
infleinf 38529 | If any element of ` B ` ca... |
xralrple4 38530 | Show that ` A ` is less th... |
xralrple3 38531 | Show that ` A ` is less th... |
eluzelzd 38532 | A member of an upper set o... |
suplesup2 38533 | If any element of ` A ` is... |
recnnltrp 38534 | ` N ` is a natural number ... |
fiminre2 38535 | A nonempty finite set of r... |
nnn0 38536 | The set of positive intege... |
fzct 38537 | A finite set of sequential... |
rpgtrecnn 38538 | Any positive real number i... |
fzossuz 38539 | A half-open integer interv... |
fzossz 38540 | A half-open integer interv... |
infrefilb 38541 | The infimum of a finite se... |
infxrrefi 38542 | The real and extended real... |
xrralrecnnle 38543 | Show that ` A ` is less th... |
fzoct 38544 | A finite set of sequential... |
frexr 38545 | A function taking real val... |
nnrecrp 38546 | The reciprocal of a positi... |
qred 38547 | A rational number is a rea... |
reclt0d 38548 | The reciprocal of a negati... |
lt0neg1dd 38549 | If a number is negative, i... |
mnfled 38550 | Minus infinity is less tha... |
xrleidd 38551 | 'Less than or equal to' is... |
negelrpd 38552 | The negation of a negative... |
infxrcld 38553 | The infimum of an arbitrar... |
xrralrecnnge 38554 | Show that ` A ` is less th... |
reclt0 38555 | The reciprocal of a negati... |
ltmulneg 38556 | Multiplying by a negative ... |
allbutfi 38557 | For all but finitely many.... |
ltdiv23neg 38558 | Swap denominator with othe... |
gtnelioc 38559 | A real number larger than ... |
ioossioc 38560 | An open interval is a subs... |
ioondisj2 38561 | A condition for two open i... |
ioondisj1 38562 | A condition for two open i... |
ioosscn 38563 | An open interval is a set ... |
ioogtlb 38564 | An element of a closed int... |
evthiccabs 38565 | Extreme Value Theorem on y... |
ltnelicc 38566 | A real number smaller than... |
eliood 38567 | Membership in an open real... |
iooabslt 38568 | An upper bound for the dis... |
gtnelicc 38569 | A real number greater than... |
iooinlbub 38570 | An open interval has empty... |
iocgtlb 38571 | An element of a left open ... |
iocleub 38572 | An element of a left open ... |
eliccd 38573 | Membership in a closed rea... |
iccssred 38574 | A closed real interval is ... |
eliccre 38575 | A member of a closed inter... |
eliooshift 38576 | Element of an open interva... |
eliocd 38577 | Membership in a left open,... |
snunioo2 38578 | The closure of one end of ... |
icoltub 38579 | An element of a left close... |
tgiooss 38580 | The restriction of the com... |
eliocre 38581 | A member of a left open, r... |
iooltub 38582 | An element of an open inte... |
ioontr 38583 | The interior of an interva... |
eliccxr 38584 | A member of a closed inter... |
snunioo1 38585 | The closure of one end of ... |
lbioc 38586 | An left open right closed ... |
ioomidp 38587 | The midpoint is an element... |
iccdifioo 38588 | If the open inverval is re... |
iccdifprioo 38589 | An open interval is the cl... |
ioossioobi 38590 | Biconditional form of ~ io... |
iccshift 38591 | A closed interval shifted ... |
iccsuble 38592 | An upper bound to the dist... |
iocopn 38593 | A left open right closed i... |
eliccelioc 38594 | Membership in a closed int... |
iooshift 38595 | An open interval shifted b... |
iccintsng 38596 | Intersection of two adiace... |
icoiccdif 38597 | Left closed, right open in... |
icoopn 38598 | A left closed right open i... |
icoub 38599 | A left-closed, right-open ... |
eliccxrd 38600 | Membership in a closed rea... |
pnfel0pnf 38601 | ` +oo ` is a nonnegative e... |
ge0nemnf2 38602 | A nonnegative extended rea... |
eliccnelico 38603 | An element of a closed int... |
eliccelicod 38604 | A member of a closed inter... |
ge0xrre 38605 | A nonnegative extended rea... |
ge0lere 38606 | A nonnegative extended Rea... |
elicores 38607 | Membership in a left-close... |
inficc 38608 | The infimum of a nonempty ... |
qinioo 38609 | The rational numbers are d... |
lenelioc 38610 | A real number smaller than... |
ioonct 38611 | C non empty open interval ... |
xrgtnelicc 38612 | A real number greater than... |
iccdificc 38613 | The difference of two clos... |
iocnct 38614 | A non empty left-open, rig... |
iccnct 38615 | A closed interval, with mo... |
iooiinicc 38616 | A closed interval expresse... |
iccgelbd 38617 | An element of a closed int... |
iooltubd 38618 | An element of an open inte... |
icoltubd 38619 | An element of a left close... |
qelioo 38620 | The rational numbers are d... |
tgqioo2 38621 | Every open set of reals is... |
iccleubd 38622 | An element of a closed int... |
elioored 38623 | A member of an open interv... |
ioogtlbd 38624 | An element of a closed int... |
ioofun 38625 | ` (,) ` is a function. (C... |
icomnfinre 38626 | A left-closed, right-open,... |
sqrlearg 38627 | The square compared with i... |
ressiocsup 38628 | If the supremum belongs to... |
ressioosup 38629 | If the supremum does not b... |
iooiinioc 38630 | A left-open, right-closed ... |
ressiooinf 38631 | If the infimum does not be... |
sumeq2ad 38632 | Equality deduction for sum... |
fsumclf 38633 | Closure of a finite sum of... |
fsumsplitf 38634 | Split a sum into two parts... |
fsummulc1f 38635 | Closure of a finite sum of... |
sumsnf 38636 | A sum of a singleton is th... |
fsumsplitsn 38637 | Separate out a term in a f... |
fsumnncl 38638 | Closure of a non empty, fi... |
fsumsplit1 38639 | Separate out a term in a f... |
fsumge0cl 38640 | The finite sum of nonnegat... |
fsumf1of 38641 | Re-index a finite sum usin... |
fsumiunss 38642 | Sum over a disjoint indexe... |
fsumreclf 38643 | Closure of a finite sum of... |
fsumlessf 38644 | A shorter sum of nonnegati... |
fsumsupp0 38645 | Finite sum of function val... |
fsumsermpt 38646 | A finite sum expressed in ... |
fmul01 38647 | Multiplying a finite numbe... |
fmulcl 38648 | If ' Y ' is closed under t... |
fmuldfeqlem1 38649 | induction step for the pro... |
fmuldfeq 38650 | X and Z are two equivalent... |
fmul01lt1lem1 38651 | Given a finite multiplicat... |
fmul01lt1lem2 38652 | Given a finite multiplicat... |
fmul01lt1 38653 | Given a finite multiplicat... |
cncfmptss 38654 | A continuous complex funct... |
rrpsscn 38655 | The positive reals are a s... |
mulc1cncfg 38656 | A version of ~ mulc1cncf u... |
infrglb 38657 | The infimum of a nonempty ... |
expcnfg 38658 | If ` F ` is a complex cont... |
prodeq2ad 38659 | Equality deduction for pro... |
fprodsplit1 38660 | Separate out a term in a f... |
fprodexp 38661 | Positive integer exponenti... |
fprodabs2 38662 | The absolute value of a fi... |
fprod0 38663 | A finite product with a ze... |
mccllem 38664 | * Induction step for ~ mcc... |
mccl 38665 | A multinomial coefficient,... |
fprodcnlem 38666 | A finite product of functi... |
fprodcn 38667 | A finite product of functi... |
clim1fr1 38668 | A class of sequences of fr... |
isumneg 38669 | Negation of a converging s... |
climrec 38670 | Limit of the reciprocal of... |
climmulf 38671 | A version of ~ climmul usi... |
climexp 38672 | The limit of natural power... |
climinf 38673 | A bounded monotonic non in... |
climsuselem1 38674 | The subsequence index ` I ... |
climsuse 38675 | A subsequence ` G ` of a c... |
climrecf 38676 | A version of ~ climrec usi... |
climneg 38677 | Complex limit of the negat... |
climinff 38678 | A version of ~ climinf usi... |
climdivf 38679 | Limit of the ratio of two ... |
climreeq 38680 | If ` F ` is a real functio... |
ellimciota 38681 | An explicit value for the ... |
climaddf 38682 | A version of ~ climadd usi... |
mullimc 38683 | Limit of the product of tw... |
ellimcabssub0 38684 | An equivalent condition fo... |
limcdm0 38685 | If a function has empty do... |
islptre 38686 | An equivalence condition f... |
limccog 38687 | Limit of the composition o... |
limciccioolb 38688 | The limit of a function at... |
climf 38689 | Express the predicate: Th... |
mullimcf 38690 | Limit of the multiplicatio... |
constlimc 38691 | Limit of constant function... |
rexlim2d 38692 | Inference removing two res... |
idlimc 38693 | Limit of the identity func... |
divcnvg 38694 | The sequence of reciprocal... |
limcperiod 38695 | If ` F ` is a periodic fun... |
limcrecl 38696 | If ` F ` is a real valued ... |
sumnnodd 38697 | A series indexed by ` NN `... |
lptioo2 38698 | The upper bound of an open... |
lptioo1 38699 | The lower bound of an open... |
elprn1 38700 | A member of an unordered p... |
elprn2 38701 | A member of an unordered p... |
limcmptdm 38702 | The domain of a map-to fun... |
clim2f 38703 | Express the predicate: Th... |
limcicciooub 38704 | The limit of a function at... |
ltmod 38705 | A sufficient condition for... |
islpcn 38706 | A characterization for a l... |
lptre2pt 38707 | If a set in the real line ... |
limsupre 38708 | If a sequence is bounded, ... |
limcresiooub 38709 | The left limit doesn't cha... |
limcresioolb 38710 | The right limit doesn't ch... |
limcleqr 38711 | If the left and the right ... |
lptioo2cn 38712 | The upper bound of an open... |
lptioo1cn 38713 | The lower bound of an open... |
neglimc 38714 | Limit of the negative func... |
addlimc 38715 | Sum of two limits. (Contr... |
0ellimcdiv 38716 | If the numerator converges... |
clim2cf 38717 | Express the predicate ` F ... |
limclner 38718 | For a limit point, both fr... |
sublimc 38719 | Subtraction of two limits.... |
reclimc 38720 | Limit of the reciprocal of... |
clim0cf 38721 | Express the predicate ` F ... |
limclr 38722 | For a limit point, both fr... |
divlimc 38723 | Limit of the quotient of t... |
expfac 38724 | Factorial grows faster tha... |
climconstmpt 38725 | A constant sequence conver... |
climresmpt 38726 | A function restricted to u... |
climsubmpt 38727 | Limit of the difference of... |
climsubc2mpt 38728 | Limit of the difference of... |
climsubc1mpt 38729 | Limit of the difference of... |
fnlimfv 38730 | The value of the limit fun... |
climreclf 38731 | The limit of a convergent ... |
climeldmeq 38732 | Two functions that are eve... |
climf2 38733 | Express the predicate: Th... |
fnlimcnv 38734 | The sequence of function v... |
climeldmeqmpt 38735 | Two functions that are eve... |
climfveq 38736 | Two functions that are eve... |
clim2f2 38737 | Express the predicate: Th... |
climfveqmpt 38738 | Two functions that are eve... |
climd 38739 | Express the predicate: Th... |
clim2d 38740 | The limit of complex numbe... |
fnlimfvre 38741 | The limit function of real... |
allbutfifvre 38742 | Given a sequence of real v... |
climleltrp 38743 | The limit of complex numbe... |
fnlimfvre2 38744 | The limit function of real... |
fnlimf 38745 | The limit function of real... |
fnlimabslt 38746 | A sequence of function val... |
coseq0 38747 | A complex number whose cos... |
sinmulcos 38748 | Multiplication formula for... |
coskpi2 38749 | The cosine of an integer m... |
cosnegpi 38750 | The cosine of negative ` _... |
sinaover2ne0 38751 | If ` A ` in ` ( 0 , 2 _pi ... |
cosknegpi 38752 | The cosine of an integer m... |
mulcncff 38753 | The multiplication of two ... |
subcncf 38754 | The addition of two contin... |
cncfmptssg 38755 | A continuous complex funct... |
constcncfg 38756 | A constant function is a c... |
idcncfg 38757 | The identity function is a... |
addcncf 38758 | The addition of two contin... |
cncfshift 38759 | A periodic continuous func... |
resincncf 38760 | ` sin ` restricted to real... |
addccncf2 38761 | Adding a constant is a con... |
0cnf 38762 | The empty set is a continu... |
fsumcncf 38763 | The finite sum of continuo... |
cncfperiod 38764 | A periodic continuous func... |
subcncff 38765 | The subtraction of two con... |
negcncfg 38766 | The opposite of a continuo... |
cnfdmsn 38767 | A function with a singleto... |
cncfcompt 38768 | Composition of continuous ... |
divcncf 38769 | The quotient of two contin... |
addcncff 38770 | The addition of two contin... |
ioccncflimc 38771 | Limit at the upper bound, ... |
cncfuni 38772 | A function is continuous i... |
icccncfext 38773 | A continuous function on a... |
cncficcgt0 38774 | A the absolute value of a ... |
icocncflimc 38775 | Limit at the lower bound, ... |
cncfdmsn 38776 | A complex function with a ... |
divcncff 38777 | The quotient of two contin... |
cncfshiftioo 38778 | A periodic continuous func... |
cncfiooicclem1 38779 | A continuous function ` F ... |
cncfiooicc 38780 | A continuous function ` F ... |
cncfiooiccre 38781 | A continuous function ` F ... |
cncfioobdlem 38782 | ` G ` actually extends ` F... |
cncfioobd 38783 | A continuous function ` F ... |
jumpncnp 38784 | Jump discontinuity or disc... |
cncfcompt2 38785 | Composition of continuous ... |
cxpcncf2 38786 | The complex power function... |
fprodcncf 38787 | The finite product of cont... |
add1cncf 38788 | Addition to a constant is ... |
add2cncf 38789 | Addition to a constant is ... |
sub1cncfd 38790 | Subtracting a constant is ... |
sub2cncfd 38791 | Subtraction from a constan... |
fprodsub2cncf 38792 | ` F ` is continuous. (Con... |
fprodadd2cncf 38793 | ` F ` is continuous. (Con... |
fprodsubrecnncnvlem 38794 | The sequence ` S ` of fini... |
fprodsubrecnncnv 38795 | The sequence ` S ` of fini... |
fprodaddrecnncnvlem 38796 | The sequence ` S ` of fini... |
fprodaddrecnncnv 38797 | The sequence ` S ` of fini... |
dvsinexp 38798 | The derivative of sin^N . ... |
dvcosre 38799 | The real derivative of the... |
dvrecg 38800 | Derivative of the reciproc... |
dvsinax 38801 | Derivative exercise: the d... |
dvsubf 38802 | The subtraction rule for e... |
dvmptconst 38803 | Function-builder for deriv... |
dvcnre 38804 | From compex differentiatio... |
dvmptidg 38805 | Function-builder for deriv... |
dvresntr 38806 | Function-builder for deriv... |
dvmptdiv 38807 | Function-builder for deriv... |
fperdvper 38808 | The derivative of a period... |
dvmptresicc 38809 | Derivative of a function r... |
dvasinbx 38810 | Derivative exercise: the d... |
dvresioo 38811 | Restriction of a derivativ... |
dvdivf 38812 | The quotient rule for ever... |
dvdivbd 38813 | A sufficient condition for... |
dvsubcncf 38814 | A sufficient condition for... |
dvmulcncf 38815 | A sufficient condition for... |
dvcosax 38816 | Derivative exercise: the d... |
dvdivcncf 38817 | A sufficient condition for... |
dvbdfbdioolem1 38818 | Given a function with boun... |
dvbdfbdioolem2 38819 | A function on an open inte... |
dvbdfbdioo 38820 | A function on an open inte... |
ioodvbdlimc1lem1 38821 | If ` F ` has bounded deriv... |
ioodvbdlimc1lem2 38822 | Limit at the lower bound o... |
ioodvbdlimc1 38823 | A real function with bound... |
ioodvbdlimc2lem 38824 | Limit at the upper bound o... |
ioodvbdlimc2 38825 | A real function with bound... |
dvdmsscn 38826 | ` X ` is a subset of ` CC ... |
dvmptmulf 38827 | Function-builder for deriv... |
dvnmptdivc 38828 | Function-builder for itera... |
dvdsn1add 38829 | If ` K ` divides ` N ` but... |
dvxpaek 38830 | Derivative of the polynomi... |
dvnmptconst 38831 | The ` N ` -th derivative o... |
dvnxpaek 38832 | The ` n ` -th derivative o... |
dvnmul 38833 | Function-builder for the `... |
dvmptfprodlem 38834 | Induction step for ~ dvmpt... |
dvmptfprod 38835 | Function-builder for deriv... |
dvnprodlem1 38836 | ` D ` is bijective. (Cont... |
dvnprodlem2 38837 | Induction step for ~ dvnpr... |
dvnprodlem3 38838 | The multinomial formula fo... |
dvnprod 38839 | The multinomial formula fo... |
volioo 38840 | The measure of an open int... |
itgsin0pilem1 38841 | Calculation of the integra... |
ibliccsinexp 38842 | sin^n on a closed interval... |
itgsin0pi 38843 | Calculation of the integra... |
iblioosinexp 38844 | sin^n on an open integral ... |
itgsinexplem1 38845 | Integration by parts is ap... |
itgsinexp 38846 | A recursive formula for th... |
iblconstmpt 38847 | A constant function is int... |
itgeq1d 38848 | Equality theorem for an in... |
mbf0 38849 | The empty set is a measura... |
mbfres2cn 38850 | Measurability of a piecewi... |
vol0 38851 | The measure of the empty s... |
ditgeqiooicc 38852 | A function ` F ` on an ope... |
volge0 38853 | The volume of a set is alw... |
cnbdibl 38854 | A continuous bounded funct... |
snmbl 38855 | A singleton is measurable.... |
ditgeq3d 38856 | Equality theorem for the d... |
iblempty 38857 | The empty function is inte... |
iblsplit 38858 | The union of two integrabl... |
volsn 38859 | A singleton has 0 Lebesgue... |
itgvol0 38860 | If the domani is negligibl... |
itgcoscmulx 38861 | Exercise: the integral of ... |
iblsplitf 38862 | A version of ~ iblsplit us... |
ibliooicc 38863 | If a function is integrabl... |
volioc 38864 | The measure of left open, ... |
iblspltprt 38865 | If a function is integrabl... |
itgsincmulx 38866 | Exercise: the integral of ... |
itgsubsticclem 38867 | lemma for ~ itgsubsticc . ... |
itgsubsticc 38868 | Integration by u-substitut... |
itgioocnicc 38869 | The integral of a piecewis... |
iblcncfioo 38870 | A continuous function ` F ... |
itgspltprt 38871 | The ` S. ` integral splits... |
itgiccshift 38872 | The integral of a function... |
itgperiod 38873 | The integral of a periodic... |
itgsbtaddcnst 38874 | Integral substitution, add... |
itgeq2d 38875 | Equality theorem for an in... |
volico 38876 | The measure of left closed... |
sublevolico 38877 | The Lebesgue measure of a ... |
dmvolss 38878 | Lebesgue measurable sets a... |
ismbl3 38879 | The predicate " ` A ` is L... |
volioof 38880 | The function that assigns ... |
ovolsplit 38881 | The Lebesgue outer measure... |
fvvolioof 38882 | The function value of the ... |
volioore 38883 | The measure of an open int... |
fvvolicof 38884 | The function value of the ... |
voliooico 38885 | An open interval and a lef... |
ismbl4 38886 | The predicate " ` A ` is L... |
volioofmpt 38887 | ` ( ( vol o. (,) ) o. F ) ... |
volicoff 38888 | ` ( ( vol o. [,) ) o. F ) ... |
voliooicof 38889 | The Lebesgue measure of op... |
volicofmpt 38890 | ` ( ( vol o. [,) ) o. F ) ... |
volicc 38891 | The Lebesgue measure of a ... |
voliccico 38892 | A closed interval and a le... |
mbfdmssre 38893 | The domain of a measurable... |
stoweidlem1 38894 | Lemma for ~ stoweid . Thi... |
stoweidlem2 38895 | lemma for ~ stoweid : here... |
stoweidlem3 38896 | Lemma for ~ stoweid : if `... |
stoweidlem4 38897 | Lemma for ~ stoweid : a cl... |
stoweidlem5 38898 | There exists a δ as ... |
stoweidlem6 38899 | Lemma for ~ stoweid : two ... |
stoweidlem7 38900 | This lemma is used to prov... |
stoweidlem8 38901 | Lemma for ~ stoweid : two ... |
stoweidlem9 38902 | Lemma for ~ stoweid : here... |
stoweidlem10 38903 | Lemma for ~ stoweid . Thi... |
stoweidlem11 38904 | This lemma is used to prov... |
stoweidlem12 38905 | Lemma for ~ stoweid . Thi... |
stoweidlem13 38906 | Lemma for ~ stoweid . Thi... |
stoweidlem14 38907 | There exists a ` k ` as in... |
stoweidlem15 38908 | This lemma is used to prov... |
stoweidlem16 38909 | Lemma for ~ stoweid . The... |
stoweidlem17 38910 | This lemma proves that the... |
stoweidlem18 38911 | This theorem proves Lemma ... |
stoweidlem19 38912 | If a set of real functions... |
stoweidlem20 38913 | If a set A of real functio... |
stoweidlem21 38914 | Once the Stone Weierstrass... |
stoweidlem22 38915 | If a set of real functions... |
stoweidlem23 38916 | This lemma is used to prov... |
stoweidlem24 38917 | This lemma proves that for... |
stoweidlem25 38918 | This lemma proves that for... |
stoweidlem26 38919 | This lemma is used to prov... |
stoweidlem27 38920 | This lemma is used to prov... |
stoweidlem28 38921 | There exists a δ as ... |
stoweidlem29 38922 | When the hypothesis for th... |
stoweidlem30 38923 | This lemma is used to prov... |
stoweidlem31 38924 | This lemma is used to prov... |
stoweidlem32 38925 | If a set A of real functio... |
stoweidlem33 38926 | If a set of real functions... |
stoweidlem34 38927 | This lemma proves that for... |
stoweidlem35 38928 | This lemma is used to prov... |
stoweidlem36 38929 | This lemma is used to prov... |
stoweidlem37 38930 | This lemma is used to prov... |
stoweidlem38 38931 | This lemma is used to prov... |
stoweidlem39 38932 | This lemma is used to prov... |
stoweidlem40 38933 | This lemma proves that q_n... |
stoweidlem41 38934 | This lemma is used to prov... |
stoweidlem42 38935 | This lemma is used to prov... |
stoweidlem43 38936 | This lemma is used to prov... |
stoweidlem44 38937 | This lemma is used to prov... |
stoweidlem45 38938 | This lemma proves that, gi... |
stoweidlem46 38939 | This lemma proves that set... |
stoweidlem47 38940 | Subtracting a constant fro... |
stoweidlem48 38941 | This lemma is used to prov... |
stoweidlem49 38942 | There exists a function q_... |
stoweidlem50 38943 | This lemma proves that set... |
stoweidlem51 38944 | There exists a function x ... |
stoweidlem52 38945 | There exists a neighborood... |
stoweidlem53 38946 | This lemma is used to prov... |
stoweidlem54 38947 | There exists a function ` ... |
stoweidlem55 38948 | This lemma proves the exis... |
stoweidlem56 38949 | This theorem proves Lemma ... |
stoweidlem57 38950 | There exists a function x ... |
stoweidlem58 38951 | This theorem proves Lemma ... |
stoweidlem59 38952 | This lemma proves that the... |
stoweidlem60 38953 | This lemma proves that the... |
stoweidlem61 38954 | This lemma proves that the... |
stoweidlem62 38955 | This theorem proves the St... |
stoweid 38956 | This theorem proves the St... |
stowei 38957 | This theorem proves the St... |
wallispilem1 38958 | ` I ` is monotone: increas... |
wallispilem2 38959 | A first set of properties ... |
wallispilem3 38960 | I maps to real values. (C... |
wallispilem4 38961 | ` F ` maps to explicit exp... |
wallispilem5 38962 | The sequence ` H ` converg... |
wallispi 38963 | Wallis' formula for π :... |
wallispi2lem1 38964 | An intermediate step betwe... |
wallispi2lem2 38965 | Two expressions are proven... |
wallispi2 38966 | An alternative version of ... |
stirlinglem1 38967 | A simple limit of fraction... |
stirlinglem2 38968 | ` A ` maps to positive rea... |
stirlinglem3 38969 | Long but simple algebraic ... |
stirlinglem4 38970 | Algebraic manipulation of ... |
stirlinglem5 38971 | If ` T ` is between ` 0 ` ... |
stirlinglem6 38972 | A series that converges to... |
stirlinglem7 38973 | Algebraic manipulation of ... |
stirlinglem8 38974 | If ` A ` converges to ` C ... |
stirlinglem9 38975 | ` ( ( B `` N ) - ( B `` ( ... |
stirlinglem10 38976 | A bound for any B(N)-B(N +... |
stirlinglem11 38977 | ` B ` is decreasing. (Con... |
stirlinglem12 38978 | The sequence ` B ` is boun... |
stirlinglem13 38979 | ` B ` is decreasing and ha... |
stirlinglem14 38980 | The sequence ` A ` converg... |
stirlinglem15 38981 | The Stirling's formula is ... |
stirling 38982 | Stirling's approximation f... |
stirlingr 38983 | Stirling's approximation f... |
dirkerval 38984 | The N_th Dirichlet Kernel.... |
dirker2re 38985 | The Dirchlet Kernel value ... |
dirkerdenne0 38986 | The Dirchlet Kernel denomi... |
dirkerval2 38987 | The N_th Dirichlet Kernel ... |
