Home Metamath Proof ExplorerTheorem List (p. 295 of 314) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21444) Hilbert Space Explorer (21445-22967) Users' Mathboxes (22968-31305)

Theorem List for Metamath Proof Explorer - 29401-29500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem4atexlemqtb 29401 Lemma for 4atexlem7 29415. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlempns 29402 Lemma for 4atexlem7 29415. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemswapqr 29403 Lemma for 4atexlem7 29415. Swap and , so that theorems involving can be reused for . Note that must be expanded because it involves . (Contributed by NM, 25-Nov-2012.)

Theorem4atexlemu 29404 Lemma for 4atexlem7 29415. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemv 29405 Lemma for 4atexlem7 29415. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemunv 29406 Lemma for 4atexlem7 29415. (Contributed by NM, 21-Nov-2012.)

Theorem4atexlemtlw 29407 Lemma for 4atexlem7 29415. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemntlpq 29408 Lemma for 4atexlem7 29415. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemc 29409 Lemma for 4atexlem7 29415. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemnclw 29410 Lemma for 4atexlem7 29415. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemex2 29411* Lemma for 4atexlem7 29415. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.)

Theorem4atexlemcnd 29412 Lemma for 4atexlem7 29415. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemex4 29413* Lemma for 4atexlem7 29415. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012.)

Theorem4atexlemex6 29414* Lemma for 4atexlem7 