dirkerre 38988 | The Dirichlet Kernel at an... |
dirkerper 38989 | the Dirichlet Kernel has p... |
dirkerf 38990 | For any natural number ` N... |
dirkertrigeqlem1 38991 | Sum of an even number of a... |
dirkertrigeqlem2 38992 | Trigonomic equality lemma ... |
dirkertrigeqlem3 38993 | Trigonometric equality lem... |
dirkertrigeq 38994 | Trigonometric equality for... |
dirkeritg 38995 | The definite integral of t... |
dirkercncflem1 38996 | If ` Y ` is a multiple of ... |
dirkercncflem2 38997 | Lemma used to prove that t... |
dirkercncflem3 38998 | The Dirichlet Kernel is co... |
dirkercncflem4 38999 | The Dirichlet Kernel is co... |
dirkercncf 39000 | For any natural number ` N... |
fourierdlem1 39001 | A partition interval is a ... |
fourierdlem2 39002 | Membership in a partition.... |
fourierdlem3 39003 | Membership in a partition.... |
fourierdlem4 39004 | ` E ` is a function that m... |
fourierdlem5 39005 | ` S ` is a function. (Con... |
fourierdlem6 39006 | ` X ` is in the periodic p... |
fourierdlem7 39007 | The difference between a p... |
fourierdlem8 39008 | A partition interval is a ... |
fourierdlem9 39009 | ` H ` is a complex functio... |
fourierdlem10 39010 | Condition on the bounds of... |
fourierdlem11 39011 | If there is a partition, t... |
fourierdlem12 39012 | A point of a partition is ... |
fourierdlem13 39013 | Value of ` V ` in terms of... |
fourierdlem14 39014 | Given the partition ` V ` ... |
fourierdlem15 39015 | The range of the partition... |
fourierdlem16 39016 | The coefficients of the fo... |
fourierdlem17 39017 | The defined ` L ` is actua... |
fourierdlem18 39018 | The function ` S ` is cont... |
fourierdlem19 39019 | If two elements of ` D ` h... |
fourierdlem20 39020 | Every interval in the part... |
fourierdlem21 39021 | The coefficients of the fo... |
fourierdlem22 39022 | The coefficients of the fo... |
fourierdlem23 39023 | If ` F ` is continuous and... |
fourierdlem24 39024 | A sufficient condition for... |
fourierdlem25 39025 | If ` C ` is not in the ran... |
fourierdlem26 39026 | Periodic image of a point ... |
fourierdlem27 39027 | A partition open interval ... |
fourierdlem28 39028 | Derivative of ` ( F `` ( X... |
fourierdlem29 39029 | Explicit function value fo... |
fourierdlem30 39030 | Sum of three small pieces ... |
fourierdlem31 39031 | If ` A ` is finite and for... |
fourierdlem32 39032 | Limit of a continuous func... |
fourierdlem33 39033 | Limit of a continuous func... |
fourierdlem34 39034 | A partition is one to one.... |
fourierdlem35 39035 | There is a single point in... |
fourierdlem36 39036 | ` F ` is an isomorphism. ... |
fourierdlem37 39037 | ` I ` is a function that m... |
fourierdlem38 39038 | The function ` F ` is cont... |
fourierdlem39 39039 | Integration by parts of ... |
fourierdlem40 39040 | ` H ` is a continuous func... |
fourierdlem41 39041 | Lemma used to prove that e... |
fourierdlem42 39042 | The set of points in a mov... |
fourierdlem43 39043 | ` K ` is a real function. ... |
fourierdlem44 39044 | A condition for having ` (... |
fourierdlem46 39045 | The function ` F ` has a l... |
fourierdlem47 39046 | For ` r ` large enough, th... |
fourierdlem48 39047 | The given periodic functio... |
fourierdlem49 39048 | The given periodic functio... |
fourierdlem50 39049 | Continuity of ` O ` and it... |
fourierdlem51 39050 | ` X ` is in the periodic p... |
fourierdlem52 39051 | d16:d17,d18:jca |- ( ph ->... |
fourierdlem53 39052 | The limit of ` F ( s ) ` a... |
fourierdlem54 39053 | Given a partition ` Q ` an... |
fourierdlem55 39054 | ` U ` is a real function. ... |
fourierdlem56 39055 | Derivative of the ` K ` fu... |
fourierdlem57 39056 | The derivative of ` O ` . ... |
fourierdlem58 39057 | The derivative of ` K ` is... |
fourierdlem59 39058 | The derivative of ` H ` is... |
fourierdlem60 39059 | Given a differentiable fun... |
fourierdlem61 39060 | Given a differentiable fun... |
fourierdlem62 39061 | The function ` K ` is cont... |
fourierdlem63 39062 | The upper bound of interva... |
fourierdlem64 39063 | The partition ` V ` is fin... |
fourierdlem65 39064 | The distance of two adjace... |
fourierdlem66 39065 | Value of the ` G ` functio... |
fourierdlem67 39066 | ` G ` is a function. (Con... |
fourierdlem68 39067 | The derivative of ` O ` is... |
fourierdlem69 39068 | A piecewise continuous fun... |
fourierdlem70 39069 | A piecewise continuous fun... |
fourierdlem71 39070 | A periodic piecewise conti... |
fourierdlem72 39071 | The derivative of ` O ` is... |
fourierdlem73 39072 | A version of the Riemann L... |
fourierdlem74 39073 | Given a piecewise smooth f... |
fourierdlem75 39074 | Given a piecewise smooth f... |
fourierdlem76 39075 | Continuity of ` O ` and it... |
fourierdlem77 39076 | If ` H ` is bounded, then ... |
fourierdlem78 39077 | ` G ` is continuous when r... |
fourierdlem79 39078 | ` E ` projects every inter... |
fourierdlem80 39079 | The derivative of ` O ` is... |
fourierdlem81 39080 | The integral of a piecewis... |
fourierdlem82 39081 | Integral by substitution, ... |
fourierdlem83 39082 | The fourier partial sum fo... |
fourierdlem84 39083 | If ` F ` is piecewise coni... |
fourierdlem85 39084 | Limit of the function ` G ... |
fourierdlem86 39085 | Continuity of ` O ` and it... |
fourierdlem87 39086 | The integral of ` G ` goes... |
fourierdlem88 39087 | Given a piecewise continuo... |
fourierdlem89 39088 | Given a piecewise continuo... |
fourierdlem90 39089 | Given a piecewise continuo... |
fourierdlem91 39090 | Given a piecewise continuo... |
fourierdlem92 39091 | The integral of a piecewis... |
fourierdlem93 39092 | Integral by substitution (... |
fourierdlem94 39093 | For a piecewise smooth fun... |
fourierdlem95 39094 | Algebraic manipulation of ... |
fourierdlem96 39095 | limit for ` F ` at the low... |
fourierdlem97 39096 | ` F ` is continuous on the... |
fourierdlem98 39097 | ` F ` is continuous on the... |
fourierdlem99 39098 | limit for ` F ` at the upp... |
fourierdlem100 39099 | A piecewise continuous fun... |
fourierdlem101 39100 | Integral by substitution f... |
fourierdlem102 39101 | For a piecewise smooth fun... |
fourierdlem103 39102 | The half lower part of the... |
fourierdlem104 39103 | The half upper part of the... |
fourierdlem105 39104 | A piecewise continuous fun... |
fourierdlem106 39105 | For a piecewise smooth fun... |
fourierdlem107 39106 | The integral of a piecewis... |
fourierdlem108 39107 | The integral of a piecewis... |
fourierdlem109 39108 | The integral of a piecewis... |
fourierdlem110 39109 | The integral of a piecewis... |
fourierdlem111 39110 | The fourier partial sum fo... |
fourierdlem112 39111 | Here abbreviations (local ... |
fourierdlem113 39112 | Fourier series convergence... |
fourierdlem114 39113 | Fourier series convergence... |
fourierdlem115 39114 | Fourier serier convergence... |
fourierd 39115 | Fourier series convergence... |
fourierclimd 39116 | Fourier series convergence... |
fourierclim 39117 | Fourier series convergence... |
fourier 39118 | Fourier series convergence... |
fouriercnp 39119 | If ` F ` is continuous at ... |
fourier2 39120 | Fourier series convergence... |
sqwvfoura 39121 | Fourier coefficients for t... |
sqwvfourb 39122 | Fourier series ` B ` coeff... |
fourierswlem 39123 | The Fourier series for the... |
fouriersw 39124 | Fourier series convergence... |
fouriercn 39125 | If the derivative of ` F `... |
elaa2lem 39126 | Elementhood in the set of ... |
elaa2 39127 | Elementhood in the set of ... |
etransclem1 39128 | ` H ` is a function. (Con... |
etransclem2 39129 | Derivative of ` G ` . (Co... |
etransclem3 39130 | The given ` if ` term is a... |
etransclem4 39131 | ` F ` expressed as a finit... |
etransclem5 39132 | A change of bound variable... |
etransclem6 39133 | A change of bound variable... |
etransclem7 39134 | The given product is an in... |
etransclem8 39135 | ` F ` is a function. (Con... |
etransclem9 39136 | If ` K ` divides ` N ` but... |
etransclem10 39137 | The given ` if ` term is a... |
etransclem11 39138 | A change of bound variable... |
etransclem12 39139 | ` C ` applied to ` N ` . ... |
etransclem13 39140 | ` F ` applied to ` Y ` . ... |
etransclem14 39141 | Value of the term ` T ` , ... |
etransclem15 39142 | Value of the term ` T ` , ... |
etransclem16 39143 | Every element in the range... |
etransclem17 39144 | The ` N ` -th derivative o... |
etransclem18 39145 | The given function is inte... |
etransclem19 39146 | The ` N ` -th derivative o... |
etransclem20 39147 | ` H ` is smooth. (Contrib... |
etransclem21 39148 | The ` N ` -th derivative o... |
etransclem22 39149 | The ` N ` -th derivative o... |
etransclem23 39150 | This is the claim proof in... |
etransclem24 39151 | ` P ` divides the I -th de... |
etransclem25 39152 | ` P ` factorial divides th... |
etransclem26 39153 | Every term in the sum of t... |
etransclem27 39154 | The ` N ` -th derivative o... |
etransclem28 39155 | ` ( P - 1 ) ` factorial di... |
etransclem29 39156 | The ` N ` -th derivative o... |
etransclem30 39157 | The ` N ` -th derivative o... |
etransclem31 39158 | The ` N ` -th derivative o... |
etransclem32 39159 | This is the proof for the ... |
etransclem33 39160 | ` F ` is smooth. (Contrib... |
etransclem34 39161 | The ` N ` -th derivative o... |
etransclem35 39162 | ` P ` does not divide the ... |
etransclem36 39163 | The ` N ` -th derivative o... |
etransclem37 39164 | ` ( P - 1 ) ` factorial di... |
etransclem38 39165 | ` P ` divides the I -th de... |
etransclem39 39166 | ` G ` is a function. (Con... |
etransclem40 39167 | The ` N ` -th derivative o... |
etransclem41 39168 | ` P ` does not divide the ... |
etransclem42 39169 | The ` N ` -th derivative o... |
etransclem43 39170 | ` G ` is a continuous func... |
etransclem44 39171 | The given finite sum is no... |
etransclem45 39172 | ` K ` is an integer. (Con... |
etransclem46 39173 | This is the proof for equa... |
etransclem47 39174 | ` _e ` is transcendental. ... |
etransclem48 39175 | ` _e ` is transcendental. ... |
etransc 39176 | ` _e ` is transcendental. ... |
rrxtopn 39177 | The topology of the genera... |
rrxngp 39178 | Generalized Euclidean real... |
rrxbasefi 39179 | The base of the generalize... |
rrxtps 39180 | Generalized Euclidean real... |
rrxdsfi 39181 | The distance over generali... |
rrxtopnfi 39182 | The topology of the n-dime... |
rrxmetfi 39183 | Euclidean space is a metri... |
rrxtopon 39184 | The topology on Generalize... |
rrxtop 39185 | The topology on Generalize... |
rrndistlt 39186 | Given two points in the sp... |
rrxtoponfi 39187 | The topology on n-dimensio... |
rrxunitopnfi 39188 | The base set of the standa... |
rrxtopn0 39189 | The topology of the zero-d... |
qndenserrnbllem 39190 | n-dimensional rational num... |
qndenserrnbl 39191 | n-dimensional rational num... |
rrxtopn0b 39192 | The topology of the zero-d... |
qndenserrnopnlem 39193 | n-dimensional rational num... |
qndenserrnopn 39194 | n-dimensional rational num... |
qndenserrn 39195 | n-dimensional rational num... |
rrxsnicc 39196 | A multidimensional singlet... |
rrnprjdstle 39197 | The distance between two p... |
rrndsmet 39198 | ` D ` is a metric for the ... |
rrndsxmet 39199 | ` D ` is an extended metri... |
ioorrnopnlem 39200 | The a point in an indexed ... |
ioorrnopn 39201 | The indexed product of ope... |
ioorrnopnxrlem 39202 | Given a point ` F ` that b... |
ioorrnopnxr 39203 | The indexed product of ope... |
issal 39210 | Express the predicate " ` ... |
pwsal 39211 | The power set of a given s... |
salunicl 39212 | SAlg sigma-algebra is clos... |
saluncl 39213 | The union of two sets in a... |
prsal 39214 | The pair of the empty set ... |
saldifcl 39215 | The complement of an eleme... |
0sal 39216 | The empty set belongs to e... |
salgenval 39217 | The sigma-algebra generate... |
saliuncl 39218 | SAlg sigma-algebra is clos... |
salincl 39219 | The intersection of two se... |
saluni 39220 | A set is an element of any... |
saliincl 39221 | SAlg sigma-algebra is clos... |
saldifcl2 39222 | The difference of two elem... |
intsaluni 39223 | The union of an arbitrary ... |
intsal 39224 | The arbitrary intersection... |
salgenn0 39225 | The set used in the defini... |
salgencl 39226 | ` SalGen ` actually genera... |
issald 39227 | Sufficient condition to pr... |
salexct 39228 | An example of non trivial ... |
sssalgen 39229 | A set is a subset of the s... |
salgenss 39230 | The sigma-algebra generate... |
salgenuni 39231 | The base set of the sigma-... |
issalgend 39232 | One side of ~ dfsalgen2 . ... |
salexct2 39233 | An example of a subset tha... |
unisalgen 39234 | The union of a set belongs... |
dfsalgen2 39235 | Alternate characterization... |
salexct3 39236 | An example of a sigma-alge... |
salgencntex 39237 | This counterexample shows ... |
salgensscntex 39238 | This counterexample shows ... |
issalnnd 39239 | Sufficient condition to pr... |
dmvolsal 39240 | Lebesgue measurable sets f... |
saldifcld 39241 | The complement of an eleme... |
saluncld 39242 | The union of two sets in a... |
salgencld 39243 | ` SalGen ` actually genera... |
0sald 39244 | The empty set belongs to e... |
iooborel 39245 | An open interval is a Bore... |
salincld 39246 | The intersection of two se... |
salunid 39247 | A set is an element of any... |
unisalgen2 39248 | The union of a set belongs... |
bor1sal 39249 | The Borel sigma-algebra on... |
iocborel 39250 | A left-open, right-closed ... |
subsaliuncllem 39251 | A subspace sigma-algebra i... |
subsaliuncl 39252 | A subspace sigma-algebra i... |
subsalsal 39253 | A subspace sigma-algebra i... |
subsaluni 39254 | A set belongs to the subsp... |
sge0rnre 39257 | When ` sum^ ` is applied t... |
fge0icoicc 39258 | If ` F ` maps to nonnegati... |
sge0val 39259 | The value of the sum of no... |
fge0npnf 39260 | If ` F ` maps to nonnegati... |
sge0rnn0 39261 | The range used in the defi... |
sge0vald 39262 | The value of the sum of no... |
fge0iccico 39263 | A range of nonnegative ext... |
gsumge0cl 39264 | Closure of group sum, for ... |
sge0reval 39265 | Value of the sum of nonneg... |
sge0pnfval 39266 | If a term in the sum of no... |
fge0iccre 39267 | A range of nonnegative ext... |
sge0z 39268 | Any nonnegative extended s... |
sge00 39269 | The sum of nonnegative ext... |
fsumlesge0 39270 | Every finite subsum of non... |
sge0revalmpt 39271 | Value of the sum of nonneg... |
sge0sn 39272 | A sum of a nonnegative ext... |
sge0tsms 39273 | ` sum^ ` applied to a nonn... |
sge0cl 39274 | The arbitrary sum of nonne... |
sge0f1o 39275 | Re-index a nonnegative ext... |
sge0snmpt 39276 | A sum of a nonnegative ext... |
sge0ge0 39277 | The sum of nonnegative ext... |
sge0xrcl 39278 | The arbitrary sum of nonne... |
sge0repnf 39279 | The of nonnegative extende... |
sge0fsum 39280 | The arbitrary sum of a fin... |
sge0rern 39281 | If the sum of nonnegative ... |
sge0supre 39282 | If the arbitrary sum of no... |
sge0fsummpt 39283 | The arbitrary sum of a fin... |
sge0sup 39284 | The arbitrary sum of nonne... |
sge0less 39285 | A shorter sum of nonnegati... |
sge0rnbnd 39286 | The range used in the defi... |
sge0pr 39287 | Sum of a pair of nonnegati... |
sge0gerp 39288 | The arbitrary sum of nonne... |
sge0pnffigt 39289 | If the sum of nonnegative ... |
sge0ssre 39290 | If a sum of nonnegative ex... |
sge0lefi 39291 | A sum of nonnegative exten... |
sge0lessmpt 39292 | A shorter sum of nonnegati... |
sge0ltfirp 39293 | If the sum of nonnegative ... |
sge0prle 39294 | The sum of a pair of nonne... |
sge0gerpmpt 39295 | The arbitrary sum of nonne... |
sge0resrnlem 39296 | The sum of nonnegative ext... |
sge0resrn 39297 | The sum of nonnegative ext... |
sge0ssrempt 39298 | If a sum of nonnegative ex... |
sge0resplit 39299 | ` sum^ ` splits into two p... |
sge0le 39300 | If all of the terms of sum... |
sge0ltfirpmpt 39301 | If the extended sum of non... |
sge0split 39302 | Split a sum of nonnegative... |
sge0lempt 39303 | If all of the terms of sum... |
sge0splitmpt 39304 | Split a sum of nonnegative... |
sge0ss 39305 | Change the index set to a ... |
sge0iunmptlemfi 39306 | Sum of nonnegative extende... |
sge0p1 39307 | The addition of the next t... |
sge0iunmptlemre 39308 | Sum of nonnegative extende... |
sge0fodjrnlem 39309 | Re-index a nonnegative ext... |
sge0fodjrn 39310 | Re-index a nonnegative ext... |
sge0iunmpt 39311 | Sum of nonnegative extende... |
sge0iun 39312 | Sum of nonnegative extende... |
sge0nemnf 39313 | The generalized sum of non... |
sge0rpcpnf 39314 | The sum of an infinite num... |
sge0rernmpt 39315 | If the sum of nonnegative ... |
sge0lefimpt 39316 | A sum of nonnegative exten... |
nn0ssge0 39317 | Nonnegative integers are n... |
sge0clmpt 39318 | The generalized sum of non... |
sge0ltfirpmpt2 39319 | If the extended sum of non... |
sge0isum 39320 | If a series of nonnegative... |
sge0xrclmpt 39321 | The generalized sum of non... |
sge0xp 39322 | Combine two generalized su... |
sge0isummpt 39323 | If a series of nonnegative... |
sge0ad2en 39324 | The value of the infinite ... |
sge0isummpt2 39325 | If a series of nonnegative... |
sge0xaddlem1 39326 | The extended addition of t... |
sge0xaddlem2 39327 | The extended addition of t... |
sge0xadd 39328 | The extended addition of t... |
sge0fsummptf 39329 | The generalized sum of a f... |
sge0snmptf 39330 | A sum of a nonnegative ext... |
sge0ge0mpt 39331 | The sum of nonnegative ext... |
sge0repnfmpt 39332 | The of nonnegative extende... |
sge0pnffigtmpt 39333 | If the generalized sum of ... |
sge0splitsn 39334 | Separate out a term in a g... |
sge0pnffsumgt 39335 | If the sum of nonnegative ... |
sge0gtfsumgt 39336 | If the generalized sum of ... |
sge0uzfsumgt 39337 | If a real number is smalle... |
sge0pnfmpt 39338 | If a term in the sum of no... |
sge0seq 39339 | A series of nonnegative re... |
sge0reuz 39340 | Value of the generalized s... |
sge0reuzb 39341 | Value of the generalized s... |
ismea 39344 | Express the predicate " ` ... |
dmmeasal 39345 | The domain of a measure is... |
meaf 39346 | A measure is a function th... |
mea0 39347 | The measure of the empty s... |
nnfoctbdjlem 39348 | There exists a mapping fro... |
nnfoctbdj 39349 | There exists a mapping fro... |
meadjuni 39350 | The measure of the disjoin... |
meacl 39351 | The measure of a set is a ... |
iundjiunlem 39352 | The sets in the sequence `... |
iundjiun 39353 | Given a sequence ` E ` of ... |
meaxrcl 39354 | The measure of a set is an... |
meadjun 39355 | The measure of the union o... |
meassle 39356 | The measure of a set is la... |
meaunle 39357 | The measure of the union o... |
meadjiunlem 39358 | The sum of nonnegative ext... |
meadjiun 39359 | The measure of the disjoin... |
ismeannd 39360 | Sufficient condition to pr... |
meaiunlelem 39361 | The measure of the union o... |
meaiunle 39362 | The measure of the union o... |
psmeasurelem 39363 | ` M ` applied to a disjoin... |
psmeasure 39364 | Point supported measure, R... |
voliunsge0lem 39365 | The Lebesgue measure funct... |
voliunsge0 39366 | The Lebesgue measure funct... |
volmea 39367 | The Lebeasgue measure on t... |
meage0 39368 | If the measure of a measur... |
meadjunre 39369 | The measure of the union o... |
meassre 39370 | If the measure of a measur... |
meale0eq0 39371 | A measure that is smaller ... |
meadif 39372 | The measure of the differe... |
meaiuninclem 39373 | Measures are continuous fr... |
meaiuninc 39374 | Measures are continuous fr... |
meaiuninc2 39375 | Measures are continuous fr... |
meaiininclem 39376 | Measures are continuous fr... |
meaiininc 39377 | Measures are continuous fr... |
meaiininc2 39378 | Measures are continuous fr... |
caragenval 39383 | The sigma-algebra generate... |
isome 39384 | Express the predicate " ` ... |
caragenel 39385 | Membership in the Caratheo... |
omef 39386 | An outer measure is a func... |
ome0 39387 | The outer measure of the e... |
omessle 39388 | The outer measure of a set... |
omedm 39389 | The domain of an outer mea... |
caragensplit 39390 | If ` E ` is in the set gen... |
caragenelss 39391 | An element of the Caratheo... |
carageneld 39392 | Membership in the Caratheo... |
omecl 39393 | The outer measure of a set... |
caragenss 39394 | The sigma-algebra generate... |
omeunile 39395 | The outer measure of the u... |
caragen0 39396 | The empty set belongs to a... |
omexrcl 39397 | The outer measure of a set... |
caragenunidm 39398 | The base set of an outer m... |
caragensspw 39399 | The sigma-algebra generate... |
omessre 39400 | If the outer measure of a ... |
caragenuni 39401 | The base set of the sigma-... |
caragenuncllem 39402 | The Caratheodory's constru... |
caragenuncl 39403 | The Caratheodory's constru... |
caragendifcl 39404 | The Caratheodory's constru... |
caragenfiiuncl 39405 | The Caratheodory's constru... |
omeunle 39406 | The outer measure of the u... |
omeiunle 39407 | The outer measure of the i... |
omelesplit 39408 | The outer measure of a set... |
omeiunltfirp 39409 | If the outer measure of a ... |
omeiunlempt 39410 | The outer measure of the i... |
carageniuncllem1 39411 | The outer measure of ` A i... |
carageniuncllem2 39412 | The Caratheodory's constru... |
carageniuncl 39413 | The Caratheodory's constru... |
caragenunicl 39414 | The Caratheodory's constru... |
caragensal 39415 | Caratheodory's method gene... |
caratheodorylem1 39416 | Lemma used to prove that C... |
caratheodorylem2 39417 | Caratheodory's constructio... |
caratheodory 39418 | Caratheodory's constructio... |
0ome 39419 | The map that assigns 0 to ... |
isomenndlem 39420 | ` O ` is sub-additive w.r.... |
isomennd 39421 | Sufficient condition to pr... |
caragenel2d 39422 | Membership in the Caratheo... |
omege0 39423 | If the outer measure of a ... |
omess0 39424 | If the outer measure of a ... |
caragencmpl 39425 | A measure built with the C... |
vonval 39430 | Value of the Lebesgue meas... |
ovnval 39431 | Value of the Lebesgue oute... |
elhoi 39432 | Membership in a multidimen... |
icoresmbl 39433 | A closed-below, open-above... |
hoissre 39434 | The projection of a half-o... |
ovnval2 39435 | Value of the Lebesgue oute... |
volicorecl 39436 | The Lebesgue measure of a ... |
hoiprodcl 39437 | The pre-measure of half-op... |
hoicvr 39438 | ` I ` is a countable set o... |
hoissrrn 39439 | A half-open interval is a ... |
ovn0val 39440 | The Lebesgue outer measure... |
ovnn0val 39441 | The value of a (multidimen... |
ovnval2b 39442 | Value of the Lebesgue oute... |
volicorescl 39443 | The Lebesgue measure of a ... |
ovnprodcl 39444 | The product used in the de... |
hoiprodcl2 39445 | The pre-measure of half-op... |
hoicvrrex 39446 | Any subset of the multidim... |
ovnsupge0 39447 | The set used in the defini... |
ovnlecvr 39448 | Given a subset of multidim... |
ovnpnfelsup 39449 | ` +oo ` is an element of t... |
ovnsslelem 39450 | The (multidimensional, non... |
ovnssle 39451 | The (multidimensional) Leb... |
ovnlerp 39452 | The Lebesgue outer measure... |
ovnf 39453 | The Lebesgue outer measure... |
ovncvrrp 39454 | The Lebesgue outer measure... |
ovn0lem 39455 | For any finite dimension, ... |
ovn0 39456 | For any finite dimension, ... |
ovncl 39457 | The Lebesgue outer measure... |
ovn02 39458 | For the zero-dimensional s... |
ovnxrcl 39459 | The Lebesgue outer measure... |
ovnsubaddlem1 39460 | The Lebesgue outer measure... |
ovnsubaddlem2 39461 | ` ( voln* `` X ) ` is suba... |
ovnsubadd 39462 | ` ( voln* `` X ) ` is suba... |
ovnome 39463 | ` ( voln* `` X ) ` is an o... |
vonmea 39464 | ` ( voln `` X ) ` is a mea... |
volicon0 39465 | The measure of a nonempty ... |
hsphoif 39466 | ` H ` is a function (that ... |
hoidmvval 39467 | The dimensional volume of ... |
hoissrrn2 39468 | A half-open interval is a ... |
hsphoival 39469 | ` H ` is a function (that ... |
hoiprodcl3 39470 | The pre-measure of half-op... |
volicore 39471 | The Lebesgue measure of a ... |
hoidmvcl 39472 | The dimensional volume of ... |
hoidmv0val 39473 | The dimensional volume of ... |
hoidmvn0val 39474 | The dimensional volume of ... |
hsphoidmvle2 39475 | The dimensional volume of ... |
hsphoidmvle 39476 | The dimensional volume of ... |
hoidmvval0 39477 | The dimensional volume of ... |
hoiprodp1 39478 | The dimensional volume of ... |
sge0hsphoire 39479 | If the generalized sum of ... |
hoidmvval0b 39480 | The dimensional volume of ... |
hoidmv1lelem1 39481 | The supremum of ` U ` belo... |
hoidmv1lelem2 39482 | This is the contradiction ... |
hoidmv1lelem3 39483 | The dimensional volume of ... |
hoidmv1le 39484 | The dimensional volume of ... |
hoidmvlelem1 39485 | The supremum of ` U ` belo... |
hoidmvlelem2 39486 | This is the contradiction ... |
hoidmvlelem3 39487 | This is the contradiction ... |
hoidmvlelem4 39488 | The dimensional volume of ... |
hoidmvlelem5 39489 | The dimensional volume of ... |
hoidmvle 39490 | The dimensional volume of ... |
ovnhoilem1 39491 | The Lebesgue outer measure... |
ovnhoilem2 39492 | The Lebesgue outer measure... |
ovnhoi 39493 | The Lebesgue outer measure... |
dmovn 39494 | The domain of the Lebesgue... |
hoicoto2 39495 | The half-open interval exp... |
dmvon 39496 | Lebesgue measurable n-dime... |
hoi2toco 39497 | The half-open interval exp... |
hoidifhspval 39498 | ` D ` is a function that r... |
hspval 39499 | The value of the half-spac... |
ovnlecvr2 39500 | Given a subset of multidim... |
ovncvr2 39501 | ` B ` and ` T ` are the le... |
dmovnsal 39502 | The domain of the Lebesgue... |
unidmovn 39503 | Base set of the n-dimensio... |
rrnmbl 39504 | The set of n-dimensional R... |
hoidifhspval2 39505 | ` D ` is a function that r... |
hspdifhsp 39506 | A n-dimensional half-open ... |
unidmvon 39507 | Base set of the n-dimensio... |
hoidifhspf 39508 | ` D ` is a function that r... |
hoidifhspval3 39509 | ` D ` is a function that r... |
hoidifhspdmvle 39510 | The dimensional volume of ... |
voncmpl 39511 | The Lebesgue measure is co... |
hoiqssbllem1 39512 | The center of the n-dimens... |
hoiqssbllem2 39513 | The center of the n-dimens... |
hoiqssbllem3 39514 | A n-dimensional ball conta... |
hoiqssbl 39515 | A n-dimensional ball conta... |
hspmbllem1 39516 | Any half-space of the n-di... |
hspmbllem2 39517 | Any half-space of the n-di... |
hspmbllem3 39518 | Any half-space of the n-di... |
hspmbl 39519 | Any half-space of the n-di... |
hoimbllem 39520 | Any n-dimensional half-ope... |
hoimbl 39521 | Any n-dimensional half-ope... |
opnvonmbllem1 39522 | The half-open interval exp... |
opnvonmbllem2 39523 | An open subset of the n-di... |
opnvonmbl 39524 | An open subset of the n-di... |
opnssborel 39525 | Open sets of a generalized... |
borelmbl 39526 | All Borel subsets of the n... |
volicorege0 39527 | The Lebesgue measure of a ... |
isvonmbl 39528 | The predicate " ` A ` is m... |
mblvon 39529 | The n-dimensional Lebesgue... |
vonmblss 39530 | n-dimensional Lebesgue mea... |
volico2 39531 | The measure of left closed... |
vonmblss2 39532 | n-dimensional Lebesgue mea... |
ovolval2lem 39533 | The value of the Lebesgue ... |
ovolval2 39534 | The value of the Lebesgue ... |
ovnsubadd2lem 39535 | ` ( voln* `` X ) ` is suba... |
ovnsubadd2 39536 | ` ( voln* `` X ) ` is suba... |
ovolval3 39537 | The value of the Lebesgue ... |
ovnsplit 39538 | The n-dimensional Lebesgue... |
ovolval4lem1 39539 | |- ( ( ph /\ n e. A ) -> ... |
ovolval4lem2 39540 | The value of the Lebesgue ... |
ovolval4 39541 | The value of the Lebesgue ... |
ovolval5lem1 39542 | |- ( ph -> ( sum^ ` ( n e.... |
ovolval5lem2 39543 | |- ( ( ph /\ n e. NN ) ->... |
ovolval5lem3 39544 | The value of the Lebesgue ... |
ovolval5 39545 | The value of the Lebesgue ... |
ovnovollem1 39546 | if ` F ` is a cover of ` B... |
ovnovollem2 39547 | if ` I ` is a cover of ` (... |
ovnovollem3 39548 | The 1-dimensional Lebesgue... |
ovnovol 39549 | The 1-dimensional Lebesgue... |
vonvolmbllem 39550 | If a subset ` B ` of real ... |
vonvolmbl 39551 | A subset of Real numbers i... |
vonvol 39552 | The 1-dimensional Lebesgue... |
vonvolmbl2 39553 | A subset ` X ` of the spac... |
vonvol2 39554 | The 1-dimensional Lebesgue... |
hoimbl2 39555 | Any n-dimensional half-ope... |
voncl 39556 | The Lebesgue measure of a ... |
vonhoi 39557 | The Lebesgue outer measure... |
vonxrcl 39558 | The Lebesgue measure of a ... |
vonval2 39559 | Value of the Lebesgue meas... |
ioosshoi 39560 | A n-dimensional open inter... |
vonn0hoi 39561 | The Lebesgue outer measure... |
von0val 39562 | The Lebesgue measure (for ... |
vonhoire 39563 | The Lebesgue measure of a ... |
iinhoiicclem 39564 | A n-dimensional closed int... |
iinhoiicc 39565 | A n-dimensional closed int... |
iunhoiioolem 39566 | A n-dimensional open inter... |
iunhoiioo 39567 | A n-dimensional open inter... |
ioovonmbl 39568 | Any n-dimensional open int... |
iccvonmbllem 39569 | Any n-dimensional closed i... |
iccvonmbl 39570 | Any n-dimensional closed i... |
vonioolem1 39571 | The sequence of the measur... |
vonioolem2 39572 | The n-dimensional Lebesgue... |
vonioo 39573 | The n-dimensional Lebesgue... |
vonicclem1 39574 | The sequence of the measur... |
vonicclem2 39575 | The n-dimensional Lebesgue... |
vonicc 39576 | The n-dimensional Lebesgue... |
snvonmbl 39577 | A n-dimensional singleton ... |
vonn0ioo 39578 | The n-dimensional Lebesgue... |
vonn0icc 39579 | The n-dimensional Lebesgue... |
ctvonmbl 39580 | Any n-dimensional countabl... |
vonn0ioo2 39581 | The n-dimensional Lebesgue... |
vonsn 39582 | The n-dimensional Lebesgue... |
vonn0icc2 39583 | The n-dimensional Lebesgue... |
vonct 39584 | The n-dimensional Lebesgue... |
vitali2 39585 | There are non-measurable s... |
pimltmnf2 39588 | Given a real valued functi... |
preimagelt 39589 | The preimage of a right-op... |
preimalegt 39590 | The preimage of a left-ope... |
pimconstlt0 39591 | Given a constant function,... |
pimconstlt1 39592 | Given a constant function,... |
pimltpnf 39593 | Given a real valued functi... |
pimgtpnf2 39594 | Given a real valued functi... |
salpreimagelt 39595 | If all the preimages of le... |
pimrecltpos 39596 | The preimage of an unbound... |
salpreimalegt 39597 | If all the preimages of ri... |
pimiooltgt 39598 | The preimage of an open in... |
preimaicomnf 39599 | Preimage of an open interv... |
pimltpnf2 39600 | Given a real valued functi... |
pimgtmnf2 39601 | Given a real valued functi... |
pimdecfgtioc 39602 | Given a non-increasing fun... |
pimincfltioc 39603 | Given a non decreasing fun... |
pimdecfgtioo 39604 | Given a non decreasing fun... |
pimincfltioo 39605 | Given a non decreasing fun... |
preimaioomnf 39606 | Preimage of an open interv... |
preimageiingt 39607 | A preimage of a left-close... |
preimaleiinlt 39608 | A preimage of a left-open,... |
pimgtmnf 39609 | Given a real valued functi... |
pimrecltneg 39610 | The preimage of an unbound... |
salpreimagtge 39611 | If all the preimages of le... |
salpreimaltle 39612 | If all the preimages of ri... |
issmflem 39613 | The predicate " ` F ` is a... |
issmf 39614 | The predicate " ` F ` is a... |
salpreimalelt 39615 | If all the preimages of ri... |
salpreimagtlt 39616 | If all the preimages of le... |
smfpreimalt 39617 | Given a function measurabl... |
smff 39618 | A function measurable w.r.... |
smfdmss 39619 | The domain of a function m... |
issmff 39620 | The predicate " ` F ` is a... |
issmfd 39621 | A sufficient condition for... |
issmfltle 39622 | The definition of a measur... |
smfpreimaltf 39623 | Given a function measurabl... |
issmfdf 39624 | A sufficient condition for... |
sssmf 39625 | The restriction of a sigma... |
mbfresmf 39626 | A Real valued, measurable ... |
cnfsmf 39627 | A continuous function is m... |
incsmflem 39628 | A non decreasing function ... |
incsmf 39629 | A real valued, non-decreas... |
smfsssmf 39630 | If a function is measurabl... |
issmflelem 39631 | The predicate " ` F ` is a... |
issmfle 39632 | The predicate " ` F ` is b... |
smfpimltmpt 39633 | Given a function measurabl... |
smfpimltxr 39634 | Given a function measurabl... |
issmfdmpt 39635 | A sufficient condition for... |
smfconst 39636 | A constant function is mea... |
sssmfmpt 39637 | The restriction of a sigma... |
cnfrrnsmf 39638 | A function, continuous fro... |
smfid 39639 | The identity function is B... |
bormflebmf 39640 | A Borel measurable functio... |
smfpreimale 39641 | Given a function measurabl... |
issmfgtlem 39642 | The predicate " ` F ` is a... |
issmfgt 39643 | The predicate " ` F ` is b... |
issmfled 39644 | A sufficient condition for... |
smfpimltxrmpt 39645 | Given a function measurabl... |
smfmbfcex 39646 | A constant function, with ... |
issmfgtd 39647 | A sufficient condition for... |
smfpreimagt 39648 | Given a function measurabl... |
smfaddlem1 39649 | Given the sum of two funct... |
smfaddlem2 39650 | The sum of two sigma-measu... |
smfadd 39651 | The sum of two sigma-measu... |
decsmflem 39652 | A non-increasing function ... |
decsmf 39653 | A real valued, non-increas... |
smfpreimagtf 39654 | Given a function measurabl... |
issmfgelem 39655 | The predicate " ` F ` is a... |
issmfge 39656 | The predicate " ` F ` is b... |
smflimlem1 39657 | Lemma for the proof that t... |
smflimlem2 39658 | Lemma for the proof that t... |
smflimlem3 39659 | The limit of sigma-measura... |
smflimlem4 39660 | Lemma for the proof that t... |
smflimlem5 39661 | Lemma for the proof that t... |
smflimlem6 39662 | Lemma for the proof that t... |
smflim 39663 | The limit of sigma-measura... |
nsssmfmbflem 39664 | The sigma-measurable funct... |
nsssmfmbf 39665 | The sigma-measurable funct... |
smfpimgtxr 39666 | Given a function measurabl... |
smfpimgtmpt 39667 | Given a function measurabl... |
smfpreimage 39668 | Given a function measurabl... |
mbfpsssmf 39669 | Real valued, measurable fu... |
smfpimgtxrmpt 39670 | Given a function measurabl... |
smfpimioompt 39671 | Given a function measurabl... |
smfpimioo 39672 | Given a function measurabl... |
smfresal 39673 | Given a sigma-measurable f... |
smfrec 39674 | The reciprocal of a sigma-... |
smfres 39675 | The restriction of sigma-m... |
smfmullem1 39676 | The multiplication of two ... |
smfmullem2 39677 | The multiplication of two ... |
smfmullem3 39678 | The multiplication of two ... |
smfmullem4 39679 | The multiplication of two ... |
smfmul 39680 | The multiplication of two ... |
smfmulc1 39681 | A sigma-measurable functio... |
smfdiv 39682 | The fraction of two sigma-... |
smfpimbor1lem1 39683 | Every open set belongs to ... |
smfpimbor1lem2 39684 | Given D sigma-measurable f... |
smfpimbor1 39685 | Given a sigma-measurable f... |
smf2id 39686 | Twice the identity functio... |
smfco 39687 | The composition of a Borel... |
sigarval 39688 | Define the signed area by ... |
sigarim 39689 | Signed area takes value in... |
sigarac 39690 | Signed area is anticommuta... |
sigaraf 39691 | Signed area is additive by... |
sigarmf 39692 | Signed area is additive (w... |
sigaras 39693 | Signed area is additive by... |
sigarms 39694 | Signed area is additive (w... |
sigarls 39695 | Signed area is linear by t... |
sigarid 39696 | Signed area of a flat para... |
sigarexp 39697 | Expand the signed area for... |
sigarperm 39698 | Signed area ` ( A - C ) G ... |
sigardiv 39699 | If signed area between vec... |
sigarimcd 39700 | Signed area takes value in... |
sigariz 39701 | If signed area is zero, th... |
sigarcol 39702 | Given three points ` A ` ,... |
sharhght 39703 | Let ` A B C ` be a triangl... |
sigaradd 39704 | Subtracting (double) area ... |
cevathlem1 39705 | Ceva's theorem first lemma... |
cevathlem2 39706 | Ceva's theorem second lemm... |
cevath 39707 | Ceva's theorem. Let ` A B... |
hirstL-ax3 39708 | The third axiom of a syste... |
ax3h 39709 | Recovery of ~ ax-3 from ~ ... |
aibandbiaiffaiffb 39710 | A closed form showing (a i... |
aibandbiaiaiffb 39711 | A closed form showing (a i... |
notatnand 39712 | Do not use. Use intnanr i... |
aistia 39713 | Given a is equivalent to `... |
aisfina 39714 | Given a is equivalent to `... |
bothtbothsame 39715 | Given both a, b are equiva... |
bothfbothsame 39716 | Given both a, b are equiva... |
aiffbbtat 39717 | Given a is equivalent to b... |
aisbbisfaisf 39718 | Given a is equivalent to b... |
axorbtnotaiffb 39719 | Given a is exclusive to b,... |
aiffnbandciffatnotciffb 39720 | Given a is equivalent to (... |
axorbciffatcxorb 39721 | Given a is equivalent to (... |
aibnbna 39722 | Given a implies b, (not b)... |
aibnbaif 39723 | Given a implies b, not b, ... |
aiffbtbat 39724 | Given a is equivalent to b... |
astbstanbst 39725 | Given a is equivalent to T... |
aistbistaandb 39726 | Given a is equivalent to T... |
aisbnaxb 39727 | Given a is equivalent to b... |
atbiffatnnb 39728 | If a implies b, then a imp... |
bisaiaisb 39729 | Application of bicom1 with... |
atbiffatnnbalt 39730 | If a implies b, then a imp... |
abnotbtaxb 39731 | Assuming a, not b, there e... |
abnotataxb 39732 | Assuming not a, b, there e... |
conimpf 39733 | Assuming a, not b, and a i... |
conimpfalt 39734 | Assuming a, not b, and a i... |
aistbisfiaxb 39735 | Given a is equivalent to T... |
aisfbistiaxb 39736 | Given a is equivalent to F... |
aifftbifffaibif 39737 | Given a is equivalent to T... |
aifftbifffaibifff 39738 | Given a is equivalent to T... |
atnaiana 39739 | Given a, it is not the cas... |
ainaiaandna 39740 | Given a, a implies it is n... |
abcdta 39741 | Given (((a and b) and c) a... |
abcdtb 39742 | Given (((a and b) and c) a... |
abcdtc 39743 | Given (((a and b) and c) a... |
abcdtd 39744 | Given (((a and b) and c) a... |
abciffcbatnabciffncba 39745 | Operands in a biconditiona... |
abciffcbatnabciffncbai 39746 | Operands in a biconditiona... |
nabctnabc 39747 | not ( a -> ( b /\ c ) ) we... |
jabtaib 39748 | For when pm3.4 lacks a pm3... |
onenotinotbothi 39749 | From one negated implicati... |
twonotinotbothi 39750 | From these two negated imp... |
clifte 39751 | show d is the same as an i... |
cliftet 39752 | show d is the same as an i... |
clifteta 39753 | show d is the same as an i... |
cliftetb 39754 | show d is the same as an i... |
confun 39755 | Given the hypotheses there... |
confun2 39756 | Confun simplified to two p... |
confun3 39757 | Confun's more complex form... |
confun4 39758 | An attempt at derivative. ... |
confun5 39759 | An attempt at derivative. ... |
plcofph 39760 | Given, a,b and a "definiti... |
pldofph 39761 | Given, a,b c, d, "definiti... |
plvcofph 39762 | Given, a,b,d, and "definit... |
plvcofphax 39763 | Given, a,b,d, and "definit... |
plvofpos 39764 | rh is derivable because ON... |
mdandyv0 39765 | Given the equivalences set... |
mdandyv1 39766 | Given the equivalences set... |
mdandyv2 39767 | Given the equivalences set... |
mdandyv3 39768 | Given the equivalences set... |
mdandyv4 39769 | Given the equivalences set... |
mdandyv5 39770 | Given the equivalences set... |
mdandyv6 39771 | Given the equivalences set... |
mdandyv7 39772 | Given the equivalences set... |
mdandyv8 39773 | Given the equivalences set... |
mdandyv9 39774 | Given the equivalences set... |
mdandyv10 39775 | Given the equivalences set... |
mdandyv11 39776 | Given the equivalences set... |
mdandyv12 39777 | Given the equivalences set... |
mdandyv13 39778 | Given the equivalences set... |
mdandyv14 39779 | Given the equivalences set... |
mdandyv15 39780 | Given the equivalences set... |
mdandyvr0 39781 | Given the equivalences set... |
mdandyvr1 39782 | Given the equivalences set... |
mdandyvr2 39783 | Given the equivalences set... |
mdandyvr3 39784 | Given the equivalences set... |
mdandyvr4 39785 | Given the equivalences set... |
mdandyvr5 39786 | Given the equivalences set... |
mdandyvr6 39787 | Given the equivalences set... |
mdandyvr7 39788 | Given the equivalences set... |
mdandyvr8 39789 | Given the equivalences set... |
mdandyvr9 39790 | Given the equivalences set... |
mdandyvr10 39791 | Given the equivalences set... |
mdandyvr11 39792 | Given the equivalences set... |
mdandyvr12 39793 | Given the equivalences set... |
mdandyvr13 39794 | Given the equivalences set... |
mdandyvr14 39795 | Given the equivalences set... |
mdandyvr15 39796 | Given the equivalences set... |
mdandyvrx0 39797 | Given the exclusivities se... |
mdandyvrx1 39798 | Given the exclusivities se... |
mdandyvrx2 39799 | Given the exclusivities se... |
mdandyvrx3 39800 | Given the exclusivities se... |
mdandyvrx4 39801 | Given the exclusivities se... |
mdandyvrx5 39802 | Given the exclusivities se... |
mdandyvrx6 39803 | Given the exclusivities se... |
mdandyvrx7 39804 | Given the exclusivities se... |
mdandyvrx8 39805 | Given the exclusivities se... |
mdandyvrx9 39806 | Given the exclusivities se... |
mdandyvrx10 39807 | Given the exclusivities se... |
mdandyvrx11 39808 | Given the exclusivities se... |
mdandyvrx12 39809 | Given the exclusivities se... |
mdandyvrx13 39810 | Given the exclusivities se... |
mdandyvrx14 39811 | Given the exclusivities se... |
mdandyvrx15 39812 | Given the exclusivities se... |
H15NH16TH15IH16 39813 | Given 15 hypotheses and a ... |
dandysum2p2e4 39814 | CONTRADICTION PRO... |
mdandysum2p2e4 39815 | CONTRADICTION PROVED AT 1 ... |
r19.32 39816 | Theorem 19.32 of [Margaris... |
rexsb 39817 | An equivalent expression f... |
rexrsb 39818 | An equivalent expression f... |
2rexsb 39819 | An equivalent expression f... |
2rexrsb 39820 | An equivalent expression f... |
cbvral2 39821 | Change bound variables of ... |
cbvrex2 39822 | Change bound variables of ... |
2ralbiim 39823 | Split a biconditional and ... |
raaan2 39824 | Rearrange restricted quant... |
rmoimi 39825 | Restricted "at most one" i... |
2reu5a 39826 | Double restricted existent... |
reuimrmo 39827 | Restricted uniqueness impl... |
rmoanim 39828 | Introduction of a conjunct... |
reuan 39829 | Introduction of a conjunct... |
2reurex 39830 | Double restricted quantifi... |
2reurmo 39831 | Double restricted quantifi... |
2reu2rex 39832 | Double restricted existent... |
2rmoswap 39833 | A condition allowing swap ... |
2rexreu 39834 | Double restricted existent... |
2reu1 39835 | Double restricted existent... |
2reu2 39836 | Double restricted existent... |
2reu3 39837 | Double restricted existent... |
2reu4a 39838 | Definition of double restr... |
2reu4 39839 | Definition of double restr... |
2reu7 39840 | Two equivalent expressions... |
2reu8 39841 | Two equivalent expressions... |
ralbinrald 39848 | Elemination of a restricte... |
nvelim 39849 | If a class is the universa... |
alneu 39850 | If a statement holds for a... |
eu2ndop1stv 39851 | If there is a unique secon... |
eldmressn 39852 | Element of the domain of a... |
fveqvfvv 39853 | If a function's value at a... |
funresfunco 39854 | Composition of two functio... |
fnresfnco 39855 | Composition of two functio... |
funcoressn 39856 | A composition restricted t... |
funressnfv 39857 | A restriction to a singlet... |
dfateq12d 39858 | Equality deduction for "de... |
nfdfat 39859 | Bound-variable hypothesis ... |
dfdfat2 39860 | Alternate definition of th... |
dfafv2 39861 | Alternative definition of ... |
afveq12d 39862 | Equality deduction for fun... |
afveq1 39863 | Equality theorem for funct... |
afveq2 39864 | Equality theorem for funct... |
nfafv 39865 | Bound-variable hypothesis ... |
csbafv12g 39866 | Move class substitution in... |
afvfundmfveq 39867 | If a class is a function r... |
afvnfundmuv 39868 | If a set is not in the dom... |
ndmafv 39869 | The value of a class outsi... |
afvvdm 39870 | If the function value of a... |
nfunsnafv 39871 | If the restriction of a cl... |
afvvfunressn 39872 | If the function value of a... |
afvprc 39873 | A function's value at a pr... |
afvvv 39874 | If a function's value at a... |
afvpcfv0 39875 | If the value of the altern... |
afvnufveq 39876 | The value of the alternati... |
afvvfveq 39877 | The value of the alternati... |
afv0fv0 39878 | If the value of the altern... |
afvfvn0fveq 39879 | If the function's value at... |
afv0nbfvbi 39880 | The function's value at an... |
afvfv0bi 39881 | The function's value at an... |
afveu 39882 | The value of a function at... |
fnbrafvb 39883 | Equivalence of function va... |
fnopafvb 39884 | Equivalence of function va... |
funbrafvb 39885 | Equivalence of function va... |
funopafvb 39886 | Equivalence of function va... |
funbrafv 39887 | The second argument of a b... |
funbrafv2b 39888 | Function value in terms of... |
dfafn5a 39889 | Representation of a functi... |
dfafn5b 39890 | Representation of a functi... |
fnrnafv 39891 | The range of a function ex... |
afvelrnb 39892 | A member of a function's r... |
afvelrnb0 39893 | A member of a function's r... |
dfaimafn 39894 | Alternate definition of th... |
dfaimafn2 39895 | Alternate definition of th... |
afvelima 39896 | Function value in an image... |
afvelrn 39897 | A function's value belongs... |
fnafvelrn 39898 | A function's value belongs... |
fafvelrn 39899 | A function's value belongs... |
ffnafv 39900 | A function maps to a class... |
afvres 39901 | The value of a restricted ... |
tz6.12-afv 39902 | Function value. Theorem 6... |
tz6.12-1-afv 39903 | Function value (Theorem 6.... |
dmfcoafv 39904 | Domains of a function comp... |
afvco2 39905 | Value of a function compos... |
rlimdmafv 39906 | Two ways to express that a... |
aoveq123d 39907 | Equality deduction for ope... |
nfaov 39908 | Bound-variable hypothesis ... |
csbaovg 39909 | Move class substitution in... |
aovfundmoveq 39910 | If a class is a function r... |
aovnfundmuv 39911 | If an ordered pair is not ... |
ndmaov 39912 | The value of an operation ... |
ndmaovg 39913 | The value of an operation ... |
aovvdm 39914 | If the operation value of ... |
nfunsnaov 39915 | If the restriction of a cl... |
aovvfunressn 39916 | If the operation value of ... |
aovprc 39917 | The value of an operation ... |
aovrcl 39918 | Reverse closure for an ope... |
aovpcov0 39919 | If the alternative value o... |
aovnuoveq 39920 | The alternative value of t... |
aovvoveq 39921 | The alternative value of t... |
aov0ov0 39922 | If the alternative value o... |
aovovn0oveq 39923 | If the operation's value a... |
aov0nbovbi 39924 | The operation's value on a... |
aovov0bi 39925 | The operation's value on a... |
rspceaov 39926 | A frequently used special ... |
fnotaovb 39927 | Equivalence of operation v... |
ffnaov 39928 | An operation maps to a cla... |
faovcl 39929 | Closure law for an operati... |
aovmpt4g 39930 | Value of a function given ... |
aoprssdm 39931 | Domain of closure of an op... |
ndmaovcl 39932 | The "closure" of an operat... |
ndmaovrcl 39933 | Reverse closure law, in co... |
ndmaovcom 39934 | Any operation is commutati... |
ndmaovass 39935 | Any operation is associati... |
ndmaovdistr 39936 | Any operation is distribut... |
1t10e1p1e11 39937 | 11 is 1 times 10 to the po... |
1t10e1p1e11OLD 39938 | Obsolete version of ~ 1t10... |
elprneb 39939 | An element of a proper uno... |
leltletr 39940 | Transitive law, weaker for... |
deccarry 39941 | Add 1 to a 2 digit number ... |
nltle2tri 39942 | Negated extended trichotom... |
zgeltp1eq 39943 | If an integer is between a... |
smonoord 39944 | Ordering relation for a st... |
fzopred 39945 | Join a predecessor to the ... |
fzopredsuc 39946 | Join a predecessor and a s... |
1fzopredsuc 39947 | Join 0 and a successor to ... |
el1fzopredsuc 39948 | An element of an open inte... |
m1mod0mod1 39949 | An integer decreased by 1 ... |
elmod2 39950 | An integer modulo 2 is eit... |
iccpval 39953 | Partition consisting of a ... |
iccpart 39954 | A special partition. Corr... |
iccpartimp 39955 | Implications for a class b... |
iccpartres 39956 | The restriction of a parti... |
iccpartxr 39957 | If there is a partition, t... |
iccpartgtprec 39958 | If there is a partition, t... |
iccpartipre 39959 | If there is a partition, t... |
iccpartiltu 39960 | If there is a partition, t... |
iccpartigtl 39961 | If there is a partition, t... |
iccpartlt 39962 | If there is a partition, t... |
iccpartltu 39963 | If there is a partition, t... |
iccpartgtl 39964 | If there is a partition, t... |
iccpartgt 39965 | If there is a partition, t... |
iccpartleu 39966 | If there is a partition, t... |
iccpartgel 39967 | If there is a partition, t... |
iccpartrn 39968 | If there is a partition, t... |
iccpartf 39969 | The range of the partition... |
iccpartel 39970 | If there is a partition, t... |
iccelpart 39971 | An element of any partitio... |
iccpartiun 39972 | A half opened interval of ... |
icceuelpartlem 39973 | Lemma for ~ icceuelpart . ... |
icceuelpart 39974 | An element of a partitione... |
iccpartdisj 39975 | The segments of a partitio... |
iccpartnel 39976 | A point of a partition is ... |
fmtno 39979 | The ` N ` th Fermat number... |
fmtnoge3 39980 | Each Fermat number is grea... |
fmtnonn 39981 | Each Fermat number is a po... |
fmtnom1nn 39982 | A Fermat number minus one ... |
fmtnoodd 39983 | Each Fermat number is odd.... |
fmtnorn 39984 | A Fermat number is a funct... |
fmtnof1 39985 | The enumeration of the Fer... |
fmtnoinf 39986 | The set of Fermat numbers ... |
fmtnorec1 39987 | The first recurrence relat... |
sqrtpwpw2p 39988 | The floor of the square ro... |
fmtnosqrt 39989 | The floor of the square ro... |
fmtno0 39990 | The ` 0 ` th Fermat number... |
fmtno1 39991 | The ` 1 ` st Fermat number... |
fmtnorec2lem 39992 | Lemma for ~ fmtnorec2 (ind... |
fmtnorec2 39993 | The second recurrence rela... |
fmtnodvds 39994 | Any Fermat number divides ... |
goldbachthlem1 39995 | Lemma 1 for ~ goldbachth .... |
goldbachthlem2 39996 | Lemma 2 for ~ goldbachth .... |
goldbachth 39997 | Goldbach's theorem: Two d... |
fmtnorec3 39998 | The third recurrence relat... |
fmtnorec4 39999 | The fourth recurrence rela... |
fmtno2 40000 | The ` 2 ` nd Fermat number... |
fmtno3 40001 | The ` 3 ` rd Fermat number... |
fmtno4 40002 | The ` 4 ` th Fermat number... |
fmtno5lem1 40003 | Lemma 1 for ~ fmtno5 . (C... |
fmtno5lem2 40004 | Lemma 2 for ~ fmtno5 . (C... |
fmtno5lem3 40005 | Lemma 3 for ~ fmtno5 . (C... |
fmtno5lem4 40006 | Lemma 4 for ~ fmtno5 . (C... |
fmtno5 40007 | The ` 5 ` th Fermat number... |
fmtno0prm 40008 | The ` 0 ` th Fermat number... |
fmtno1prm 40009 | The ` 1 ` st Fermat number... |
fmtno2prm 40010 | The ` 2 ` nd Fermat number... |
257prm 40011 | 257 is a prime number (the... |
fmtno3prm 40012 | The ` 3 ` rd Fermat number... |
odz2prm2pw 40013 | Any power of two is coprim... |
fmtnoprmfac1lem 40014 | Lemma for ~ fmtnoprmfac1 :... |
fmtnoprmfac1 40015 | Divisor of Fermat number (... |
fmtnoprmfac2lem1 40016 | Lemma for ~ fmtnoprmfac2 .... |
fmtnoprmfac2 40017 | Divisor of Fermat number (... |
fmtnofac2lem 40018 | Lemma for ~ fmtnofac2 (Ind... |
fmtnofac2 40019 | Divisor of Fermat number (... |
fmtnofac1 40020 | Divisor of Fermat number (... |
fmtno4sqrt 40021 | The floor of the square ro... |
fmtno4prmfac 40022 | If P was a (prime) factor ... |
fmtno4prmfac193 40023 | If P was a (prime) factor ... |
fmtno4nprmfac193 40024 | 193 is not a (prime) facto... |
fmtno4prm 40025 | The ` 4 `-th Fermat number... |
65537prm 40026 | 65537 is a prime number (t... |
fmtnofz04prm 40027 | The first five Fermat numb... |
fmtnole4prm 40028 | The first five Fermat numb... |
fmtno5faclem1 40029 | Lemma 1 for ~ fmtno5fac . ... |
fmtno5faclem2 40030 | Lemma 2 for ~ fmtno5fac . ... |
fmtno5faclem3 40031 | Lemma 3 for ~ fmtno5fac . ... |
fmtno5fac 40032 | The factorisation of the `... |
fmtno5nprm 40033 | The ` 5 ` th Fermat number... |
prmdvdsfmtnof1lem1 40034 | Lemma 1 for ~ prmdvdsfmtno... |
prmdvdsfmtnof1lem2 40035 | Lemma 2 for ~ prmdvdsfmtno... |
prmdvdsfmtnof 40036 | The mapping of a Fermat nu... |
prmdvdsfmtnof1 40037 | The mapping of a Fermat nu... |
prminf2 40038 | The set of prime numbers i... |
pwdif 40039 | The difference of two numb... |
pwm1geoserALT 40040 | The n-th power of a number... |
2pwp1prm 40041 | For every prime number of ... |
2pwp1prmfmtno 40042 | Every prime number of the ... |
m2prm 40043 | The second Mersenne number... |
m3prm 40044 | The third Mersenne number ... |
2exp5 40045 | Two to the fifth power is ... |
flsqrt 40046 | A condition equivalent to ... |
flsqrt5 40047 | The floor of the square ro... |
3ndvds4 40048 | 3 does not divide 4. (Con... |
139prmALT 40049 | 139 is a prime number. In... |
31prm 40050 | 31 is a prime number. In ... |
m5prm 40051 | The fifth Mersenne number ... |
2exp7 40052 | Two to the seventh power i... |
127prm 40053 | 127 is a prime number. (C... |
m7prm 40054 | The seventh Mersenne numbe... |
2exp11 40055 | Two to the eleventh power ... |
m11nprm 40056 | The eleventh Mersenne numb... |
mod42tp1mod8 40057 | If a number is ` 3 ` modul... |
sfprmdvdsmersenne 40058 | If ` Q ` is a safe prime (... |
sgprmdvdsmersenne 40059 | If ` P ` is a Sophie Germa... |
lighneallem1 40060 | Lemma 1 for ~ lighneal . ... |
lighneallem2 40061 | Lemma 2 for ~ lighneal . ... |
lighneallem3 40062 | Lemma 3 for ~ lighneal . ... |
lighneallem4a 40063 | Lemma 1 for ~ lighneallem4... |
lighneallem4b 40064 | Lemma 2 for ~ lighneallem4... |
lighneallem4 40065 | Lemma 3 for ~ lighneal . ... |
lighneal 40066 | If a power of a prime ` P ... |
modexp2m1d 40067 | The square of an integer w... |
proththdlem 40068 | Lemma for ~ proththd . (C... |
proththd 40069 | Proth's theorem (1878). I... |
5tcu2e40 40070 | 5 times the cube of 2 is 4... |
3exp4mod41 40071 | 3 to the fourth power is -... |
41prothprmlem1 40072 | Lemma 1 for ~ 41prothprm .... |
41prothprmlem2 40073 | Lemma 2 for ~ 41prothprm .... |
41prothprm 40074 | 41 is a _Proth prime_. (C... |
iseven 40079 | The predicate "is an even ... |
isodd 40080 | The predicate "is an odd n... |
evenz 40081 | An even number is an integ... |
oddz 40082 | An odd number is an intege... |
evendiv2z 40083 | The result of dividing an ... |
oddp1div2z 40084 | The result of dividing an ... |
oddm1div2z 40085 | The result of dividing an ... |
isodd2 40086 | The predicate "is an odd n... |
dfodd2 40087 | Alternate definition for o... |
dfodd6 40088 | Alternate definition for o... |
dfeven4 40089 | Alternate definition for e... |
evenm1odd 40090 | The predecessor of an even... |
evenp1odd 40091 | The successor of an even n... |
oddp1eveni 40092 | The successor of an odd nu... |
oddm1eveni 40093 | The predecessor of an odd ... |
evennodd 40094 | An even number is not an o... |
oddneven 40095 | An odd number is not an ev... |
enege 40096 | The negative of an even nu... |
onego 40097 | The negative of an odd num... |
m1expevenALTV 40098 | Exponentiation of -1 by an... |
m1expoddALTV 40099 | Exponentiation of -1 by an... |
dfeven2 40100 | Alternate definition for e... |
dfodd3 40101 | Alternate definition for o... |
iseven2 40102 | The predicate "is an even ... |
isodd3 40103 | The predicate "is an odd n... |
2dvdseven 40104 | 2 divides an even number. ... |
2ndvdsodd 40105 | 2 does not divide an odd n... |
2dvdsoddp1 40106 | 2 divides an odd number in... |
2dvdsoddm1 40107 | 2 divides an odd number de... |
dfeven3 40108 | Alternate definition for e... |
dfodd4 40109 | Alternate definition for o... |
dfodd5 40110 | Alternate definition for o... |
zefldiv2ALTV 40111 | The floor of an even numbe... |
zofldiv2ALTV 40112 | The floor of an odd numer ... |
oddflALTV 40113 | Odd number representation ... |
iseven5 40114 | The predicate "is an even ... |
isodd7 40115 | The predicate "is an odd n... |
dfeven5 40116 | Alternate definition for e... |
dfodd7 40117 | Alternate definition for o... |
zneoALTV 40118 | No even integer equals an ... |
zeoALTV 40119 | An integer is even or odd.... |
zeo2ALTV 40120 | An integer is even or odd ... |
nneoALTV 40121 | A positive integer is even... |
nneoiALTV 40122 | A positive integer is even... |
odd2np1ALTV 40123 | An integer is odd iff it i... |
oddm1evenALTV 40124 | An integer is odd iff its ... |
oddp1evenALTV 40125 | An integer is odd iff its ... |
oexpnegALTV 40126 | The exponential of the neg... |
oexpnegnz 40127 | The exponential of the neg... |
bits0ALTV 40128 | Value of the zeroth bit. ... |
bits0eALTV 40129 | The zeroth bit of an even ... |
bits0oALTV 40130 | The zeroth bit of an odd n... |
divgcdoddALTV 40131 | Either ` A / ( A gcd B ) `... |
opoeALTV 40132 | The sum of two odds is eve... |
opeoALTV 40133 | The sum of an odd and an e... |
omoeALTV 40134 | The difference of two odds... |
omeoALTV 40135 | The difference of an odd a... |
oddprmALTV 40136 | A prime not equal to ` 2 `... |
0evenALTV 40137 | 0 is an even number. (Con... |
0noddALTV 40138 | 0 is not an odd number. (... |
1oddALTV 40139 | 1 is an odd number. (Cont... |
1nevenALTV 40140 | 1 is not an even number. ... |
2evenALTV 40141 | 2 is an even number. (Con... |
2noddALTV 40142 | 2 is not an odd number. (... |
nn0o1gt2ALTV 40143 | An odd nonnegative integer... |
nnoALTV 40144 | An alternate characterizat... |
nn0oALTV 40145 | An alternate characterizat... |
nn0e 40146 | An alternate characterizat... |
nn0onn0exALTV 40147 | For each odd nonnegative i... |
nn0enn0exALTV 40148 | For each even nonnegative ... |
nnpw2evenALTV 40149 | 2 to the power of a positi... |
epoo 40150 | The sum of an even and an ... |
emoo 40151 | The difference of an even ... |
epee 40152 | The sum of two even number... |
emee 40153 | The difference of two even... |
evensumeven 40154 | If a summand is even, the ... |
3odd 40155 | 3 is an odd number. (Cont... |
4even 40156 | 4 is an even number. (Con... |
5odd 40157 | 5 is an odd number. (Cont... |
6even 40158 | 6 is an even number. (Con... |
7odd 40159 | 7 is an odd number. (Cont... |
8even 40160 | 8 is an even number. (Con... |
evenprm2 40161 | A prime number is even iff... |
oddprmne2 40162 | Every prime number not bei... |
oddprmuzge3 40163 | A prime number which is od... |
perfectALTVlem1 40164 | Lemma for ~ perfectALTV . ... |
perfectALTVlem2 40165 | Lemma for ~ perfectALTV . ... |
perfectALTV 40166 | The Euclid-Euler theorem, ... |
isgbe 40173 | The predicate "is an even ... |
isgbo 40174 | The predicate "is an odd G... |
isgboa 40175 | The predicate "is an odd G... |
gbeeven 40176 | An even Goldbach number is... |
gboodd 40177 | An odd Goldbach number is ... |
gboagbo 40178 | An odd Goldbach number (st... |
gboaodd 40179 | An odd Goldbach number is ... |
gbepos 40180 | Any even Goldbach number i... |
gbopos 40181 | Any odd Goldbach number is... |
gboapos 40182 | Any odd Goldbach number is... |
gbegt5 40183 | Any even Goldbach number i... |
gbogt5 40184 | Any odd Goldbach number is... |
gboge7 40185 | Any odd Goldbach number is... |
gboage9 40186 | Any odd Goldbach number (s... |
gbege6 40187 | Any even Goldbach number i... |
gbpart6 40188 | The Goldbach partition of ... |
gbpart7 40189 | The (weak) Goldbach partit... |
gbpart8 40190 | The Goldbach partition of ... |
gbpart9 40191 | The (strong) Goldbach part... |
gbpart11 40192 | The (strong) Goldbach part... |
6gbe 40193 | 6 is an even Goldbach numb... |
7gbo 40194 | 7 is an odd Goldbach numbe... |
8gbe 40195 | 8 is an even Goldbach numb... |
9gboa 40196 | 9 is an odd Goldbach numbe... |
11gboa 40197 | 11 is an odd Goldbach numb... |
stgoldbwt 40198 | If the strong ternary Gold... |
bgoldbwt 40199 | If the binary Goldbach con... |
bgoldbst 40200 | If the binary Goldbach con... |
sgoldbaltlem1 40201 | Lemma 1 for ~ sgoldbalt : ... |
sgoldbaltlem2 40202 | Lemma 2 for ~ sgoldbalt : ... |
sgoldbalt 40203 | An alternate (the original... |
nnsum3primes4 40204 | 4 is the sum of at most 3 ... |
nnsum4primes4 40205 | 4 is the sum of at most 4 ... |
nnsum3primesprm 40206 | Every prime is "the sum of... |
nnsum4primesprm 40207 | Every prime is "the sum of... |
nnsum3primesgbe 40208 | Any even Goldbach number i... |
nnsum4primesgbe 40209 | Any even Goldbach number i... |
nnsum3primesle9 40210 | Every integer greater than... |
nnsum4primesle9 40211 | Every integer greater than... |
nnsum4primesodd 40212 | If the (weak) ternary Gold... |
nnsum4primesoddALTV 40213 | If the (strong) ternary Go... |
evengpop3 40214 | If the (weak) ternary Gold... |
evengpoap3 40215 | If the (strong) ternary Go... |
nnsum4primeseven 40216 | If the (weak) ternary Gold... |
nnsum4primesevenALTV 40217 | If the (strong) ternary Go... |
wtgoldbnnsum4prm 40218 | If the (weak) ternary Gold... |
stgoldbnnsum4prm 40219 | If the (strong) ternary Go... |
bgoldbnnsum3prm 40220 | If the binary Goldbach con... |
bgoldbtbndlem1 40221 | Lemma 1 for ~ bgoldbtbnd :... |
bgoldbtbndlem2 40222 | Lemma 2 for ~ bgoldbtbnd .... |
bgoldbtbndlem3 40223 | Lemma 3 for ~ bgoldbtbnd .... |
bgoldbtbndlem4 40224 | Lemma 4 for ~ bgoldbtbnd .... |
bgoldbtbnd 40225 | If the binary Goldbach con... |
bgoldbachlt 40227 | The binary Goldbach conjec... |
tgblthelfgott 40229 | The ternary Goldbach conje... |
tgoldbachlt 40230 | The ternary Goldbach conje... |
tgoldbach 40232 | The ternary Goldbach conje... |
bgoldbachltOLD 40234 | Obsolete version of ~ bgol... |
tgblthelfgottOLD 40236 | Obsolete version of ~ tgbl... |
tgoldbachltOLD 40237 | Obsolete version of ~ tgol... |
tgoldbachOLD 40239 | Obsolete version of ~ tgol... |
wrdred1 40240 | A word truncated by a symb... |
wrdred1hash 40241 | The length of a word trunc... |
lswn0 40242 | The last symbol of a not e... |
ccatw2s1cl 40243 | The concatenation of a wor... |
pfxval 40246 | Value of a prefix. (Contr... |
pfx00 40247 | A zero length prefix. (Co... |
pfx0 40248 | A prefix of an empty set i... |
pfxcl 40249 | Closure of the prefix extr... |
pfxmpt 40250 | Value of the prefix extrac... |
pfxres 40251 | Value of the prefix extrac... |
pfxf 40252 | A prefix of a word is a fu... |
pfxfn 40253 | Value of the prefix extrac... |
pfxlen 40254 | Length of a prefix. Could... |
pfxid 40255 | A word is a prefix of itse... |
pfxrn 40256 | The range of a prefix of a... |
pfxn0 40257 | A prefix consisting of at ... |
pfxnd 40258 | The value of the prefix ex... |
pfxlen0 40259 | Length of a prefix of a wo... |
addlenrevpfx 40260 | The sum of the lengths of ... |
addlenpfx 40261 | The sum of the lengths of ... |
pfxfv 40262 | A symbol in a prefix of a ... |
pfxfv0 40263 | The first symbol in a pref... |
pfxtrcfv 40264 | A symbol in a word truncat... |
pfxtrcfv0 40265 | The first symbol in a word... |
pfxfvlsw 40266 | The last symbol in a (not ... |
pfxeq 40267 | The prefixes of two words ... |
pfxtrcfvl 40268 | The last symbol in a word ... |
pfxsuffeqwrdeq 40269 | Two words are equal if and... |
pfxsuff1eqwrdeq 40270 | Two (nonempty) words are e... |
disjwrdpfx 40271 | Sets of words are disjoint... |
ccatpfx 40272 | Joining a prefix with an a... |
pfxccat1 40273 | Recover the left half of a... |
pfx1 40274 | A prefix of length 1. (Co... |
pfx2 40275 | A prefix of length 2. (Co... |
pfxswrd 40276 | A prefix of a subword. Co... |
swrdpfx 40277 | A subword of a prefix. Co... |
pfxpfx 40278 | A prefix of a prefix. Cou... |
pfxpfxid 40279 | A prefix of a prefix with ... |
pfxcctswrd 40280 | The concatenation of the p... |
lenpfxcctswrd 40281 | The length of the concaten... |
lenrevpfxcctswrd 40282 | The length of the concaten... |
pfxlswccat 40283 | Reconstruct a nonempty wor... |
ccats1pfxeq 40284 | The last symbol of a word ... |
ccats1pfxeqrex 40285 | There exists a symbol such... |
pfxccatin12lem1 40286 | Lemma 1 for ~ pfxccatin12 ... |
pfxccatin12lem2 40287 | Lemma 2 for ~ pfxccatin12 ... |
pfxccatin12 40288 | The subword of a concatena... |
pfxccat3 40289 | The subword of a concatena... |
pfxccatpfx1 40290 | A prefix of a concatenatio... |
pfxccatpfx2 40291 | A prefix of a concatenatio... |
pfxccat3a 40292 | A prefix of a concatenatio... |
pfxccatid 40293 | A prefix of a concatenatio... |
ccats1pfxeqbi 40294 | A word is a prefix of a wo... |
pfxccatin12d 40295 | The subword of a concatena... |
reuccatpfxs1lem 40296 | Lemma for ~ reuccatpfxs1 .... |
reuccatpfxs1 40297 | There is a unique word hav... |
splvalpfx 40298 | Value of the substring rep... |
repswpfx 40299 | A prefix of a repeated sym... |
cshword2 40300 | Perform a cyclical shift f... |
pfxco 40301 | Mapping of words commutes ... |
elnelall 40302 | A contradiction concerning... |
clel5 40303 | Alternate definition of cl... |
dfss7 40304 | Alternate definition of su... |
sssseq 40305 | If a class is a subclass o... |
prcssprc 40306 | The superclass of a proper... |
ralnralall 40307 | A contradiction concerning... |
falseral0 40308 | A false statement can only... |
ralralimp 40309 | Selecting one of two alter... |
n0rex 40310 | There is an element in a n... |
ssn0rex 40311 | There is an element in a c... |
elpwdifsn 40312 | A subset of a set is an el... |
pr1eqbg 40313 | A (proper) pair is equal t... |
pr1nebg 40314 | A (proper) pair is not equ... |
rexdifpr 40315 | Restricted existential qua... |
opidg 40316 | The ordered pair ` <. A , ... |
otiunsndisjX 40317 | The union of singletons co... |
opabn1stprc 40318 | An ordered-pair class abst... |
resresdm 40319 | A restriction by an arbitr... |
resisresindm 40320 | The restriction of a relat... |
fvifeq 40321 | Equality of function value... |
2f1fvneq 40322 | If two one-to-one function... |
f1cofveqaeq 40323 | If the values of a composi... |
f1cofveqaeqALT 40324 | Alternate proof of ~ f1cof... |
rnfdmpr 40325 | The range of a one-to-one ... |
imarnf1pr 40326 | The image of the range of ... |
funop1 40327 | A function is an ordered p... |
f1ssf1 40328 | A subset of an injective f... |
fun2dmnopgexmpl 40329 | A function with a domain c... |
opabresex0d 40330 | A collection of ordered pa... |
opabbrfex0d 40331 | A collection of ordered pa... |
opabresexd 40332 | A collection of ordered pa... |
opabbrfexd 40333 | A collection of ordered pa... |
opabresex2d 40334 | Restrictions of a collecti... |
mptmpt2opabbrd 40335 | The operation value of a f... |
mptmpt2opabovd 40336 | The operation value of a f... |
fpropnf1 40337 | A function, given by an un... |
riotaeqimp 40338 | If two restricted iota des... |
resfnfinfin 40339 | The restriction of a funct... |
residfi 40340 | A restricted identity func... |
cnambpcma 40341 | ((a-b)+c)-a = c-a holds fo... |
cnapbmcpd 40342 | ((a+b)-c)+d = ((a+d)+b)-c ... |
leaddsuble 40343 | Addition and subtraction o... |
2leaddle2 40344 | If two real numbers are le... |
ltnltne 40345 | Variant of trichotomy law ... |
p1lep2 40346 | A real number increasd by ... |
ltsubsubaddltsub 40347 | If the result of subtracti... |
zm1nn 40348 | An integer minus 1 is posi... |
nn0resubcl 40349 | Closure law for subtractio... |
eluzge0nn0 40350 | If an integer is greater t... |
ssfz12 40351 | Subset relationship for fi... |
elfz2z 40352 | Membership of an integer i... |
2elfz3nn0 40353 | If there are two elements ... |
fz0addcom 40354 | The addition of two member... |
2elfz2melfz 40355 | If the sum of two integers... |
fz0addge0 40356 | The sum of two integers in... |
elfzlble 40357 | Membership of an integer i... |
elfzelfzlble 40358 | Membership of an element o... |
subsubelfzo0 40359 | Subtracting a difference f... |
fzoopth 40360 | A half-open integer range ... |
2ffzoeq 40361 | Two functions over a half-... |
fzosplitpr 40362 | Extending a half-open inte... |
prinfzo0 40363 | The intersection of a half... |
elfzr 40364 | A member of a finite inter... |
elfzo0l 40365 | A member of a half-open ra... |
elfzlmr 40366 | A member of a finite inter... |
elfz0lmr 40367 | A member of a finite inter... |
resunimafz0 40368 | Formerly part of proof of ... |
nfile 40369 | The size of any infinite s... |
hash1n0 40370 | If the size of a set is 1 ... |
fsummsndifre 40371 | A finite sum with one of i... |
fsumsplitsndif 40372 | Separate out a term in a f... |
fsummmodsndifre 40373 | A finite sum of summands m... |
fsummmodsnunz 40374 | A finite sum of summands m... |
uhgruhgra 40375 | Equivalence of the definit... |
uhgrauhgr 40376 | Equivalence of the definit... |
uhgrauhgrbi 40377 | Equivalence of the definit... |
isuspgr 40382 | The property of being a si... |
isusgr 40383 | The property of being a si... |
uspgrf 40384 | The edge function of a sim... |
usgrf 40385 | The edge function of a sim... |
isusgrs 40386 | The property of being a si... |
usgrfs 40387 | The edge function of a sim... |
usgrfun 40388 | The edge function of a sim... |
usgrusgra 40389 | A simple graph represented... |
usgredgss 40390 | The set of edges of a simp... |
edgusgr 40391 | An edge of a simple graph ... |
isusgrop 40392 | The property of being an u... |
usgrop 40393 | A simple graph represented... |
isausgr 40394 | The property of an unorder... |
ausgrusgrb 40395 | The equivalence of the def... |
usgrausgri 40396 | A simple graph represented... |
ausgrumgri 40397 | If an alternatively define... |
ausgrusgri 40398 | The equivalence of the def... |
usgrausgrb 40399 | The equivalence of the def... |
usgredgop 40400 | An edge of a simple graph ... |
usgrf1o 40401 | The edge function of a sim... |
usgrf1 40402 | The edge function of a sim... |
uspgrf1oedg 40403 | The edge function of a sim... |
usgrss 40404 | An edge is a subset of ver... |
uspgrushgr 40405 | A simple pseudograph is an... |
uspgrupgr 40406 | A simple pseudograph is an... |
uspgrupgrushgr 40407 | A graph is a simple pseudo... |
usgruspgr 40408 | A simple graph is a simple... |
usgrumgr 40409 | A simple graph is an undir... |
usgrumgruspgr 40410 | A graph is a simple graph ... |
usgruspgrb 40411 | A class is a simple graph ... |
usgrupgr 40412 | A simple graph is an undir... |
usgruhgr 40413 | A simple graph is an undir... |
usgrislfuspgr 40414 | A simple graph is a loop-f... |
uspgrun 40415 | The union ` U ` of two sim... |
uspgrunop 40416 | The union of two simple ps... |
usgrun 40417 | The union ` U ` of two sim... |
usgrunop 40418 | The union of two simple gr... |
usgredg2 40419 | The value of the "edge fun... |
usgredg2ALT 40420 | Alternate proof of ~ usgre... |
usgredgprv 40421 | In a simple graph, an edge... |
usgredgprvALT 40422 | Alternate proof of ~ usgre... |
usgredgappr 40423 | An edge of a simple graph ... |
usgrpredgav 40424 | An edge of a simple graph ... |
edgassv2 40425 | An edge of a simple graph ... |
usgredg 40426 | For each edge in a simple ... |
usgrnloopv 40427 | In a simple graph, there i... |
usgrnloopvALT 40428 | Alternate proof of ~ usgrn... |
usgrnloop 40429 | In a simple graph, there i... |
usgrnloopALT 40430 | Alternate proof of ~ usgrn... |
usgrnloop0 40431 | A simple graph has no loop... |
usgrnloop0ALT 40432 | Alternate proof of ~ usgrn... |
usgredgne 40433 | An edge of a simple graph ... |
usgrf1oedg 40434 | The edge function of a sim... |
uhgr2edg 40435 | If a vertex is adjacent to... |
umgr2edg 40436 | If a vertex is adjacent to... |
usgr2edg 40437 | If a vertex is adjacent to... |
umgr2edg1 40438 | If a vertex is adjacent to... |
usgr2edg1 40439 | If a vertex is adjacent to... |
umgrvad2edg 40440 | If a vertex is adjacent to... |
umgr2edgneu 40441 | If a vertex is adjacent to... |
usgrsizedg 40442 | In a simple graph, the siz... |
usgredg3 40443 | The value of the "edge fun... |
usgredg4 40444 | For a vertex incident to a... |
usgredgreu 40445 | For a vertex incident to a... |
usgredg2vtx 40446 | For a vertex incident to a... |
uspgredg2vtxeu 40447 | For a vertex incident to a... |
usgredg2vtxeu 40448 | For a vertex incident to a... |
usgredg2vtxeuALT 40449 | Alternate proof of ~ usgre... |
uspgredg2vlem 40450 | Lemma for ~ uspgredg2v . ... |
uspgredg2v 40451 | In a simple pseudograph, t... |
usgredg2vlem1 40452 | Lemma 1 for ~ usgredg2v . ... |
usgredg2vlem2 40453 | Lemma 2 for ~ usgredg2v . ... |
usgredg2v 40454 | In a simple graph, the map... |
usgredgleord 40455 | In a simple graph the numb... |
ushgredgedga 40456 | In a simple hypergraph the... |
usgredgedga 40457 | In a simple graph there is... |
ushgredgedgaloop 40458 | In a simple hypergraph the... |
uspgredgaleord 40459 | In a simple pseudograph th... |
usgredgaleord 40460 | In a simple graph the numb... |
usgredgaleordALT 40461 | In a simple graph the numb... |
usgr0e 40462 | The empty graph, with vert... |
usgr0vb 40463 | The null graph, with no ve... |
uhgr0v0e 40464 | The null graph, with no ve... |
uhgr0vsize0 40465 | The size of a hypergraph w... |
uhgr0edgfi 40466 | A graph of order 0 (i.e. w... |
usgr0v 40467 | The null graph, with no ve... |
uhgr0vusgr 40468 | The null graph, with no ve... |
usgr0 40469 | The null graph represented... |
uspgr1e 40470 | A simple pseudograph with ... |
usgr1e 40471 | A simple graph with one ed... |
usgr0eop 40472 | The empty graph, with vert... |
uspgr1eop 40473 | A simple pseudograph with ... |
uspgr1ewop 40474 | A simple pseudograph with ... |
uspgr1v1eop 40475 | A simple pseudograph with ... |
usgr1eop 40476 | A simple graph with (at le... |
uspgr2v1e2w 40477 | A simple pseudograph with ... |
usgr2v1e2w 40478 | A simple graph with two ve... |
edg0usgr 40479 | A class without edges is a... |
lfuhgr1v0e 40480 | A loop-free hypergraph wit... |
usgr1vr 40481 | A simple graph with one ve... |
usgr1v 40482 | A class with one (or no) v... |
usgr1v0edg 40483 | A class with one (or no) v... |
usgrexmpllem 40484 | Lemma for ~ usgrexmpl . (... |
usgrexmplvtx 40485 | The vertices ` 0 , 1 , 2 ,... |
usgrexmpledg 40486 | The edges ` { 0 , 1 } , { ... |
usgrexmpl 40487 | ` G ` is a simple graph of... |
griedg0prc 40488 | The class of empty graphs ... |
griedg0ssusgr 40489 | The class of all simple gr... |
usgrprc 40490 | The class of simple graphs... |
relsubgr 40493 | The class of the subgraph ... |
subgrv 40494 | If a class is a subgraph o... |
issubgr 40495 | The property of a set to b... |
issubgr2 40496 | The property of a set to b... |
subgrprop 40497 | The properties of a subgra... |
subgrprop2 40498 | The properties of a subgra... |
uhgrissubgr 40499 | The property of a hypergra... |
subgrprop3 40500 | The properties of a subgra... |
egrsubgr 40501 | An empty graph consisting ... |
0grsubgr 40502 | The null graph (represente... |
0uhgrsubgr 40503 | The null graph (as hypergr... |
uhgrsubgrself 40504 | A hypergraph is a subgraph... |
subgrfun 40505 | The edge function of a sub... |
subgruhgrfun 40506 | The edge function of a sub... |
subgreldmiedg 40507 | An element of the domain o... |
subgruhgredgd 40508 | An edge of a subgraph of a... |
subumgredg2 40509 | An edge of a subgraph of a... |
subuhgr 40510 | A subgraph of a hypergraph... |
subupgr 40511 | A subgraph of a pseudograp... |
subumgr 40512 | A subgraph of a multigraph... |
subusgr 40513 | A subgraph of a simple gra... |
uhgrspansubgrlem 40514 | Lemma for ~ uhgrspansubgr ... |
uhgrspansubgr 40515 | A spanning subgraph ` S ` ... |
uhgrspan 40516 | A spanning subgraph ` S ` ... |
upgrspan 40517 | A spanning subgraph ` S ` ... |
umgrspan 40518 | A spanning subgraph ` S ` ... |
usgrspan 40519 | A spanning subgraph ` S ` ... |
uhgrspanop 40520 | A spanning subgraph of a h... |
upgrspanop 40521 | A spanning subgraph of a p... |
umgrspanop 40522 | A spanning subgraph of a m... |
usgrspanop 40523 | A spanning subgraph of a s... |
uhgrspan1lem1 40524 | Lemma 1 for ~ uhgrspan1 . ... |
uhgrspan1lem2 40525 | Lemma 2 for ~ uhgrspan1 . ... |
uhgrspan1lem3 40526 | Lemma 3 for ~ uhgrspan1 . ... |
uhgrspan1 40527 | The induced subgraph ` S `... |
upgrres1lem1 40528 | Lemma 1 for ~ upgrres1 . ... |
umgrres1lem 40529 | Lemma for ~ umgrres1 . (C... |
upgrres1lem2 40530 | Lemma 2 for ~ upgrres1 . ... |
upgrres1lem3 40531 | Lemma 3 for ~ upgrres1 . ... |
upgrres1 40532 | A pseudograph obtained by ... |
umgrres1 40533 | A multigraph obtained by r... |
usgrres1 40534 | Restricting a simple graph... |
isfusgr 40537 | The property of being a fi... |
fusgrvtxfi 40538 | A finite simple graph has ... |
isfusgrf1 40539 | The property of being a fi... |
isfusgrcl 40540 | The property of being a fi... |
fusgrusgr 40541 | A finite simple graph is a... |
opfusgr 40542 | A finite simple graph repr... |
usgredgffibi 40543 | The number of edges in a s... |
fusgredgfi 40544 | In a finite simple graph t... |
usgr1v0e 40545 | The size of a (finite) sim... |
usgrfilem 40546 | In a finite simple graph, ... |
fusgrfisbase 40547 | Induction base for ~ fusgr... |
fusgrfisstep 40548 | Induction step in ~ fusgrf... |
fusgrfis 40549 | A finite simple graph is o... |
nbgrprc0 40555 | The set of neighbors is em... |
nbgrcl 40559 | If a class has at least on... |
nbgrval 40560 | The set of neighbors of a ... |
dfnbgr2 40561 | Alternate definition of th... |
dfnbgr3 40562 | Alternate definition of th... |
nbgrnvtx0 40563 | There are no neighbors of ... |
nbgrel 40564 | Characterization of a neig... |
nbuhgr 40565 | The set of neighbors of a ... |
nbupgr 40566 | The set of neighbors of a ... |
nbupgrel 40567 | A neighbor of a vertex in ... |
nbumgrvtx 40568 | The set of neighbors of a ... |
nbumgr 40569 | The set of neighbors of an... |
nbusgrvtx 40570 | The set of neighbors of a ... |
nbusgr 40571 | The set of neighbors of an... |
nbgr2vtx1edg 40572 | If a graph has two vertice... |
nbuhgr2vtx1edgblem 40573 | Lemma for ~ nbuhgr2vtx1edg... |
nbuhgr2vtx1edgb 40574 | If a hypergraph has two ve... |
nbusgreledg 40575 | A class/vertex is a neighb... |
uhgrnbgr0nb 40576 | A vertex which is not endp... |
nbgr0vtxlem 40577 | Lemma for ~ nbgr0vtx and ~... |
nbgr0vtx 40578 | In a null graph (with no v... |
nbgr0edg 40579 | In an empty graph (with no... |
nbgr1vtx 40580 | In a graph with one vertex... |
nbgrisvtx 40581 | Every neighbor of a class/... |
nbgrssvtx 40582 | The neighbors of a vertex ... |
nbgrnself 40583 | A vertex in a graph is not... |
usgrnbnself 40584 | A vertex in a simple graph... |
nbgrnself2 40585 | A class is not a neighbor ... |
nbgrssovtx 40586 | The neighbors of a vertex ... |
nbgrssvwo2 40587 | The neighbors of a vertex ... |
usgrnbnself2 40588 | In a simple graph, a class... |
usgrnbssovtx 40589 | The neighbors of a vertex ... |
usgrnbssvwo2 40590 | The neighbors of a vertex ... |
nbgrsym 40591 | A vertex in a graph is a n... |
nbupgrres 40592 | The neighborhood of a vert... |
usgrnbcnvfv 40593 | Applying the edge function... |
nbusgredgeu 40594 | For each neighbor of a ver... |
edgnbusgreu 40595 | For each edge incident to ... |
nbusgredgeu0 40596 | For each neighbor of a ver... |
nbusgrf1o0 40597 | The mapping of neighbors o... |
nbusgrf1o1 40598 | The set of neighbors of a ... |
nbusgrf1o 40599 | The set of neighbors of a ... |
nbedgusgr 40600 | The number of neighbors of... |
edgusgrnbfin 40601 | The number of neighbors of... |
nbusgrfi 40602 | The class of neighbors of ... |
nbfiusgrfi 40603 | The class of neighbors of ... |
hashnbusgrnn0 40604 | The number of neighbors of... |
nbfusgrlevtxm1 40605 | The number of neighbors of... |
nbfusgrlevtxm2 40606 | If there is a vertex which... |
nbusgrvtxm1 40607 | If the number of neighbors... |
nb3grprlem1 40608 | Lemma 1 for ~ nb3grapr . ... |
nb3grprlem2 40609 | Lemma 2 for ~ nb3grapr . ... |
nb3grpr 40610 | The neighbors of a vertex ... |
nb3grpr2 40611 | The neighbors of a vertex ... |
nb3gr2nb 40612 | If the neighbors of two ve... |
uvtxaval 40613 | The set of all universal v... |
uvtxael 40614 | A universal vertex, i.e. a... |
uvtxaisvtx 40615 | A universal vertex is a ve... |
uvtxassvtx 40616 | The set of the universal v... |
vtxnbuvtx 40617 | A universal vertex has all... |
uvtxanbgr 40618 | A universal vertex has all... |
uvtxanbgrvtx 40619 | A universal vertex is neig... |
uvtxa0 40620 | There is no universal vert... |
isuvtxa 40621 | The set of all universal v... |
uvtxael1 40622 | A universal vertex, i.e. a... |
uvtxa01vtx0 40623 | If a graph/class has no ed... |
uvtxa01vtx 40624 | If a graph/class has no ed... |
uvtx2vtx1edg 40625 | If a graph has two vertice... |
uvtx2vtx1edgb 40626 | If a hypergraph has two ve... |
uvtxnbgr 40627 | A universal vertex has all... |
uvtxnbgrb 40628 | A vertex is universal iff ... |
uvtxusgr 40629 | The set of all universal v... |
uvtxusgrel 40630 | A universal vertex, i.e. a... |
uvtxanm1nbgr 40631 | A universal vertex has ` n... |
nbusgrvtxm1uvtx 40632 | If the number of neighbors... |
uvtxnbvtxm1 40633 | A universal vertex has ` n... |
nbupgruvtxres 40634 | The neighborhood of a univ... |
uvtxupgrres 40635 | A universal vertex is univ... |
iscplgr 40636 | The property of being a co... |
cplgruvtxb 40637 | An graph is complete iff e... |
iscplgrnb 40638 | A graph is complete iff al... |
iscplgredg 40639 | A graph is complete iff al... |
iscusgr 40640 | The property of being a co... |
cusgrusgr 40641 | A complete simple graph is... |
cusgrcplgr 40642 | A complete simple graph is... |
iscusgrvtx 40643 | A simple graph is complete... |
cusgruvtxb 40644 | A simple graph is complete... |
iscusgredg 40645 | A simple graph is complete... |
cusgredg 40646 | In a complete simple graph... |
cplgr0 40647 | The null graph (with no ve... |
cusgr0 40648 | The null graph (with no ve... |
cplgr0v 40649 | A graph with no vertices (... |
cusgr0v 40650 | A graph with no vertices (... |
cplgr1vlem 40651 | Lemma for ~ cplgr1v and ~ ... |
cplgr1v 40652 | A graph with one vertex is... |
cusgr1v 40653 | A graph with one vertex an... |
cplgr2v 40654 | An undirected hypergraph w... |
cplgr2vpr 40655 | An undirected hypergraph w... |
nbcplgr 40656 | In a complete graph, each ... |
cplgr3v 40657 | A pseudograph with three (... |
cusgr3vnbpr 40658 | The neighbors of a vertex ... |
cplgrop 40659 | A complete graph represent... |
cusgrop 40660 | A complete simple graph re... |
usgrexi 40661 | An arbitrary set regarded ... |
cusgrexi 40662 | An arbitrary set regarded ... |
cusgrexg 40663 | For each set there is a se... |
cusgrres 40664 | Restricting a complete sim... |
cusgrsizeindb0 40665 | Base case of the induction... |
cusgrsizeindb1 40666 | Base case of the induction... |
cusgrsizeindslem 40667 | Lemma for ~ cusgrsizeinds ... |
cusgrsizeinds 40668 | Part 1 of induction step i... |
cusgrsize2inds 40669 | Induction step in ~ cusgra... |
cusgrsize 40670 | The size of a finite compl... |
cusgrfilem1 40671 | Lemma 1 for ~ cusgrfi . (... |
cusgrfilem2 40672 | Lemma 2 for ~ cusgrfi . (... |
cusgrfilem3 40673 | Lemma 3 for ~ cusgrfi . (... |
cusgrfi 40674 | If the size of a complete ... |
usgredgsscusgredg 40675 | A simple graph is a subgra... |
usgrsscusgr 40676 | A simple graph is a subgra... |
sizusglecusglem1 40677 | Lemma 1 for ~ sizusglecusg... |
sizusglecusglem2 40678 | Lemma 2 for ~ sizusglecusg... |
sizusglecusg 40679 | The size of a simple graph... |
fusgrmaxsize 40680 | The maximum size of a fini... |
vtxdgfval 40683 | The value of the vertex de... |
vtxdgval 40684 | The degree of a vertex. (... |
vtxdgfival 40685 | The degree of a vertex for... |
vtxdgf 40686 | The vertex degree function... |
vtxdgelxnn0 40687 | The degree of a vertex is ... |
vtxdg0v 40688 | The degree of a vertex in ... |
vtxdg0e 40689 | The degree of a vertex in ... |
vtxdgfisnn0 40690 | The degree of a vertex in ... |
vtxdgfisf 40691 | The vertex degree function... |
vtxdeqd 40692 | Equality theorem for the v... |
vtxduhgr0e 40693 | The degree of a vertex in ... |
vtxdlfuhgr1v 40694 | The degree of the vertex i... |
vdumgr0 40695 | A vertex in a multigraph h... |
vtxdun 40696 | The degree of a vertex in ... |
vtxdfiun 40697 | The degree of a vertex in ... |
vtxduhgrun 40698 | The degree of a vertex in ... |
vtxduhgrfiun 40699 | The degree of a vertex in ... |
vtxdlfgrval 40700 | The value of the vertex de... |
vtxdumgrval 40701 | The value of the vertex de... |
vtxdusgrval 40702 | The value of the vertex de... |
vtxd0nedgb 40703 | A vertex has degree 0 iff ... |
vtxdushgrfvedglem 40704 | Lemma for ~ vtxdushgrfvedg... |
vtxdushgrfvedg 40705 | The value of the vertex de... |
vtxdusgrfvedg 40706 | The value of the vertex de... |
vtxduhgr0nedg 40707 | If a vertex in a hypergrap... |
vtxdumgr0nedg 40708 | If a vertex in a multigrap... |
vtxduhgr0edgnel 40709 | A vertex in a hypergraph h... |
vtxdusgr0edgnel 40710 | A vertex in a simple graph... |
vtxdusgr0edgnelALT 40711 | Alternate proof of ~ vtxdu... |
vtxdgfusgrf 40712 | The vertex degree function... |
vtxdgfusgr 40713 | In a finite simple graph, ... |
fusgrn0degnn0 40714 | In a nonempty, finite grap... |
1loopgruspgr 40715 | A graph with one edge whic... |
1loopgredg 40716 | The set of edges in a grap... |
1loopgrnb0 40717 | In a graph (simple pseudog... |
1loopgrvd2 40718 | The vertex degree of a one... |
1loopgrvd0 40719 | The vertex degree of a one... |
1hevtxdg0 40720 | The vertex degree of verte... |
1hevtxdg1 40721 | The vertex degree of verte... |
1hegrlfgr 40722 | --- TODO-AV: not used anym... |
1hegrvtxdg1 40723 | The vertex degree of a gra... |
1hegrvtxdg1r 40724 | The vertex degree of a gra... |
1egrvtxdg1 40725 | The vertex degree of a one... |
1egrvtxdg1r 40726 | The vertex degree of a one... |
1egrvtxdg0 40727 | The vertex degree of a one... |
p1evtxdeqlem 40728 | Lemma for ~ p1evtxdeq and ... |
p1evtxdeq 40729 | If an edge ` E ` which doe... |
p1evtxdp1 40730 | If an edge ` E ` (not bein... |
uspgrloopvtx 40731 | The set of vertices in a g... |
uspgrloopvtxel 40732 | A vertex in a graph (simpl... |
uspgrloopiedg 40733 | The set of edges in a grap... |
uspgrloopedg 40734 | The set of edges in a grap... |
uspgrloopnb0 40735 | In a graph (simple pseudog... |
uspgrloopvd2 40736 | The vertex degree of a one... |
umgr2v2evtx 40737 | The set of vertices in a m... |
umgr2v2evtxel 40738 | A vertex in a multigraph w... |
umgr2v2eiedg 40739 | The edge function in a mul... |
umgr2v2eedg 40740 | The set of edges in a mult... |
umgr2v2e 40741 | A multigraph with two edge... |
umgr2v2enb1 40742 | In a multigraph with two e... |
umgr2v2evd2 40743 | In a multigraph with two e... |
hashnbusgrvd 40744 | In a simple graph, the num... |
usgruvtxvdb 40745 | In a finite simple graph w... |
vdiscusgrb 40746 | A finite simple graph with... |
vdiscusgr 40747 | In a finite complete simpl... |
vtxdusgradjvtx 40748 | The degree of a vertex in ... |
usgrvd0nedg 40749 | If a vertex in a simple gr... |
uhgrvd00 40750 | If every vertex in a hyper... |
usgrvd00 40751 | If every vertex in a simpl... |
vdegp1ai-av 40752 | The induction step for a v... |
vdegp1bi-av 40753 | The induction step for a v... |
vdegp1ci-av 40754 | The induction step for a v... |
isrgr 40759 | The property of a class be... |
rgrprop 40760 | The properties of a k-regu... |
isrusgr 40761 | The property of being a k-... |
rusgrprop 40762 | The properties of a k-regu... |
rusgrrgr 40763 | A k-regular simple graph i... |
rusgrusgr 40764 | A k-regular simple graph i... |
finrusgrfusgr 40765 | A finite regular simple gr... |
isrusgr0 40766 | The property of being a k-... |
rusgrprop0 40767 | The properties of a k-regu... |
usgreqdrusgr 40768 | If all vertices in a simpl... |
fusgrregdegfi 40769 | In a nonempty finite simpl... |
fusgrn0eqdrusgr 40770 | If all vertices in a nonem... |
frusgrnn0 40771 | In a nonempty finite k-reg... |
0edg0rgr 40772 | A graph is 0-regular if it... |
uhgr0edg0rgr 40773 | A hypergraph is 0-regular ... |
uhgr0edg0rgrb 40774 | A hypergraph is 0-regular ... |
usgr0edg0rusgr 40775 | A simple graph is 0-regula... |
0vtxrgr 40776 | A graph with no vertices i... |
0vtxrusgr 40777 | A graph with no vertices a... |
0uhgrrusgr 40778 | The null graph as hypergra... |
0grrusgr 40779 | The null graph represented... |
0grrgr 40780 | The null graph represented... |
cusgrrusgr 40781 | A complete simple graph wi... |
cusgrm1rusgr 40782 | A finite simple graph with... |
rusgrpropnb 40783 | The properties of a k-regu... |
rusgrpropedg 40784 | The properties of a k-regu... |
rusgrpropadjvtx 40785 | The properties of a k-regu... |
rusgrnumwrdl2 40786 | In a k-regular simple grap... |
rusgr1vtxlem 40787 | Lemma for ~ rusgr1vtx . (... |
rusgr1vtx 40788 | If a k-regular simple grap... |
rgrusgrprc 40789 | The class of 0-regular sim... |
rusgrprc 40790 | The class of 0-regular sim... |
rgrprc 40791 | The class of 0-regular gra... |
rgrprcx 40792 | The class of 0-regular gra... |
rgrx0ndm 40793 | 0 is not in the domain of ... |
rgrx0nd 40794 | The potentially alternativ... |
ewlksfval 40803 | The set of s-walks of edge... |
isewlk 40804 | Conditions for a function ... |
ewlkprop 40805 | Properties of an s-walk of... |
ewlkinedg 40806 | The intersection (common v... |
ewlkle 40807 | An s-walk of edges is also... |
upgrewlkle2 40808 | In a pseudograph, there is... |
1wlkslem1 40809 | Lemma 1 for 1-walks to sub... |
1wlkslem2 40810 | Lemma 2 for 1-walks to sub... |
1wlksfval 40811 | The set of 1-walks (in an ... |
wlksfval 40812 | The set of walks (in an un... |
is1wlk 40813 | Properties of a pair of fu... |
isWlk 40814 | Properties of a pair of fu... |
wlkv 40815 | The classes involved in a ... |
is1wlkg 40816 | Generalisation of ~ is1wlk... |
wlkbProp 40817 | Basic properties of a walk... |
2m1wlk 40818 | The two mappings determini... |
1wlkf 40819 | The mapping enumerating th... |
1wlkcl 40820 | A 1-walk has length ` # ( ... |
1wlkp 40821 | The mapping enumerating th... |
1wlkpwrd 40822 | The sequence of vertices o... |
1wlklenvp1 40823 | The number of vertices of ... |
1wlksv 40824 | The class of 1-walks is a ... |
1wlkn0 40825 | The sequence of vertices o... |
1wlklenvm1 40826 | The number of edges of a w... |
1wlkvtxeledglem 40827 | Lemma for ~ 1wlkvtxeledg :... |
1wlkvtxeledg 40828 | Each pair of adjacent vert... |
1wlkvtxiedg 40829 | The vertices of a walk are... |
rel1wlk 40830 | The set ` ( 1Walks `` G ) ... |
1wlkvv 40831 | If there is at least one w... |
1wlkop 40832 | A walk is an ordered pair.... |
1wlkcpr 40833 | A walk as class with two c... |
1wlk2f 40834 | If there is a 1-walk ` W `... |
1wlkcomp 40835 | A walk expressed by proper... |
1wlkcompim 40836 | Implications for the prope... |
1wlkelwrd 40837 | The components of a walk a... |
1wlkeq 40838 | Conditions for two walks (... |
edginwlk 40839 | The value of the edge func... |
upgredginwlk 40840 | The value of the edge func... |
iedginwlk 40841 | The value of the edge func... |
1wlkl1loop 40842 | A 1-walk of length 1 from ... |
1wlk1walk 40843 | A 1-walk is a 1-walk "on t... |
1wlk1ewlk 40844 | A 1-walk is an s-walk "on ... |
ifpprsnss 40845 | An unordered pair is a sin... |
wlk1wlk 40846 | A walk is a 1-walk. (Cont... |
upgr1wlkwlk 40847 | In a pseudograph, a 1-walk... |
upgr1wlkwlkb 40848 | In a pseudograph, the defi... |
upgriswlk 40849 | Properties of a pair of fu... |
upgrwlkedg 40850 | The edges of a walk in a p... |
upgr1wlkcompim 40851 | Implications for the prope... |
1wlkvtxedg 40852 | The vertices of a walk are... |
upgr1wlkvtxedg 40853 | The pairs of connected ver... |
uspgr2wlkeq 40854 | Conditions for two walks w... |
uspgr2wlkeq2 40855 | Conditions for two walks w... |
uspgr2wlkeqi 40856 | Conditions for two walks w... |
umgr1wlknloop 40857 | In a multigraph, each walk... |
wlkRes 40858 | Restrictions of walks (i.e... |
1wlkv0 40859 | If there is a walk in the ... |
g01wlk0 40860 | There is no walk in a null... |
01wlk0 40861 | There is no walk for the e... |
1wlk0prc 40862 | There is no walk in a null... |
1wlklenvclwlk 40863 | The number of vertices in ... |
wlkson 40864 | The set of walks between t... |
iswlkOn 40865 | Properties of a pair of fu... |
wlkOnprop 40866 | Properties of a walk betwe... |
1wlkpvtx 40867 | A 1-walk connects vertices... |
1wlkepvtx 40868 | The endpoints of a walk ar... |
wlkOniswlk 40869 | A walk between two vertice... |
wlkOnwlk 40870 | A walk is a walk between i... |
wlkOnwlk1l 40871 | A walk is a walk from its ... |
wlksoneq1eq2 40872 | Two walks with identical s... |
wlkOnl1iedg 40873 | If there is a walk between... |
wlkOn2n0 40874 | The length of a walk betwe... |
2Wlklem 40875 | Lemma for ~ upgr2wlk and ~... |
upgr2wlk 40876 | Properties of a pair of fu... |
1wlkreslem0 40877 | Lemma for ~ 1wlkres . TOD... |
1wlkreslem 40878 | Lemma for ~ 1wlkres . (Co... |
1wlkres 40879 | The restriction ` <. H , Q... |
red1wlklem 40880 | Lemma for ~ red1wlk . (Co... |
red1wlk 40881 | A 1-walk ending at the las... |
1wlkp1lem1 40882 | Lemma for ~ 1wlkp1 . (Con... |
1wlkp1lem2 40883 | Lemma for ~ 1wlkp1 . (Con... |
1wlkp1lem3 40884 | Lemma for ~ 1wlkp1 . (Con... |
1wlkp1lem4 40885 | Lemma for ~ 1wlkp1 . (Con... |
1wlkp1lem5 40886 | Lemma for ~ 1wlkp1 . (Con... |
1wlkp1lem6 40887 | Lemma for ~ 1wlkp1 . (Con... |
1wlkp1lem7 40888 | Lemma for ~ 1wlkp1 . (Con... |
1wlkp1lem8 40889 | Lemma for ~ 1wlkp1 . (Con... |
1wlkp1 40890 | Append one path segment (e... |
1wlkdlem1 40891 | Lemma 1 for ~ 1wlkd . (Co... |
1wlkdlem2 40892 | Lemma 2 for ~ 1wlkd . (Co... |
1wlkdlem3 40893 | Lemma 3 for ~ 1wlkd . (Co... |
1wlkdlem4 40894 | Lemma 4 for ~ 1wlkd . (Co... |
1wlkd 40895 | Two words representing a w... |
lfgrwlkprop 40896 | Two adjacent vertices in a... |
lfgriswlk 40897 | Conditions for a pair of f... |
lfgr1wlknloop 40898 | In a loop-free graph, each... |
trlsfval 40903 | The set of trails (in an u... |
isTrl 40904 | Conditions for a pair of f... |
trlis1wlk 40905 | A trail is a walk. (Contr... |
trlf1 40906 | The enumeration ` F ` of a... |
trlreslem 40907 | Lemma for ~ trlres . Form... |
trlres 40908 | The restriction ` <. H , Q... |
upgrtrls 40909 | The set of trails in a pse... |
upgristrl 40910 | Properties of a pair of fu... |
upgrf1istrl 40911 | Properties of a pair of a ... |
1wlksonproplem 40912 | Lemma for theorems for pro... |
trlsonfval 40913 | The set of trails between ... |
istrlson 40914 | Properties of a pair of fu... |
trlsonprop 40915 | Properties of a trail betw... |
trlsonistrl 40916 | A trail between two vertic... |
trlsonwlkon 40917 | A trail between two vertic... |
trlOntrl 40918 | A trail is a trail between... |
pthsfval 40927 | The set of paths (in an un... |
spthsfval 40928 | The set of simple paths (i... |
isPth 40929 | Conditions for a pair of f... |
issPth 40930 | Conditions for a pair of f... |
PthisTrl 40931 | A path is a trail (in an u... |
sPthisPth 40932 | A simple path is a path (i... |
pthis1wlk 40933 | A path is a 1-walk (in an ... |
sPthis1wlk 40934 | A simple path is a 1-walk ... |
pthdivtx 40935 | The inner vertices of a pa... |
pthdadjvtx 40936 | The adjacent vertices of a... |
2pthnloop 40937 | A path of length at least ... |
upgr2pthnlp 40938 | A path of length at least ... |
sPthdifv 40939 | The vertices of a simple p... |
spthdep 40940 | A simple path (at least of... |
pthdepissPth 40941 | A path with different star... |
upgrwlkdvdelem 40942 | Lemma for ~ upgrwlkdvde . ... |
upgrwlkdvde 40943 | In a pseudograph, all edge... |
upgrspths1wlk 40944 | The set of simple paths in... |
upgrwlkdvspth 40945 | A walk consisting of diffe... |
pthsonfval 40946 | The set of paths between t... |
spthson 40947 | The set of simple paths be... |
ispthson 40948 | Properties of a pair of fu... |
isspthson 40949 | Properties of a pair of fu... |
pthsonprop 40950 | Properties of a path betwe... |
spthonprop 40951 | Properties of a simple pat... |
pthonispth-av 40952 | A path between two vertice... |
pthontrlon 40953 | A path between two vertice... |
pthOnpth 40954 | A path is a path between i... |
isspthonpth-av 40955 | A pair of functions is a s... |
spthonisspth-av 40956 | A simple path between to v... |
spthonpthon 40957 | A simple path between two ... |
spthonepeq-av 40958 | The endpoints of a simple ... |
uhgr1wlkspthlem1 40959 | Lemma 1 for ~ uhgr1wlkspth... |
uhgr1wlkspthlem2 40960 | Lemma 2 for ~ uhgr1wlkspth... |
uhgr1wlkspth 40961 | Any walk of length 1 betwe... |
usgr2wlkneq 40962 | The vertices and edges are... |
usgr2wlkspthlem1 40963 | Lemma 1 for ~ usgr2wlkspth... |
usgr2wlkspthlem2 40964 | Lemma 2 for ~ usgr2wlkspth... |
usgr2wlkspth 40965 | In a simple graph, any wal... |
usgr2trlncl 40966 | In a simple graph, any tra... |
usgr2trlspth 40967 | In a simple graph, any tra... |
usgr2pthspth 40968 | In a simple graph, any pat... |
usgr2pthlem 40969 | Lemma for ~ usgr2pth . (C... |
usgr2pth 40970 | In a simple graph, there i... |
usgr2pth0 40971 | In a simply graph, there i... |
pthdlem1 40972 | Lemma 1 for ~ pthd . (Con... |
pthdlem2lem 40973 | Lemma for ~ pthdlem2 . (C... |
pthdlem2 40974 | Lemma 2 for ~ pthd . (Con... |
pthd 40975 | Two words representing a t... |
clwlkS 40978 | The set of closed walks (i... |
isclWlk 40979 | Properties of a pair of fu... |
isclWlkb 40980 | Generalisation of ~ isclwl... |
clwlkis1wlk 40981 | A closed walk is a walk (i... |
clwlk1wlk 40982 | Closed walks are walks (in... |
clwlks1wlks 40983 | Closed walks are walks (in... |
isclWlke 40984 | Properties of a pair of fu... |
isclWlkupgr 40985 | Properties of a pair of fu... |
clWlkcomp 40986 | A closed walk expressed by... |
clWlkcompim 40987 | Implications for the prope... |
upgrclwlkcompim 40988 | Implications for the prope... |
clwlkl1loop 40989 | A closed walk of length 1 ... |
crctS 40994 | The set of circuits (in an... |
cyclS 40995 | The set of cycles (in an u... |
isCrct 40996 | Sufficient and necessary c... |
isCycl 40997 | Sufficient and necessary c... |
crctprop 40998 | The properties of a circui... |
cyclprop 40999 | The properties of a cycle:... |
crctisclwlk 41000 | A circuit is a closed walk... |
crctisTrl 41001 | A circuit is a trail. (Co... |
crctis1wlk 41002 | A circuit is a walk. (Con... |
cyclisPth 41003 | A cycle is a path. (Contr... |
cyclisWlk 41004 | A cycle is a walk. (Contr... |
cyclisCrct 41005 | A cycle is a circuit. (Co... |
cyclnsPth 41006 | A (non trivial) cycle is n... |
cyclisPthon 41007 | A cycle is a path starting... |
lfgrn1cycl 41008 | In a loop-free graph there... |
usgr2trlncrct 41009 | In a simple graph, any tra... |
umgrn1cycl 41010 | In a multigraph graph (wit... |
uspgrn2crct 41011 | In a simple pseudograph th... |
usgrn2cycl 41012 | In a simple graph there ar... |
crctcsh1wlkn0lem1 41013 | Lemma for ~ crctcsh1wlkn0 ... |
crctcsh1wlkn0lem2 41014 | Lemma for ~ crctcsh1wlkn0 ... |
crctcsh1wlkn0lem3 41015 | Lemma for ~ crctcsh1wlkn0 ... |
crctcsh1wlkn0lem4 41016 | Lemma for ~ crctcsh1wlkn0 ... |
crctcsh1wlkn0lem5 41017 | Lemma for ~ crctcsh1wlkn0 ... |
crctcsh1wlkn0lem6 41018 | Lemma for ~ crctcsh1wlkn0 ... |
crctcsh1wlkn0lem7 41019 | Lemma for ~ crctcsh1wlkn0 ... |
crctcshlem1 41020 | Lemma for ~ crctcsh . (Co... |
crctcshlem2 41021 | Lemma for ~ crctcsh . (Co... |
crctcshlem3 41022 | Lemma for ~ crctcsh . (Co... |
crctcshlem4 41023 | Lemma for ~ crctcsh . (Co... |
crctcsh1wlkn0 41024 | Cyclically shifting the in... |
crctcsh1wlk 41025 | Cyclically shifting the in... |
crctcshtrl 41026 | Cyclically shifting the in... |
crctcsh 41027 | Cyclically shifting the in... |
wwlks 41038 | The set of walks (in an un... |
iswwlks 41039 | A word over the set of ver... |
wwlksn 41040 | The set of walks (in an un... |
iswwlksn 41041 | A word over the set of ver... |
iswwlksnx 41042 | Properties of a word to re... |
wwlkbp 41043 | Basic properties of a walk... |
wwlknbp 41044 | Basic properties of a walk... |
wwlknp 41045 | Properties of a set being ... |
wspthsn 41046 | The set of simple paths of... |
iswspthn 41047 | An element of the set of s... |
wspthnp 41048 | Properties of a set being ... |
wwlksnon 41049 | The set of walks of a fixe... |
wspthsnon 41050 | The set of simple paths of... |
iswwlksnon 41051 | The set of walks of a fixe... |
iswspthsnon 41052 | The set of simple paths of... |
wwlknon 41053 | An element of the set of w... |
wspthnon 41054 | An element of the set of s... |
wspthnonp 41055 | Properties of a set being ... |
wspthneq1eq2 41056 | Two simple paths with iden... |
wwlksn0s 41057 | The set of all walks as wo... |
wwlkssswrd 41058 | Walks (represented by word... |
wwlksn0 41059 | A walk of length 0 is repr... |
0enwwlksnge1 41060 | In graphs without edges, t... |
wwlkswwlksn 41061 | A walk of a fixed length a... |
wwlkssswwlksn 41062 | The walks of a fixed lengt... |
wwlknbp2 41063 | Other basic properties of ... |
1wlkiswwlks1 41064 | The sequence of vertices i... |
1wlklnwwlkln1 41065 | The sequence of vertices i... |
1wlkiswwlks2lem1 41066 | Lemma 1 for ~ 1wlkiswwlks2... |
1wlkiswwlks2lem2 41067 | Lemma 2 for ~ 1wlkiswwlks2... |
1wlkiswwlks2lem3 41068 | Lemma 3 for ~ 1wlkiswwlks2... |
1wlkiswwlks2lem4 41069 | Lemma 4 for ~ 1wlkiswwlks2... |
1wlkiswwlks2lem5 41070 | Lemma 5 for ~ 1wlkiswwlks2... |
1wlkiswwlks2lem6 41071 | Lemma 6 for ~ 1wlkiswwlks2... |
1wlkiswwlks2 41072 | A walk as word corresponds... |
1wlkiswwlks 41073 | A walk as word corresponds... |
1wlkiswwlksupgr2 41074 | A walk as word corresponds... |
1wlkiswwlkupgr 41075 | A walk as word corresponds... |
1wlkpwwlkf1ouspgr 41076 | The mapping of (ordinary) ... |
1wlkisowwlkupgr 41077 | The set of walks as words ... |
wwlksm1edg 41078 | Removing the trailing edge... |
1wlklnwwlkln2lem 41079 | Lemma for ~ 1wlklnwwlkln2 ... |
1wlklnwwlkln2 41080 | A walk of length ` N ` as ... |
1wlklnwwlkn 41081 | A walk of length ` N ` as ... |
1wlklnwwlklnupgr2 41082 | A walk of length ` N ` as ... |
1wlklnwwlknupgr 41083 | A walk of length ` N ` as ... |
wlknewwlksn 41084 | If a walk in a pseudograph... |
wlknwwlksnfun 41085 | Lemma 1 for ~ wlknwwlksnbi... |
wlknwwlksninj 41086 | Lemma 2 for ~ wlknwwlksnbi... |
wlknwwlksnsur 41087 | Lemma 3 for ~ wlknwwlksnbi... |
wlknwwlksnbij 41088 | Lemma 4 for ~ wlknwwlksnbi... |
wlknwwlksnbij2 41089 | There is a bijection betwe... |
wlknwwlksnen 41090 | In a simple pseudograph, t... |
wlknwwlksneqs 41091 | The set of walks of a fixe... |
wlkwwlkfun 41092 | Lemma 1 for ~ wlkwwlkbij2 ... |
wlkwwlkinj 41093 | Lemma 2 for ~ wlkwwlkbij2 ... |
wlkwwlksur 41094 | Lemma 3 for ~ wlkwwlkbij2 ... |
wlkwwlkbij 41095 | Lemma 4 for ~ wlkwwlkbij2 ... |
wlkwwlkbij2 41096 | There is a bijection betwe... |
wwlkseq 41097 | Equality of two walks (as ... |
wwlksnred 41098 | Reduction of a walk (as wo... |
wwlksnext 41099 | Extension of a walk (as wo... |
wwlksnextbi 41100 | Extension of a walk (as wo... |
wwlksnredwwlkn 41101 | For each walk (as word) of... |
wwlksnredwwlkn0 41102 | For each walk (as word) of... |
wwlksnextwrd 41103 | Lemma for ~ wwlksnextbij .... |
wwlksnextfun 41104 | Lemma for ~ wwlksnextbij .... |
wwlksnextinj 41105 | Lemma for ~ wwlksnextbij .... |
wwlksnextsur 41106 | Lemma for ~ wwlksnextbij .... |
wwlksnextbij0 41107 | Lemma for ~ wwlksnextbij .... |
wwlksnextbij 41108 | There is a bijection betwe... |
wwlksnexthasheq 41109 | The number of the extensio... |
disjxwwlksn 41110 | Sets of walks (as words) e... |
wwlksnndef 41111 | Conditions for ` WWalkSN `... |
wwlksnfi 41112 | The number of walks repres... |
wlksnfi 41113 | The number of walks of fix... |
wlksnwwlknvbij 41114 | There is a bijection betwe... |
wwlksnextproplem1 41115 | Lemma 1 for ~ wwlkextprop ... |
wwlksnextproplem2 41116 | Lemma 2 for ~ wwlkextprop ... |
wwlksnextproplem3 41117 | Lemma 3 for ~ wwlkextprop ... |
wwlksnextprop 41118 | Adding additional properti... |
av-disjxwwlkn 41119 | Sets of walks (as words) e... |
hashwwlksnext 41120 | Number of walks (as words)... |
wwlksnwwlksnon 41121 | A walk of fixed length is ... |
wspthsnwspthsnon 41122 | A simple path of fixed len... |
wwlksnon0 41123 | Conditions for a set of wa... |
wspthsnonn0vne 41124 | If the set of simple paths... |
wspthsswwlkn 41125 | The set of simple paths of... |
wspthnfi 41126 | In a finite graph, the set... |
wwlksnonfi 41127 | In a finite graph, the set... |
wspthsswwlknon 41128 | The set of simple paths of... |
wspthnonfi 41129 | In a finite graph, the set... |
wspniunwspnon 41130 | The set of nonempty simple... |
wspn0 41131 | If there are no vertices, ... |
21wlkdlem1 41132 | Lemma 1 for ~ 21wlkd . (C... |
21wlkdlem2 41133 | Lemma 2 for ~ 21wlkd . (C... |
21wlkdlem3 41134 | Lemma 3 for ~ 21wlkd . (C... |
21wlkdlem4 41135 | Lemma 4 for ~ 21wlkd . (C... |
21wlkdlem5 41136 | Lemma 5 for ~ 21wlkd . (C... |
2pthdlem1 41137 | Lemma 1 for ~ 2pthd . (Co... |
21wlkdlem6 41138 | Lemma 6 for ~ 21wlkd . (C... |
21wlkdlem7 41139 | Lemma 7 for ~ 21wlkd . (C... |
21wlkdlem8 41140 | Lemma 8 for ~ 21wlkd . (C... |
21wlkdlem9 41141 | Lemma 9 for ~ 21wlkd . (C... |
21wlkdlem10 41142 | Lemma 10 for ~ 31wlkd . (... |
21wlkd 41143 | Construction of a walk fro... |
21wlkond 41144 | A 1-walk of length 2 from ... |
2trld 41145 | Construction of a trail fr... |
2trlond 41146 | A trail of length 2 from o... |
2pthd 41147 | A path of length 2 from on... |
2spthd 41148 | A simple path of length 2 ... |
2pthond 41149 | A simple path of length 2 ... |
2pthon3v-av 41150 | For a vertex adjacent to t... |
umgr2adedgwlklem 41151 | Lemma for ~ umgr2adedgwlk ... |
umgr2adedgwlk 41152 | In a multigraph, two adjac... |
umgr2adedgwlkon 41153 | In a multigraph, two adjac... |
umgr2adedgwlkonALT 41154 | Alternate proof for ~ umgr... |
umgr2adedgspth 41155 | In a multigraph, two adjac... |
umgr2wlk 41156 | In a multigraph, there is ... |
umgr2wlkon 41157 | For each pair of adjacent ... |
wwlks2onv 41158 | If a length 3 string repre... |
elwwlks2ons3 41159 | For each walk of length 2 ... |
s3wwlks2on 41160 | A length 3 string which re... |
umgrwwlks2on 41161 | A walk of length 2 between... |
elwwlks2on 41162 | A walk of length 2 between... |
elwspths2on 41163 | A simple path of length 2 ... |
wpthswwlks2on 41164 | For two different vertices... |
2wspdisj 41165 | All simple paths of length... |
2wspiundisj 41166 | All simple paths of length... |
usgr2wspthons3 41167 | A simple path of length 2 ... |
usgr2wspthon 41168 | A simple path of length 2 ... |
elwwlks2s3 41169 | A walk of length 2 between... |
elwwlks2 41170 | A walk of length 2 between... |
elwspths2spth 41171 | A simple path of length 2 ... |
rusgrnumwwlkl1 41172 | In a k-regular graph, ther... |
rusgrnumwwlklem 41173 | Lemma for ~ rusgrnumwwlk e... |
rusgrnumwwlkb0 41174 | Induction base 0 for ~ rus... |
rusgrnumwwlkb1 41175 | Induction base 1 for ~ rus... |
rusgr0edg 41176 | Special case for graphs wi... |
rusgrnumwwlks 41177 | Induction step for ~ rusgr... |
rusgrnumwwlk 41178 | In a ` K `-regular graph, ... |
rusgrnumwwlkg 41179 | In a ` K `-regular graph, ... |
rusgrnumwlkg 41180 | In a k-regular graph, the ... |
clwwlknclwwlkdifs 41181 | The set of walks of length... |
clwwlknclwwlkdifnum 41182 | In a k-regular graph, the ... |
clwwlks 41187 | The set of closed walks (i... |
isclwwlks 41188 | Properties of a word to re... |
clwwlksn 41189 | The set of closed walks (i... |
isclwwlksn 41190 | A word over the set of ver... |
clwwlkbp 41191 | Basic properties of a clos... |
clwwlknbp0 41192 | Basic properties of a clos... |
clwwlknbp 41193 | Basic properties of a clos... |
clwwlksnwrd 41194 | A closed walk of a fixed l... |
clwwlknp 41195 | Properties of a set being ... |
isclwwlksng 41196 | Properties of a word to re... |
isclwwlksnx 41197 | Properties of a word to re... |
clwwlksnndef 41198 | Conditions for ` ClWWalkSN... |
clwwlkclwwlkn 41199 | A closed walk of a fixed l... |
clwwlkssclwwlksn 41200 | The closed walks of a fixe... |
clwlkclwwlklem2a1 41201 | Lemma 1 for ~ clwlkclwwlkl... |
clwlkclwwlklem2a2 41202 | Lemma 2 for ~ clwlkclwwlkl... |
clwlkclwwlklem2a3 41203 | Lemma 3 for ~ clwlkclwwlkl... |
clwlkclwwlklem2fv1 41204 | Lemma 4a for ~ clwlkclwwlk... |
clwlkclwwlklem2fv2 41205 | Lemma 4b for ~ clwlkclwwlk... |
clwlkclwwlklem2a4 41206 | Lemma 4 for ~ clwlkclwwlkl... |
clwlkclwwlklem2a 41207 | Lemma for ~ clwlkclwwlklem... |
clwlkclwwlklem1 41208 | Lemma 1 for ~ clwlkclwwlk ... |
clwlkclwwlklem2 41209 | Lemma 2 for ~ clwlkclwwlk ... |
clwlkclwwlklem3 41210 | Lemma 3 for ~ clwlkclwwlk ... |
clwlkclwwlk 41211 | A closed walk as word of l... |
clwlkclwwlk2 41212 | A closed walk corresponds ... |
clwwlksgt0 41213 | There is no empty closed w... |
clwwlksn0 41214 | There is no closed walk of... |
clwwlks1loop 41215 | A closed walk of length 1 ... |
clwwlksn1loop 41216 | A closed walk of length 1 ... |
clwwlksn2 41217 | A closed walk of length 2 ... |
clwwlkssswrd 41218 | Closed walks (represented ... |
umgrclwwlksge2 41219 | A closed walk in a multigr... |
clwwlksnfi 41220 | If there is only a finite ... |
clwwlksel 41221 | Obtaining a closed walk (a... |
clwwlksf 41222 | Lemma 1 for ~ clwwlksbij :... |
clwwlksfv 41223 | Lemma 2 for ~ clwwlksbij :... |
clwwlksf1 41224 | Lemma 3 for ~ clwwlksbij :... |
clwwlksfo 41225 | Lemma 4 for ~ clwwlksbij :... |
clwwlksf1o 41226 | Lemma 5 for ~ clwwlksbij :... |
clwwlksbij 41227 | There is a bijection betwe... |
clwwlksnwwlkncl 41228 | Obtaining a closed walk (a... |
clwwlksvbij 41229 | There is a bijection betwe... |
clwwlksext2edg 41230 | If a word concatenated wit... |
wwlksext2clwwlk 41231 | If a word represents a wal... |
wwlksubclwwlks 41232 | Any prefix of a word repre... |
clwwisshclwwslemlem 41233 | Lemma for ~ clwwisshclwwle... |
clwwisshclwwslem 41234 | Lemma for ~ clwwisshclww .... |
clwwisshclwws 41235 | Cyclically shifting a clos... |
clwwisshclwwsn 41236 | Cyclically shifting a clos... |
clwwnisshclwwsn 41237 | Cyclically shifting a clos... |
erclwwlksrel 41238 | ` .~ ` is a relation. (Co... |
erclwwlkseq 41239 | Two classes are equivalent... |
erclwwlkseqlen 41240 | If two classes are equival... |
erclwwlksref 41241 | ` .~ ` is a reflexive rela... |
erclwwlkssym 41242 | ` .~ ` is a symmetric rela... |
erclwwlkstr 41243 | ` .~ ` is a transitive rel... |
erclwwlks 41244 | ` .~ ` is an equivalence r... |
eleclclwwlksnlem1 41245 | Lemma 1 for ~ eleclclwwlks... |
eleclclwwlksnlem2 41246 | Lemma 2 for ~ eleclclwwlks... |
clwwlksnscsh 41247 | The set of cyclical shifts... |
umgr2cwwk2dif 41248 | If a word represents a clo... |
umgr2cwwkdifex 41249 | If a word represents a clo... |
erclwwlksnrel 41250 | ` .~ ` is a relation. (Co... |
erclwwlksneq 41251 | Two classes are equivalent... |
erclwwlksneqlen 41252 | If two classes are equival... |
erclwwlksnref 41253 | ` .~ ` is a reflexive rela... |
erclwwlksnsym 41254 | ` .~ ` is a symmetric rela... |
erclwwlksntr 41255 | ` .~ ` is a transitive rel... |
erclwwlksn 41256 | ` .~ ` is an equivalence r... |
qerclwwlksnfi 41257 | The quotient set of the se... |
hashclwwlksn0 41258 | The number of closed walks... |
eclclwwlksn1 41259 | An equivalence class accor... |
eleclclwwlksn 41260 | A member of an equivalence... |
hashecclwwlksn1 41261 | The size of every equivale... |
umgrhashecclwwlk 41262 | The size of every equivale... |
fusgrhashclwwlkn 41263 | The size of the set of clo... |
clwwlksndivn 41264 | The size of the set of clo... |
clwlksfclwwlk2wrd 41265 | The second component of a ... |
clwlksfclwwlk1hashn 41266 | The size of the first comp... |
clwlksfclwwlk1hash 41267 | The size of the first comp... |
clwlksfclwwlk2sswd 41268 | The size of a subword of t... |
clwlksfclwwlk 41269 | There is a function betwee... |
clwlksfoclwwlk 41270 | There is an onto function ... |
clwlksf1clwwlklem0 41271 | Lemma 1 for ~ clwlksf1clww... |
clwlksf1clwwlklem1 41272 | Lemma 1 for ~ clwlksf1clww... |
clwlksf1clwwlklem2 41273 | Lemma 2 for ~ clwlksf1clww... |
clwlksf1clwwlklem3 41274 | Lemma 3 for ~ clwlksf1clww... |
clwlksf1clwwlklem 41275 | Lemma for ~ clwlksf1clwwlk... |
clwlksf1clwwlk 41276 | There is a one-to-one func... |
clwlksf1oclwwlk 41277 | There is a one-to-one onto... |
clwlkssizeeq 41278 | The size of the set of clo... |
clwlksndivn 41279 | The size of the set of clo... |
clwwlksndisj 41280 | The sets of closed walks s... |
clwwlksnun 41281 | The set of closed walks of... |
0ewlk 41282 | The empty set (empty seque... |
1ewlk 41283 | A sequence of 1 edge is an... |
01wlk 41284 | A pair of an empty set (of... |
is01wlk 41285 | A pair of an empty set (of... |
0wlkOnlem1 41286 | Lemma 1 for ~ 0wlkOn and ~... |
0wlkOnlem2 41287 | Lemma 2 for ~ 0wlkOn and ~... |
0wlkOn 41288 | A walk of length 0 from a ... |
0wlkOns1 41289 | A walk of length 0 from a ... |
0Trl 41290 | A pair of an empty set (of... |
is0Trl 41291 | A pair of an empty set (of... |
0TrlOn 41292 | A trail of length 0 from a... |
0pth-av 41293 | A pair of an empty set (of... |
0spth-av 41294 | A pair of an empty set (of... |
0pthon-av 41295 | A path of length 0 from a ... |
0pthon1-av 41296 | A path of length 0 from a ... |
0pthonv-av 41297 | For each vertex there is a... |
0clWlk 41298 | A pair of an empty set (of... |
0clwlk0 41299 | There is no closed walk in... |
0Crct 41300 | A pair of an empty set (of... |
0Cycl 41301 | A pair of an empty set (of... |
1pthdlem1 41302 | Lemma 1 for ~ 1pthd . (Co... |
1pthdlem2 41303 | Lemma 2 for ~ 1pthd . (Co... |
11wlkdlem1 41304 | Lemma 1 for ~ 11wlkd . (C... |
11wlkdlem2 41305 | Lemma 2 for ~ 11wlkd . (C... |
11wlkdlem3 41306 | Lemma 3 for ~ 11wlkd . (C... |
11wlkdlem4 41307 | Lemma 4 for ~ 11wlkd . (C... |
11wlkd 41308 | In a graph with two vertic... |
1trld 41309 | In a graph with two vertic... |
1pthd 41310 | In a graph with two vertic... |
1pthond 41311 | In a graph with two vertic... |
upgr11wlkdlem1 41312 | Lemma 1 for ~ upgr11wlkd .... |
upgr11wlkdlem2 41313 | Lemma 2 for ~ upgr11wlkd .... |
upgr11wlkd 41314 | In a pseudograph with two ... |
upgr1trld 41315 | In a pseudograph with two ... |
upgr1pthd 41316 | In a pseudograph with two ... |
upgr1pthond 41317 | In a pseudograph with two ... |
lppthon 41318 | A loop (which is an edge a... |
lp1cycl 41319 | A loop (which is an edge a... |
1pthon2v-av 41320 | For each pair of adjacent ... |
1pthon2ve 41321 | For each pair of adjacent ... |
1wlk2v2elem1 41322 | Lemma 1 for ~ 1wlk2v2e : `... |
1wlk2v2elem2 41323 | Lemma 2 for ~ 1wlk2v2e : ... |
1wlk2v2e 41324 | In a graph with two vertic... |
ntrl2v2e 41325 | A walk which is not a trai... |
31wlkdlem1 41326 | Lemma 1 for ~ 31wlkd . (C... |
31wlkdlem2 41327 | Lemma 2 for ~ 31wlkd . (C... |
31wlkdlem3 41328 | Lemma 3 for ~ 31wlkd . (C... |
31wlkdlem4 41329 | Lemma 4 for ~ 31wlkd . (C... |
31wlkdlem5 41330 | Lemma 5 for ~ 31wlkd . (C... |
3pthdlem1 41331 | Lemma 1 for ~ 3pthd . (Co... |
31wlkdlem6 41332 | Lemma 6 for ~ 31wlkd . (C... |
31wlkdlem7 41333 | Lemma 7 for ~ 31wlkd . (C... |
31wlkdlem8 41334 | Lemma 8 for ~ 31wlkd . (C... |
31wlkdlem9 41335 | Lemma 9 for ~ 31wlkd . (C... |
31wlkdlem10 41336 | Lemma 10 for ~ 31wlkd . (... |
31wlkd 41337 | Construction of a walk fro... |
31wlkond 41338 | A 1-walk of length 3 from ... |
3trld 41339 | Construction of a trail fr... |
3trlond 41340 | A trail of length 3 from o... |
3pthd 41341 | A path of length 3 from on... |
3pthond 41342 | A path of length 3 from on... |
3spthd 41343 | A simple path of length 3 ... |
3spthond 41344 | A simple path of length 3 ... |
3cycld 41345 | Construction of a 3-cycle ... |
3cyclpd 41346 | Construction of a 3-cycle ... |
upgr3v3e3cycl 41347 | If there is a cycle of len... |
uhgr3cyclexlem 41348 | Lemma for ~ uhgr3cyclex . ... |
uhgr3cyclex 41349 | If there are three differe... |
umgr3cyclex 41350 | If there are three (differ... |
umgr3v3e3cycl 41351 | If and only if there is a ... |
upgr4cycl4dv4e 41352 | If there is a cycle of len... |
dfconngr1 41355 | Alternative definition of ... |
isconngr 41356 | The property of being a co... |
isconngr1 41357 | The property of being a co... |
cusconngr 41358 | A complete hypergraph is c... |
0conngr 41359 | A graph without vertices i... |
0vconngr 41360 | A graph without vertices i... |
1conngr 41361 | A graph with (at most) one... |
conngrv2edg 41362 | A vertex in a connected gr... |
vdn0conngrumgrv2 41363 | A vertex in a connected mu... |
releupth 41366 | The set ` ( EulerPaths `` ... |
eupths 41367 | The Eulerian paths on the ... |
iseupth 41368 | The property " ` <. F , P ... |
iseupthf1o 41369 | The property " ` <. F , P ... |
eupthtrli 41370 | Properties of an Eulerian ... |
eupthi 41371 | Properties of an Eulerian ... |
eupthf1o 41372 | The ` F ` function in an E... |
eupthfi 41373 | Any graph with an Eulerian... |
eupthseg 41374 | The ` N ` -th edge in an e... |
upgriseupth 41375 | The property " ` <. F , P ... |
upgreupthi 41376 | Properties of an Eulerian ... |
upgreupthseg 41377 | The ` N ` -th edge in an e... |
eupthcl 41378 | An Eulerian path has lengt... |
eupthistrl 41379 | An Eulerian path is a trai... |
eupthis1wlk 41380 | An Eulerian path is a walk... |
eupthpf 41381 | The ` P ` function in an E... |
eupth0 41382 | There is an Eulerian path ... |
eupthres 41383 | The restriction ` <. H , Q... |
eupthp1 41384 | Append one path segment to... |
eupth2eucrct 41385 | Append one path segment to... |
eupth2lem1 41386 | TODO-AV: Duplicate of ~ e... |
eupth2lem2 41387 | TODO-AV: Duplicate of ~ e... |
trlsegvdeglem1 41388 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem2 41389 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem3 41390 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem4 41391 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem5 41392 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem6 41393 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeglem7 41394 | Lemma for ~ trlsegvdeg . ... |
trlsegvdeg 41395 | Formerly part of proof of ... |
eupth2lem3lem1 41396 | Lemma for ~ eupth2lem3 . ... |
eupth2lem3lem2 41397 | Lemma for ~ eupth2lem3 . ... |
eupth2lem3lem3 41398 | Lemma for ~ eupth2lem3 , f... |
eupth2lem3lem4 41399 | Lemma for ~ eupth2lem3 , f... |
eupth2lem3lem5 41400 | Lemma for ~ eupath2 . (Co... |
eupth2lem3lem6 41401 | Formerly part of proof of ... |
eupth2lem3lem7 41402 | Lemma for ~ eupath2lem3 : ... |
eupthvdres 41403 | Formerly part of proof of ... |
eupth2lem3 41404 | Lemma for ~ eupath2 . (Co... |
eupth2lemb 41405 | Lemma for ~ eupth2 (induct... |
eupth2lems 41406 | Lemma for ~ eupth2 (induct... |
eupth2 41407 | The only vertices of odd d... |
eulerpathpr 41408 | A graph with an Eulerian p... |
eulerpath 41409 | A pseudograph with an Eule... |
eulercrct 41410 | A pseudograph with an Eule... |
eucrctshift 41411 | Cyclically shifting the in... |
eucrct2eupth1 41412 | Removing one edge ` ( I ``... |
eucrct2eupth 41413 | Removing one edge ` ( I ``... |
konigsbergvtx 41414 | The set of vertices of the... |
konigsbergiedg 41415 | The indexed edges of the K... |
konigsbergiedgw 41416 | The indexed edges of the K... |
konigsbergiedgwOLD 41417 | The indexed edges of the K... |
konigsbergssiedgwpr 41418 | Each subset of the indexed... |
konigsbergssiedgw 41419 | Each subset of the indexed... |
konigsbergumgr 41420 | The Königsberg graph ... |
konigsbergupgrOLD 41421 | The Königsberg graph ... |
konigsberglem1 41422 | Lemma 1 for ~ konigsberg-a... |
konigsberglem2 41423 | Lemma 2 for ~ konigsberg-a... |
konigsberglem3 41424 | Lemma 3 for ~ konigsberg-a... |
konigsberglem4 41425 | Lemma 4 for ~ konigsberg-a... |
konigsberglem5 41426 | Lemma 5 for ~ konigsberg-a... |
konigsberg-av 41427 | The Königsberg Bridge... |
isfrgr 41430 | The property of being a fr... |
frgrusgrfrcond 41431 | A friendship graph is a si... |
frgrusgr 41432 | A friendship graph is a si... |
frgr0v 41433 | Any null graph (set with n... |
frgr0vb 41434 | Any null graph (without ve... |
frgruhgr0v 41435 | Any null graph (without ve... |
frgr0 41436 | The null graph (graph with... |
rspc2vd 41437 | Deduction version of 2-var... |
frcond1 41438 | The friendship condition: ... |
frcond2 41439 | The friendship condition: ... |
frcond3 41440 | The friendship condition, ... |
frgr1v 41441 | Any graph with (at most) o... |
nfrgr2v 41442 | Any graph with two (differ... |
frgr3vlem1 41443 | Lemma 1 for ~ frgra3v . (... |
frgr3vlem2 41444 | Lemma 2 for ~ frgra3v . (... |
frgr3v 41445 | Any graph with three verti... |
1vwmgr 41446 | Every graph with one verte... |
3vfriswmgrlem 41447 | Lemma for ~ 3vfriswmgra . ... |
3vfriswmgr 41448 | Every friendship graph wit... |
1to2vfriswmgr 41449 | Every friendship graph wit... |
1to3vfriswmgr 41450 | Every friendship graph wit... |
1to3vfriendship-av 41451 | The friendship theorem for... |
2pthfrgrrn 41452 | Between any two (different... |
2pthfrgrrn2 41453 | Between any two (different... |
2pthfrgr 41454 | Between any two (different... |
3cyclfrgrrn1 41455 | Every vertex in a friendsh... |
3cyclfrgrrn 41456 | Every vertex in a friendsh... |
3cyclfrgrrn2 41457 | Every vertex in a friendsh... |
3cyclfrgr 41458 | Every vertex in a friendsh... |
4cycl2v2nb-av 41459 | In a (maybe degenerated) 4... |
4cycl2vnunb-av 41460 | In a 4-cycle, two distinct... |
n4cyclfrgr 41461 | There is no 4-cycle in a f... |
4cyclusnfrgr 41462 | A graph with a 4-cycle is ... |
frgrnbnb 41463 | If two neighbors ` U ` and... |
frgrconngr 41464 | A friendship graph is conn... |
vdgn0frgrv2 41465 | A vertex in a friendship g... |
vdgn1frgrv2 41466 | Any vertex in a friendship... |
vdgn1frgrv3 41467 | Any vertex in a friendship... |
vdgfrgrgt2 41468 | Any vertex in a friendship... |
frgrncvvdeqlem1 41469 | Lemma 1 for ~ frgrncvvdeq ... |
frgrncvvdeqlem2 41470 | Lemma 2 for ~ frgrncvvdeq ... |
frgrncvvdeqlem3 41471 | Lemma 3 for ~ frgrncvvdeq ... |
frgrncvvdeqlem4 41472 | Lemma 4 for ~ frgrncvvdeq ... |
frgrncvvdeqlem5 41473 | Lemma 5 for ~ frgrncvvdeq ... |
frgrncvvdeqlem6 41474 | Lemma 6 for ~ frgrncvvdeq ... |
frgrncvvdeqlem7 41475 | Lemma 7 for ~ frgrncvvdeq ... |
frgrncvvdeqlemA 41476 | Lemma A for ~ frgrncvvdeq ... |
frgrncvvdeqlemB 41477 | Lemma B for ~ frgrncvvdeq ... |
frgrncvvdeqlemC 41478 | Lemma C for ~ frgrncvvdeq ... |
frgrncvvdeqlem8 41479 | Lemma 8 for ~ frgrncvvdeq ... |
frgrncvvdeq 41480 | In a friendship graph, two... |
frgrwopreglem1 41481 | Lemma 1 for ~ frgrwopreg :... |
frgrwopreglem2 41482 | Lemma 2 for ~ frgrwopreg .... |
frgrwopreglem3 41483 | Lemma 3 for ~ frgrwopreg .... |
frgrwopreglem4 41484 | Lemma 4 for ~ frgrwopreg .... |
frgrwopreglem5 41485 | Lemma 5 for ~ frgrwopreg .... |
frgrwopreg 41486 | In a friendship graph ther... |
frgrwopreg1 41487 | According to statement 5 i... |
frgrwopreg2 41488 | According to statement 5 i... |
frgrregorufr0 41489 | In a friendship graph ther... |
frgrregorufr 41490 | If there is a vertex havin... |
frgreu 41491 | Any two (different) vertic... |
frgr2wwlkeu 41492 | For two different vertices... |
frgr2wwlkn0 41493 | In a friendship graph, the... |
frgr2wwlk1 41494 | In a friendship graph, the... |
frgr2wsp1 41495 | In a friendship graph, the... |
frgr2wwlkeqm 41496 | If there is a (simple) pat... |
frgrhash2wsp 41497 | The number of simple paths... |
fusgr2wsp2nb 41498 | The set of paths of length... |
fusgreghash2wspv 41499 | According to statement 7 i... |
fusgreg2wsp 41500 | In a finite simple graph, ... |
2wspmdisj 41501 | The sets of paths of lengt... |
fusgreghash2wsp 41502 | In a finite k-regular grap... |
frrusgrord0 41503 | If a nonempty finite frien... |
frrusgrord 41504 | If a nonempty finite frien... |
frgrregorufrg 41505 | If there is a vertex havin... |
av-numclwlk3lem3 41506 | Lemma 3 for ~ numclwwlk3 .... |
av-extwwlkfablem2lem 41507 | Lemma for ~ extwwlkfablem2... |
av-extwwlkfablem1 41508 | Lemma 1 for ~ av-extwwlkfa... |
av-clwwlkextfrlem1 41509 | Lemma for ~ av-numclwwlk2l... |
av-extwwlkfablem2 41510 | Lemma 2 for ~ av-extwwlkfa... |
av-numclwwlkovf 41511 | Value of operation ` F ` ,... |
av-numclwwlkffin 41512 | In a finite graph, the val... |
av-numclwwlkffin0 41513 | In a finite graph, the val... |
av-numclwwlkovfel2 41514 | Properties of an element o... |
av-numclwwlkovf2 41515 | Value of operation ` F ` f... |
av-numclwwlkovf2num 41516 | In a ` K `-regular graph, ... |
av-numclwwlkovf2ex 41517 | Extending a closed walk st... |
av-numclwwlkovg 41518 | Value of operation ` C ` ,... |
av-numclwwlkovgel 41519 | Properties of an element o... |
av-extwwlkfab 41520 | The set ` ( X C N ) ` of c... |
av-numclwlk1lem2foa 41521 | Going forth and back form ... |
av-numclwlk1lem2f 41522 | ` T ` is a function, mappi... |
av-numclwlk1lem2fv 41523 | Value of the function ` T ... |
av-numclwlk1lem2f1 41524 | ` T ` is a 1-1 function. ... |
av-numclwlk1lem2fo 41525 | ` T ` is an onto function.... |
av-numclwlk1lem2f1o 41526 | ` T ` is a 1-1 onto functi... |
av-numclwlk1lem2 41527 | There is a bijection betwe... |
av-numclwwlk1 41528 | Statement 9 in [Huneke] p.... |
av-numclwwlkovq 41529 | Value of operation ` Q ` ,... |
av-numclwwlkqhash 41530 | In a ` K `-regular graph, ... |
av-numclwwlkovh 41531 | Value of operation ` H ` ,... |
av-numclwwlk2lem1 41532 | In a friendship graph, for... |
av-numclwlk2lem2f 41533 | ` R ` is a function mappin... |
av-numclwlk2lem2fv 41534 | Value of the function R. (... |
av-numclwlk2lem2f1o 41535 | R is a 1-1 onto function. ... |
av-numclwwlk2lem3 41536 | In a friendship graph, the... |
av-numclwwlk2 41537 | Statement 10 in [Huneke] p... |
av-numclwwlk3lem 41538 | Lemma for ~ av-numclwwlk3 ... |
av-numclwwlk3 41539 | Statement 12 in [Huneke] p... |
av-numclwwlk4 41540 | The total number of closed... |
av-numclwwlk5lem 41541 | Lemma for ~ av-numclwwlk5 ... |
av-numclwwlk5 41542 | Statement 13 in [Huneke] p... |
av-numclwwlk7lem 41543 | Lemma for ~ av-numclwwlk7 ... |
av-numclwwlk6 41544 | For a prime divisor ` P ` ... |
av-numclwwlk7 41545 | Statement 14 in [Huneke] p... |
av-numclwwlk8 41546 | The size of the set of clo... |
av-frgrareggt1 41547 | If a finite nonempty frien... |
av-frgrareg 41548 | If a finite nonempty frien... |
av-frgraregord013 41549 | If a finite friendship gra... |
av-frgraregord13 41550 | If a nonempty finite frien... |
av-frgraogt3nreg 41551 | If a finite friendship gra... |
av-friendshipgt3 41552 | The friendship theorem for... |
av-friendship 41553 | The friendship theorem: I... |
ovn0dmfun 41554 | If a class' operation valu... |
xpsnopab 41555 | A Cartesian product with a... |
xpiun 41556 | A Cartesian product expres... |
ovn0ssdmfun 41557 | If a class' operation valu... |
fnxpdmdm 41558 | The domain of the domain o... |
cnfldsrngbas 41559 | The base set of a subring ... |
cnfldsrngadd 41560 | The group addition operati... |
cnfldsrngmul 41561 | The ring multiplication op... |
plusfreseq 41562 | If the empty set is not co... |
mgmplusfreseq 41563 | If the empty set is not co... |
0mgm 41564 | A set with an empty base s... |
mgmpropd 41565 | If two structures have the... |
ismgmd 41566 | Deduce a magma from its pr... |
mgmhmrcl 41571 | Reverse closure of a magma... |
submgmrcl 41572 | Reverse closure for submag... |
ismgmhm 41573 | Property of a magma homomo... |
mgmhmf 41574 | A magma homomorphism is a ... |
mgmhmpropd 41575 | Magma homomorphism depends... |
mgmhmlin 41576 | A magma homomorphism prese... |
mgmhmf1o 41577 | A magma homomorphism is bi... |
idmgmhm 41578 | The identity homomorphism ... |
issubmgm 41579 | Expand definition of a sub... |
issubmgm2 41580 | Submagmas are subsets that... |
rabsubmgmd 41581 | Deduction for proving that... |
submgmss 41582 | Submagmas are subsets of t... |
submgmid 41583 | Every magma is trivially a... |
submgmcl 41584 | Submagmas are closed under... |
submgmmgm 41585 | Submagmas are themselves m... |
submgmbas 41586 | The base set of a submagma... |
subsubmgm 41587 | A submagma of a submagma i... |
resmgmhm 41588 | Restriction of a magma hom... |
resmgmhm2 41589 | One direction of ~ resmgmh... |
resmgmhm2b 41590 | Restriction of the codomai... |
mgmhmco 41591 | The composition of magma h... |
mgmhmima 41592 | The homomorphic image of a... |
mgmhmeql 41593 | The equalizer of two magma... |
submgmacs 41594 | Submagmas are an algebraic... |
ismhm0 41595 | Property of a monoid homom... |
mhmismgmhm 41596 | Each monoid homomorphism i... |
opmpt2ismgm 41597 | A structure with a group a... |
copissgrp 41598 | A structure with a constan... |
copisnmnd 41599 | A structure with a constan... |
0nodd 41600 | 0 is not an odd integer. ... |
1odd 41601 | 1 is an odd integer. (Con... |
2nodd 41602 | 2 is not an odd integer. ... |
oddibas 41603 | Lemma 1 for ~ oddinmgm : ... |
oddiadd 41604 | Lemma 2 for ~ oddinmgm : ... |
oddinmgm 41605 | The structure of all odd i... |
nnsgrpmgm 41606 | The structure of positive ... |
nnsgrp 41607 | The structure of positive ... |
nnsgrpnmnd 41608 | The structure of positive ... |
iscllaw 41615 | The predicate "is a closed... |
iscomlaw 41616 | The predicate "is a commut... |
clcllaw 41617 | Closure of a closed operat... |
isasslaw 41618 | The predicate "is an assoc... |
asslawass 41619 | Associativity of an associ... |
mgmplusgiopALT 41620 | Slot 2 (group operation) o... |
sgrpplusgaopALT 41621 | Slot 2 (group operation) o... |
intopval 41628 | The internal (binary) oper... |
intop 41629 | An internal (binary) opera... |
clintopval 41630 | The closed (internal binar... |
assintopval 41631 | The associative (closed in... |
assintopmap 41632 | The associative (closed in... |
isclintop 41633 | The predicate "is a closed... |
clintop 41634 | A closed (internal binary)... |
assintop 41635 | An associative (closed int... |
isassintop 41636 | The predicate "is an assoc... |
clintopcllaw 41637 | The closure law holds for ... |
assintopcllaw 41638 | The closure low holds for ... |
assintopasslaw 41639 | The associative low holds ... |
assintopass 41640 | An associative (closed int... |
ismgmALT 41649 | The predicate "is a magma.... |
iscmgmALT 41650 | The predicate "is a commut... |
issgrpALT 41651 | The predicate "is a semigr... |
iscsgrpALT 41652 | The predicate "is a commut... |
mgm2mgm 41653 | Equivalence of the two def... |
sgrp2sgrp 41654 | Equivalence of the two def... |
idfusubc0 41655 | The identity functor for a... |
idfusubc 41656 | The identity functor for a... |
inclfusubc 41657 | The "inclusion functor" fr... |
lmod0rng 41658 | If the scalar ring of a mo... |
nzrneg1ne0 41659 | The additive inverse of th... |
0ringdif 41660 | A zero ring is a ring whic... |
0ringbas 41661 | The base set of a zero rin... |
0ring1eq0 41662 | In a zero ring, a ring whi... |
nrhmzr 41663 | There is no ring homomorph... |
isrng 41666 | The predicate "is a non-un... |
rngabl 41667 | A non-unital ring is an (a... |
rngmgp 41668 | A non-unital ring is a sem... |
ringrng 41669 | A unital ring is a (non-un... |
ringssrng 41670 | The unital rings are (non-... |
isringrng 41671 | The predicate "is a unital... |
rngdir 41672 | Distributive law for the m... |
rngcl 41673 | Closure of the multiplicat... |
rnglz 41674 | The zero of a nonunital ri... |
rnghmrcl 41679 | Reverse closure of a non-u... |
rnghmfn 41680 | The mapping of two non-uni... |
rnghmval 41681 | The set of the non-unital ... |
isrnghm 41682 | A function is a non-unital... |
isrnghmmul 41683 | A function is a non-unital... |
rnghmmgmhm 41684 | A non-unital ring homomorp... |
rnghmval2 41685 | The non-unital ring homomo... |
isrngisom 41686 | An isomorphism of non-unit... |
rngimrcl 41687 | Reverse closure for an iso... |
rnghmghm 41688 | A non-unital ring homomorp... |
rnghmf 41689 | A ring homomorphism is a f... |
rnghmmul 41690 | A homomorphism of non-unit... |
isrnghm2d 41691 | Demonstration of non-unita... |
isrnghmd 41692 | Demonstration of non-unita... |
rnghmf1o 41693 | A non-unital ring homomorp... |
isrngim 41694 | An isomorphism of non-unit... |
rngimf1o 41695 | An isomorphism of non-unit... |
rngimrnghm 41696 | An isomorphism of non-unit... |
rnghmco 41697 | The composition of non-uni... |
idrnghm 41698 | The identity homomorphism ... |
c0mgm 41699 | The constant mapping to ze... |
c0mhm 41700 | The constant mapping to ze... |
c0ghm 41701 | The constant mapping to ze... |
c0rhm 41702 | The constant mapping to ze... |
c0rnghm 41703 | The constant mapping to ze... |
c0snmgmhm 41704 | The constant mapping to ze... |
c0snmhm 41705 | The constant mapping to ze... |
c0snghm 41706 | The constant mapping to ze... |
zrrnghm 41707 | The constant mapping to ze... |
rhmfn 41708 | The mapping of two rings t... |
rhmval 41709 | The ring homomorphisms bet... |
rhmisrnghm 41710 | Each unital ring homomorph... |
lidldomn1 41711 | If a (left) ideal (which i... |
lidlssbas 41712 | The base set of the restri... |
lidlbas 41713 | A (left) ideal of a ring i... |
lidlabl 41714 | A (left) ideal of a ring i... |
lidlmmgm 41715 | The multiplicative group o... |
lidlmsgrp 41716 | The multiplicative group o... |
lidlrng 41717 | A (left) ideal of a ring i... |
zlidlring 41718 | The zero (left) ideal of a... |
uzlidlring 41719 | Only the zero (left) ideal... |
lidldomnnring 41720 | A (left) ideal of a domain... |
0even 41721 | 0 is an even integer. (Co... |
1neven 41722 | 1 is not an even integer. ... |
2even 41723 | 2 is an even integer. (Co... |
2zlidl 41724 | The even integers are a (l... |
2zrng 41725 | The ring of integers restr... |
2zrngbas 41726 | The base set of R is the s... |
2zrngadd 41727 | The group addition operati... |
2zrng0 41728 | The additive identity of R... |
2zrngamgm 41729 | R is an (additive) magma. ... |
2zrngasgrp 41730 | R is an (additive) semigro... |
2zrngamnd 41731 | R is an (additive) monoid.... |
2zrngacmnd 41732 | R is a commutative (additi... |
2zrngagrp 41733 | R is an (additive) group. ... |
2zrngaabl 41734 | R is an (additive) abelian... |
2zrngmul 41735 | The ring multiplication op... |
2zrngmmgm 41736 | R is a (multiplicative) ma... |
2zrngmsgrp 41737 | R is a (multiplicative) se... |
2zrngALT 41738 | The ring of integers restr... |
2zrngnmlid 41739 | R has no multiplicative (l... |
2zrngnmrid 41740 | R has no multiplicative (r... |
2zrngnmlid2 41741 | R has no multiplicative (l... |
2zrngnring 41742 | R is not a unital ring. (... |
plusgndxnmulrndx 41743 | The slot for the group (ad... |
basendxnmulrndx 41744 | The slot for the base set ... |
cznrnglem 41745 | Lemma for ~ cznrng : The ... |
cznabel 41746 | The ring constructed from ... |
cznrng 41747 | The ring constructed from ... |
cznnring 41748 | The ring constructed from ... |
rngcvalALTV 41753 | Value of the category of n... |
rngcval 41754 | Value of the category of n... |
rnghmresfn 41755 | The class of non-unital ri... |
rnghmresel 41756 | An element of the non-unit... |
rngcbas 41757 | Set of objects of the cate... |
rngchomfval 41758 | Set of arrows of the categ... |
rngchom 41759 | Set of arrows of the categ... |
elrngchom 41760 | A morphism of non-unital r... |
rngchomfeqhom 41761 | The functionalized Hom-set... |
rngccofval 41762 | Composition in the categor... |
rngcco 41763 | Composition in the categor... |
dfrngc2 41764 | Alternate definition of th... |
rnghmsscmap2 41765 | The non-unital ring homomo... |
rnghmsscmap 41766 | The non-unital ring homomo... |
rnghmsubcsetclem1 41767 | Lemma 1 for ~ rnghmsubcset... |
rnghmsubcsetclem2 41768 | Lemma 2 for ~ rnghmsubcset... |
rnghmsubcsetc 41769 | The non-unital ring homomo... |
rngccat 41770 | The category of non-unital... |
rngcid 41771 | The identity arrow in the ... |
rngcsect 41772 | A section in the category ... |
rngcinv 41773 | An inverse in the category... |
rngciso 41774 | An isomorphism in the cate... |
rngcbasALTV 41775 | Set of objects of the cate... |
rngchomfvalALTV 41776 | Set of arrows of the categ... |
rngchomALTV 41777 | Set of arrows of the categ... |
elrngchomALTV 41778 | A morphism of non-unital r... |
rngccofvalALTV 41779 | Composition in the categor... |
rngccoALTV 41780 | Composition in the categor... |
rngccatidALTV 41781 | Lemma for ~ rngccatALTV . ... |
rngccatALTV 41782 | The category of non-unital... |
rngcidALTV 41783 | The identity arrow in the ... |
rngcsectALTV 41784 | A section in the category ... |
rngcinvALTV 41785 | An inverse in the category... |
rngcisoALTV 41786 | An isomorphism in the cate... |
rngchomffvalALTV 41787 | The value of the functiona... |
rngchomrnghmresALTV 41788 | The value of the functiona... |
rngcifuestrc 41789 | The "inclusion functor" fr... |
funcrngcsetc 41790 | The "natural forgetful fun... |
funcrngcsetcALT 41791 | Alternate proof of ~ funcr... |
zrinitorngc 41792 | The zero ring is an initia... |
zrtermorngc 41793 | The zero ring is a termina... |
zrzeroorngc 41794 | The zero ring is a zero ob... |
ringcvalALTV 41799 | Value of the category of r... |
ringcval 41800 | Value of the category of u... |
rhmresfn 41801 | The class of unital ring h... |
rhmresel 41802 | An element of the unital r... |
ringcbas 41803 | Set of objects of the cate... |
ringchomfval 41804 | Set of arrows of the categ... |
ringchom 41805 | Set of arrows of the categ... |
elringchom 41806 | A morphism of unital rings... |
ringchomfeqhom 41807 | The functionalized Hom-set... |
ringccofval 41808 | Composition in the categor... |
ringcco 41809 | Composition in the categor... |
dfringc2 41810 | Alternate definition of th... |
rhmsscmap2 41811 | The unital ring homomorphi... |
rhmsscmap 41812 | The unital ring homomorphi... |
rhmsubcsetclem1 41813 | Lemma 1 for ~ rhmsubcsetc ... |
rhmsubcsetclem2 41814 | Lemma 2 for ~ rhmsubcsetc ... |
rhmsubcsetc 41815 | The unital ring homomorphi... |
ringccat 41816 | The category of unital rin... |
ringcid 41817 | The identity arrow in the ... |
rhmsscrnghm 41818 | The unital ring homomorphi... |
rhmsubcrngclem1 41819 | Lemma 1 for ~ rhmsubcrngc ... |
rhmsubcrngclem2 41820 | Lemma 2 for ~ rhmsubcrngc ... |
rhmsubcrngc 41821 | The unital ring homomorphi... |
rngcresringcat 41822 | The restriction of the cat... |
ringcsect 41823 | A section in the category ... |
ringcinv 41824 | An inverse in the category... |
ringciso 41825 | An isomorphism in the cate... |
ringcbasbas 41826 | An element of the base set... |
funcringcsetc 41827 | The "natural forgetful fun... |
funcringcsetcALTV2lem1 41828 | Lemma 1 for ~ funcringcset... |
funcringcsetcALTV2lem2 41829 | Lemma 2 for ~ funcringcset... |
funcringcsetcALTV2lem3 41830 | Lemma 3 for ~ funcringcset... |
funcringcsetcALTV2lem4 41831 | Lemma 4 for ~ funcringcset... |
funcringcsetcALTV2lem5 41832 | Lemma 5 for ~ funcringcset... |
funcringcsetcALTV2lem6 41833 | Lemma 6 for ~ funcringcset... |
funcringcsetcALTV2lem7 41834 | Lemma 7 for ~ funcringcset... |
funcringcsetcALTV2lem8 41835 | Lemma 8 for ~ funcringcset... |
funcringcsetcALTV2lem9 41836 | Lemma 9 for ~ funcringcset... |
funcringcsetcALTV2 41837 | The "natural forgetful fun... |
ringcbasALTV 41838 | Set of objects of the cate... |
ringchomfvalALTV 41839 | Set of arrows of the categ... |
ringchomALTV 41840 | Set of arrows of the categ... |
elringchomALTV 41841 | A morphism of rings is a f... |
ringccofvalALTV 41842 | Composition in the categor... |
ringccoALTV 41843 | Composition in the categor... |
ringccatidALTV 41844 | Lemma for ~ ringccatALTV .... |
ringccatALTV 41845 | The category of rings is a... |
ringcidALTV 41846 | The identity arrow in the ... |
ringcsectALTV 41847 | A section in the category ... |
ringcinvALTV 41848 | An inverse in the category... |
ringcisoALTV 41849 | An isomorphism in the cate... |
ringcbasbasALTV 41850 | An element of the base set... |
funcringcsetclem1ALTV 41851 | Lemma 1 for ~ funcringcset... |
funcringcsetclem2ALTV 41852 | Lemma 2 for ~ funcringcset... |
funcringcsetclem3ALTV 41853 | Lemma 3 for ~ funcringcset... |
funcringcsetclem4ALTV 41854 | Lemma 4 for ~ funcringcset... |
funcringcsetclem5ALTV 41855 | Lemma 5 for ~ funcringcset... |
funcringcsetclem6ALTV 41856 | Lemma 6 for ~ funcringcset... |
funcringcsetclem7ALTV 41857 | Lemma 7 for ~ funcringcset... |
funcringcsetclem8ALTV 41858 | Lemma 8 for ~ funcringcset... |
funcringcsetclem9ALTV 41859 | Lemma 9 for ~ funcringcset... |
funcringcsetcALTV 41860 | The "natural forgetful fun... |
irinitoringc 41861 | The ring of integers is an... |
zrtermoringc 41862 | The zero ring is a termina... |
zrninitoringc 41863 | The zero ring is not an in... |
nzerooringczr 41864 | There is no zero object in... |
srhmsubclem1 41865 | Lemma 1 for ~ srhmsubc . ... |
srhmsubclem2 41866 | Lemma 2 for ~ srhmsubc . ... |
srhmsubclem3 41867 | Lemma 3 for ~ srhmsubc . ... |
srhmsubc 41868 | According to ~ df-subc , t... |
sringcat 41869 | The restriction of the cat... |
crhmsubc 41870 | According to ~ df-subc , t... |
cringcat 41871 | The restriction of the cat... |
drhmsubc 41872 | According to ~ df-subc , t... |
drngcat 41873 | The restriction of the cat... |
fldcat 41874 | The restriction of the cat... |
fldc 41875 | The restriction of the cat... |
fldhmsubc 41876 | According to ~ df-subc , t... |
rngcrescrhm 41877 | The category of non-unital... |
rhmsubclem1 41878 | Lemma 1 for ~ rhmsubc . (... |
rhmsubclem2 41879 | Lemma 2 for ~ rhmsubc . (... |
rhmsubclem3 41880 | Lemma 3 for ~ rhmsubc . (... |
rhmsubclem4 41881 | Lemma 4 for ~ rhmsubc . (... |
rhmsubc 41882 | According to ~ df-subc , t... |
rhmsubccat 41883 | The restriction of the cat... |
srhmsubcALTVlem1 41884 | Lemma 1 for ~ srhmsubcALTV... |
srhmsubcALTVlem2 41885 | Lemma 2 for ~ srhmsubcALTV... |
srhmsubcALTVlem3 41886 | Lemma 3 for ~ srhmsubcALTV... |
srhmsubcALTV 41887 | According to ~ df-subc , t... |
sringcatALTV 41888 | The restriction of the cat... |
crhmsubcALTV 41889 | According to ~ df-subc , t... |
cringcatALTV 41890 | The restriction of the cat... |
drhmsubcALTV 41891 | According to ~ df-subc , t... |
drngcatALTV 41892 | The restriction of the cat... |
fldcatALTV 41893 | The restriction of the cat... |
fldcALTV 41894 | The restriction of the cat... |
fldhmsubcALTV 41895 | According to ~ df-subc , t... |
rngcrescrhmALTV 41896 | The category of non-unital... |
rhmsubcALTVlem1 41897 | Lemma 1 for ~ rhmsubcALTV ... |
rhmsubcALTVlem2 41898 | Lemma 2 for ~ rhmsubcALTV ... |
rhmsubcALTVlem3 41899 | Lemma 3 for ~ rhmsubcALTV ... |
rhmsubcALTVlem4 41900 | Lemma 4 for ~ rhmsubcALTV ... |
rhmsubcALTV 41901 | According to ~ df-subc , t... |
rhmsubcALTVcat 41902 | The restriction of the cat... |
xpprsng 41903 | The Cartesian product of a... |
opeliun2xp 41904 | Membership of an ordered p... |
eliunxp2 41905 | Membership in a union of C... |
mpt2mptx2 41906 | Express a two-argument fun... |
cbvmpt2x2 41907 | Rule to change the bound v... |
dmmpt2ssx2 41908 | The domain of a mapping is... |
mpt2exxg2 41909 | Existence of an operation ... |
ovmpt2rdxf 41910 | Value of an operation give... |
ovmpt2rdx 41911 | Value of an operation give... |
ovmpt2x2 41912 | The value of an operation ... |
fdmdifeqresdif 41913 | The restriction of a condi... |
offvalfv 41914 | The function operation exp... |
ofaddmndmap 41915 | The function operation app... |
mapsnop 41916 | A singleton of an ordered ... |
mapprop 41917 | An unordered pair containi... |
ztprmneprm 41918 | A prime is not an integer ... |
2t6m3t4e0 41919 | 2 times 6 minus 3 times 4 ... |
ssnn0ssfz 41920 | For any finite subset of `... |
nn0sumltlt 41921 | If the sum of two nonnegat... |
bcpascm1 41922 | Pascal's rule for the bino... |
altgsumbc 41923 | The sum of binomial coeffi... |
altgsumbcALT 41924 | Alternate proof of ~ altgs... |
zlmodzxzlmod 41925 | The ` ZZ `-module ` ZZ X. ... |
zlmodzxzel 41926 | An element of the (base se... |
zlmodzxz0 41927 | The ` 0 ` of the ` ZZ `-mo... |
zlmodzxzscm 41928 | The scalar multiplication ... |
zlmodzxzadd 41929 | The addition of the ` ZZ `... |
zlmodzxzsubm 41930 | The subtraction of the ` Z... |
zlmodzxzsub 41931 | The subtraction of the ` Z... |
gsumpr 41932 | Group sum of a pair. (Con... |
mgpsumunsn 41933 | Extract a summand/factor f... |
mgpsumz 41934 | If the group sum for the m... |
mgpsumn 41935 | If the group sum for the m... |
gsumsplit2f 41936 | Split a group sum into two... |
gsumdifsndf 41937 | Extract a summand from a f... |
nn0le2is012 41938 | A nonnegative integer whic... |
exple2lt6 41939 | A nonnegative integer to t... |
pgrple2abl 41940 | Every symmetric group on a... |
pgrpgt2nabl 41941 | Every symmetric group on a... |
invginvrid 41942 | Identity for a multiplicat... |
rmsupp0 41943 | The support of a mapping o... |
domnmsuppn0 41944 | The support of a mapping o... |
rmsuppss 41945 | The support of a mapping o... |
mndpsuppss 41946 | The support of a mapping o... |
scmsuppss 41947 | The support of a mapping o... |
rmsuppfi 41948 | The support of a mapping o... |
rmfsupp 41949 | A mapping of a multiplicat... |
mndpsuppfi 41950 | The support of a mapping o... |
mndpfsupp 41951 | A mapping of a scalar mult... |
scmsuppfi 41952 | The support of a mapping o... |
scmfsupp 41953 | A mapping of a scalar mult... |
suppmptcfin 41954 | The support of a mapping w... |
mptcfsupp 41955 | A mapping with value 0 exc... |
fsuppmptdmf 41956 | A mapping with a finite do... |
lmodvsmdi 41957 | Multiple distributive law ... |
gsumlsscl 41958 | Closure of a group sum in ... |
ascl0 41959 | The scalar 0 embedded into... |
ascl1 41960 | The scalar 1 embedded into... |
assaascl0 41961 | The scalar 0 embedded into... |
assaascl1 41962 | The scalar 1 embedded into... |
ply1vr1smo 41963 | The variable in a polynomi... |
ply1ass23l 41964 | Associative identity with ... |
ply1sclrmsm 41965 | The ring multiplication of... |
coe1id 41966 | Coefficient vector of the ... |
coe1sclmulval 41967 | The value of the coefficie... |
ply1mulgsumlem1 41968 | Lemma 1 for ~ ply1mulgsum ... |
ply1mulgsumlem2 41969 | Lemma 2 for ~ ply1mulgsum ... |
ply1mulgsumlem3 41970 | Lemma 3 for ~ ply1mulgsum ... |
ply1mulgsumlem4 41971 | Lemma 4 for ~ ply1mulgsum ... |
ply1mulgsum 41972 | The product of two polynom... |
evl1at0 41973 | Polynomial evaluation for ... |
evl1at1 41974 | Polynomial evaluation for ... |
linply1 41975 | A term of the form ` x - C... |
lineval 41976 | A term of the form ` x - C... |
zringsubgval 41977 | Subtraction in the ring of... |
linevalexample 41978 | The polynomial ` x - 3 ` o... |
dmatALTval 41983 | The algebra of ` N ` x ` N... |
dmatALTbas 41984 | The base set of the algebr... |
dmatALTbasel 41985 | An element of the base set... |
dmatbas 41986 | The set of all ` N ` x ` N... |
lincop 41991 | A linear combination as op... |
lincval 41992 | The value of a linear comb... |
dflinc2 41993 | Alternative definition of ... |
lcoop 41994 | A linear combination as op... |
lcoval 41995 | The value of a linear comb... |
lincfsuppcl 41996 | A linear combination of ve... |
linccl 41997 | A linear combination of ve... |
lincval0 41998 | The value of an empty line... |
lincvalsng 41999 | The linear combination ove... |
lincvalsn 42000 | The linear combination ove... |
lincvalpr 42001 | The linear combination ove... |
lincval1 42002 | The linear combination ove... |
lcosn0 42003 | Properties of a linear com... |
lincvalsc0 42004 | The linear combination whe... |
lcoc0 42005 | Properties of a linear com... |
linc0scn0 42006 | If a set contains the zero... |
lincdifsn 42007 | A vector is a linear combi... |
linc1 42008 | A vector is a linear combi... |
lincellss 42009 | A linear combination of a ... |
lco0 42010 | The set of empty linear co... |
lcoel0 42011 | The zero vector is always ... |
lincsum 42012 | The sum of two linear comb... |
lincscm 42013 | A linear combinations mult... |
lincsumcl 42014 | The sum of two linear comb... |
lincscmcl 42015 | The multiplication of a li... |
lincsumscmcl 42016 | The sum of a linear combin... |
lincolss 42017 | According to the statement... |
ellcoellss 42018 | Every linear combination o... |
lcoss 42019 | A set of vectors of a modu... |
lspsslco 42020 | Lemma for ~ lspeqlco . (C... |
lcosslsp 42021 | Lemma for ~ lspeqlco . (C... |
lspeqlco 42022 | Equivalence of a _span_ of... |
rellininds 42026 | The class defining the rel... |
linindsv 42028 | The classes of the module ... |
islininds 42029 | The property of being a li... |
linindsi 42030 | The implications of being ... |
linindslinci 42031 | The implications of being ... |
islinindfis 42032 | The property of being a li... |
islinindfiss 42033 | The property of being a li... |
linindscl 42034 | A linearly independent set... |
lindepsnlininds 42035 | A linearly dependent subse... |
islindeps 42036 | The property of being a li... |
lincext1 42037 | Property 1 of an extension... |
lincext2 42038 | Property 2 of an extension... |
lincext3 42039 | Property 3 of an extension... |
lindslinindsimp1 42040 | Implication 1 for ~ lindsl... |
lindslinindimp2lem1 42041 | Lemma 1 for ~ lindslininds... |
lindslinindimp2lem2 42042 | Lemma 2 for ~ lindslininds... |
lindslinindimp2lem3 42043 | Lemma 3 for ~ lindslininds... |
lindslinindimp2lem4 42044 | Lemma 4 for ~ lindslininds... |
lindslinindsimp2lem5 42045 | Lemma 5 for ~ lindslininds... |
lindslinindsimp2 42046 | Implication 2 for ~ lindsl... |
lindslininds 42047 | Equivalence of definitions... |
linds0 42048 | The empty set is always a ... |
el0ldep 42049 | A set containing the zero ... |
el0ldepsnzr 42050 | A set containing the zero ... |
lindsrng01 42051 | Any subset of a module is ... |
lindszr 42052 | Any subset of a module ove... |
snlindsntorlem 42053 | Lemma for ~ snlindsntor . ... |
snlindsntor 42054 | A singleton is linearly in... |
ldepsprlem 42055 | Lemma for ~ ldepspr . (Co... |
ldepspr 42056 | If a vector is a scalar mu... |
lincresunit3lem3 42057 | Lemma 3 for ~ lincresunit3... |
lincresunitlem1 42058 | Lemma 1 for properties of ... |
lincresunitlem2 42059 | Lemma for properties of a ... |
lincresunit1 42060 | Property 1 of a specially ... |
lincresunit2 42061 | Property 2 of a specially ... |
lincresunit3lem1 42062 | Lemma 1 for ~ lincresunit3... |
lincresunit3lem2 42063 | Lemma 2 for ~ lincresunit3... |
lincresunit3 42064 | Property 3 of a specially ... |
lincreslvec3 42065 | Property 3 of a specially ... |
islindeps2 42066 | Conditions for being a lin... |
islininds2 42067 | Implication of being a lin... |
isldepslvec2 42068 | Alternative definition of ... |
lindssnlvec 42069 | A singleton not containing... |
lmod1lem1 42070 | Lemma 1 for ~ lmod1 . (Co... |
lmod1lem2 42071 | Lemma 2 for ~ lmod1 . (Co... |
lmod1lem3 42072 | Lemma 3 for ~ lmod1 . (Co... |
lmod1lem4 42073 | Lemma 4 for ~ lmod1 . (Co... |
lmod1lem5 42074 | Lemma 5 for ~ lmod1 . (Co... |
lmod1 42075 | The (smallest) structure r... |
lmod1zr 42076 | The (smallest) structure r... |
lmod1zrnlvec 42077 | There is a (left) module (... |
lmodn0 42078 | Left modules exist. (Cont... |
zlmodzxzequa 42079 | Example of an equation wit... |
zlmodzxznm 42080 | Example of a linearly depe... |
zlmodzxzldeplem 42081 | A and B are not equal. (C... |
zlmodzxzequap 42082 | Example of an equation wit... |
zlmodzxzldeplem1 42083 | Lemma 1 for ~ zlmodzxzldep... |
zlmodzxzldeplem2 42084 | Lemma 2 for ~ zlmodzxzldep... |
zlmodzxzldeplem3 42085 | Lemma 3 for ~ zlmodzxzldep... |
zlmodzxzldeplem4 42086 | Lemma 4 for ~ zlmodzxzldep... |
zlmodzxzldep 42087 | { A , B } is a linearly de... |
ldepsnlinclem1 42088 | Lemma 1 for ~ ldepsnlinc .... |
ldepsnlinclem2 42089 | Lemma 2 for ~ ldepsnlinc .... |
lvecpsslmod 42090 | The class of all (left) ve... |
ldepsnlinc 42091 | The reverse implication of... |
ldepslinc 42092 | For (left) vector spaces, ... |
offval0 42093 | Value of an operation appl... |
suppdm 42094 | If the range of a function... |
eluz2cnn0n1 42095 | An integer greater than 1 ... |
divge1b 42096 | The ratio of a real number... |
divgt1b 42097 | The ratio of a real number... |
ltsubaddb 42098 | Equivalence for the "less ... |
ltsubsubb 42099 | Equivalence for the "less ... |
ltsubadd2b 42100 | Equivalence for the "less ... |
divsub1dir 42101 | Distribution of division o... |
expnegico01 42102 | An integer greater than 1 ... |
elfzolborelfzop1 42103 | An element of a half-open ... |
pw2m1lepw2m1 42104 | 2 to the power of a positi... |
zgtp1leeq 42105 | If an integer is between a... |
flsubz 42106 | An integer can be moved in... |
fldivmod 42107 | Expressing the floor of a ... |
mod0mul 42108 | If an integer is 0 modulo ... |
modn0mul 42109 | If an integer is not 0 mod... |
m1modmmod 42110 | An integer decreased by 1 ... |
difmodm1lt 42111 | The difference between an ... |
nn0onn0ex 42112 | For each odd nonnegative i... |
nn0enn0ex 42113 | For each even nonnegative ... |
nneop 42114 | A positive integer is even... |
nneom 42115 | A positive integer is even... |
nn0eo 42116 | A nonnegative integer is e... |
nnpw2even 42117 | 2 to the power of a positi... |
zefldiv2 42118 | The floor of an even integ... |
zofldiv2 42119 | The floor of an odd intege... |
nn0ofldiv2 42120 | The floor of an odd nonneg... |
flnn0div2ge 42121 | The floor of a positive in... |
flnn0ohalf 42122 | The floor of the half of a... |
logge0b 42123 | The logarithm of a number ... |
loggt0b 42124 | The logarithm of a number ... |
logle1b 42125 | The logarithm of a number ... |
loglt1b 42126 | The logarithm of a number ... |
logcxp0 42127 | Logarithm of a complex pow... |
regt1loggt0 42128 | The natural logarithm for ... |
fdivval 42131 | The quotient of two functi... |
fdivmpt 42132 | The quotient of two functi... |
fdivmptf 42133 | The quotient of two functi... |
refdivmptf 42134 | The quotient of two functi... |
fdivpm 42135 | The quotient of two functi... |
refdivpm 42136 | The quotient of two functi... |
fdivmptfv 42137 | The function value of a qu... |
refdivmptfv 42138 | The function value of a qu... |
bigoval 42141 | Set of functions of order ... |
elbigofrcl 42142 | Reverse closure of the "bi... |
elbigo 42143 | Properties of a function o... |
elbigo2 42144 | Properties of a function o... |
elbigo2r 42145 | Sufficient condition for a... |
elbigof 42146 | A function of order G(x) i... |
elbigodm 42147 | The domain of a function o... |
elbigoimp 42148 | The defining property of a... |
elbigolo1 42149 | A function (into the posit... |
rege1logbrege0 42150 | The general logarithm, wit... |
rege1logbzge0 42151 | The general logarithm, wit... |
fllogbd 42152 | A real number is between t... |
relogbmulbexp 42153 | The logarithm of the produ... |
relogbdivb 42154 | The logarithm of the quoti... |
logbge0b 42155 | The logarithm of a number ... |
logblt1b 42156 | The logarithm of a number ... |
fldivexpfllog2 42157 | The floor of a positive re... |
nnlog2ge0lt1 42158 | A positive integer is 1 if... |
logbpw2m1 42159 | The floor of the binary lo... |
fllog2 42160 | The floor of the binary lo... |
blenval 42163 | The binary length of an in... |
blen0 42164 | The binary length of 0. (... |
blenn0 42165 | The binary length of a "nu... |
blenre 42166 | The binary length of a pos... |
blennn 42167 | The binary length of a pos... |
blennnelnn 42168 | The binary length of a pos... |
blennn0elnn 42169 | The binary length of a non... |
blenpw2 42170 | The binary length of a pow... |
blenpw2m1 42171 | The binary length of a pow... |
nnpw2blen 42172 | A positive integer is betw... |
nnpw2blenfzo 42173 | A positive integer is betw... |
nnpw2blenfzo2 42174 | A positive integer is eith... |
nnpw2pmod 42175 | Every positive integer can... |
blen1 42176 | The binary length of 1. (... |
blen2 42177 | The binary length of 2. (... |
nnpw2p 42178 | Every positive integer can... |
nnpw2pb 42179 | A number is a positive int... |
blen1b 42180 | The binary length of a non... |
blennnt2 42181 | The binary length of a pos... |
nnolog2flm1 42182 | The floor of the binary lo... |
blennn0em1 42183 | The binary length of the h... |
blennngt2o2 42184 | The binary length of an od... |
blengt1fldiv2p1 42185 | The binary length of an in... |
blennn0e2 42186 | The binary length of an ev... |
digfval 42189 | Operation to obtain the ` ... |
digval 42190 | The ` K ` th digit of a no... |
digvalnn0 42191 | The ` K ` th digit of a no... |
nn0digval 42192 | The ` K ` th digit of a no... |
dignn0fr 42193 | The digits of the fraction... |
dignn0ldlem 42194 | Lemma for ~ dignnld . (Co... |
dignnld 42195 | The leading digits of a po... |
dig2nn0ld 42196 | The leading digits of a po... |
dig2nn1st 42197 | The first (relevant) digit... |
dig0 42198 | All digits of 0 are 0. (C... |
digexp 42199 | The ` K ` th digit of a po... |
dig1 42200 | All but one digits of 1 ar... |
0dig1 42201 | The ` 0 ` th digit of 1 is... |
0dig2pr01 42202 | The integers 0 and 1 corre... |
dig2nn0 42203 | A digit of a nonnegative i... |
0dig2nn0e 42204 | The last bit of an even in... |
0dig2nn0o 42205 | The last bit of an odd int... |
dig2bits 42206 | The ` K ` th digit of a no... |
dignn0flhalflem1 42207 | Lemma 1 for ~ dignn0flhalf... |
dignn0flhalflem2 42208 | Lemma 2 for ~ dignn0flhalf... |
dignn0ehalf 42209 | The digits of the half of ... |
dignn0flhalf 42210 | The digits of the rounded ... |
nn0sumshdiglemA 42211 | Lemma for ~ nn0sumshdig (i... |
nn0sumshdiglemB 42212 | Lemma for ~ nn0sumshdig (i... |
nn0sumshdiglem1 42213 | Lemma 1 for ~ nn0sumshdig ... |
nn0sumshdiglem2 42214 | Lemma 2 for ~ nn0sumshdig ... |
nn0sumshdig 42215 | A nonnegative integer can ... |
nn0mulfsum 42216 | Trivial algorithm to calcu... |
nn0mullong 42217 | Standard algorithm (also k... |
nfintd 42218 | Bound-variable hypothesis ... |
nfiund 42219 | Bound-variable hypothesis ... |
iunord 42220 | The indexed union of a col... |
iunordi 42221 | The indexed union of a col... |
rspcdf 42222 | Restricted specialization,... |
spd 42223 | Specialization deduction, ... |
spcdvw 42224 | A version of ~ spcdv where... |
tfis2d 42225 | Transfinite Induction Sche... |
bnd2d 42226 | Deduction form of ~ bnd2 .... |
dffun3f 42227 | Alternate definition of fu... |
ssdifsn 42228 | Subset of a set with an el... |
setrecseq 42231 | Equality theorem for set r... |
nfsetrecs 42232 | Bound-variable hypothesis ... |
setrec1lem1 42233 | Lemma for ~ setrec1 . Thi... |
setrec1lem2 42234 | Lemma for ~ setrec1 . If ... |
setrec1lem3 42235 | Lemma for ~ setrec1 . If ... |
setrec1lem4 42236 | Lemma for ~ setrec1 . If ... |
setrec1 42237 | This is the first of two f... |
setrec2fun 42238 | This is the second of two ... |
setrec2lem1 42239 | Lemma for ~ setrec2 . The... |
setrec2lem2 42240 | Lemma for ~ setrec2 . The... |
setrec2 42241 | This is the second of two ... |
setrec2v 42242 | Version of ~ setrec2 with ... |
elsetrecslem 42243 | Lemma for ~ elsetrecs . A... |
elsetrecs 42244 | A set ` A ` is an element ... |
vsetrec 42245 | Construct ` _V ` using set... |
0setrec 42246 | If a function sends the em... |
onsetreclem1 42247 | Lemma for ~ onsetrec . (C... |
onsetreclem2 42248 | Lemma for ~ onsetrec . (C... |
onsetreclem3 42249 | Lemma for ~ onsetrec . (C... |
onsetrec 42250 | Construct ` On ` using set... |
elpglem1 42253 | Lemma for ~ elpg . (Contr... |
elpglem2 42254 | Lemma for ~ elpg . (Contr... |
elpglem3 42255 | Lemma for ~ elpg . (Contr... |
elpg 42256 | Membership in the class of... |
19.8ad 42257 | If a wff is true, it is tr... |
sbidd 42258 | An identity theorem for su... |
sbidd-misc 42259 | An identity theorem for su... |
gte-lte 42264 | Simple relationship betwee... |
gt-lt 42265 | Simple relationship betwee... |
gte-lteh 42266 | Relationship between ` <_ ... |
gt-lth 42267 | Relationship between ` < `... |
ex-gt 42268 | Simple example of ` > ` , ... |
ex-gte 42269 | Simple example of ` >_ ` ,... |
sinhval-named 42276 | Value of the named sinh fu... |
coshval-named 42277 | Value of the named cosh fu... |
tanhval-named 42278 | Value of the named tanh fu... |
sinh-conventional 42279 | Conventional definition of... |
sinhpcosh 42280 | Prove that ` ( sinh `` A )... |
secval 42287 | Value of the secant functi... |
cscval 42288 | Value of the cosecant func... |
cotval 42289 | Value of the cotangent fun... |
seccl 42290 | The closure of the secant ... |
csccl 42291 | The closure of the cosecan... |
cotcl 42292 | The closure of the cotange... |
reseccl 42293 | The closure of the secant ... |
recsccl 42294 | The closure of the cosecan... |
recotcl 42295 | The closure of the cotange... |
recsec 42296 | The reciprocal of secant i... |
reccsc 42297 | The reciprocal of cosecant... |
reccot 42298 | The reciprocal of cotangen... |
rectan 42299 | The reciprocal of tangent ... |
sec0 42300 | The value of the secant fu... |
onetansqsecsq 42301 | Prove the tangent squared ... |
cotsqcscsq 42302 | Prove the tangent squared ... |
ifnmfalse 42303 | If A is not a member of B,... |
dfdp2OLD 42307 | Obsolete version of ~ df-d... |
dp2cl 42309 | Define the closure for the... |
dpval 42310 | Define the value of the de... |
dpcl 42311 | Prove that the closure of ... |
dpfrac1 42312 | Prove a simple equivalence... |
dpfrac1OLD 42313 | Obsolete version of ~ dpfr... |
logb2aval 42314 | Define the value of the ` ... |
comraddi 42321 | Commute RHS addition. See... |
mvlladdd 42322 | Move LHS left addition to ... |
mvlraddi 42323 | Move LHS right addition to... |
mvrladdd 42324 | Move RHS left addition to ... |
mvrladdi 42325 | Move RHS left addition to ... |
assraddsubd 42326 | Associate RHS addition-sub... |
assraddsubi 42327 | Associate RHS addition-sub... |
joinlmuladdmuli 42328 | Join AB+CB into (A+C) on L... |
joinlmulsubmuld 42329 | Join AB-CB into (A-C) on L... |
joinlmulsubmuli 42330 | Join AB-CB into (A-C) on L... |
mvlrmuld 42331 | Move LHS right multiplicat... |
mvlrmuli 42332 | Move LHS right multiplicat... |
i2linesi 42333 | Solve for the intersection... |
i2linesd 42334 | Solve for the intersection... |
alimp-surprise 42335 | Demonstrate that when usin... |
alimp-no-surprise 42336 | There is no "surprise" in ... |
empty-surprise 42337 | Demonstrate that when usin... |
empty-surprise2 42338 | "Prove" that false is true... |
eximp-surprise 42339 | Show what implication insi... |
eximp-surprise2 42340 | Show that "there exists" w... |
alsconv 42345 | There is an equivalence be... |
alsi1d 42346 | Deduction rule: Given "al... |
alsi2d 42347 | Deduction rule: Given "al... |
alsc1d 42348 | Deduction rule: Given "al... |
alsc2d 42349 | Deduction rule: Given "al... |
alscn0d 42350 | Deduction rule: Given "al... |
alsi-no-surprise 42351 | Demonstrate that there is ... |
5m4e1 42352 | Prove that 5 - 4 = 1. (Co... |
2p2ne5 42353 | Prove that ` 2 + 2 =/= 5 `... |
resolution 42354 | Resolution rule. This is ... |
testable 42355 | In classical logic all wff... |
aacllem 42356 | Lemma for other theorems a... |
amgmwlem 42357 | Weighted version of ~ amgm... |
amgmlemALT 42358 | Alternative proof of ~ amg... |
amgmw2d 42359 | Weighted arithmetic-geomet... |
young2d 42360 | Young's inequality for ` n... |
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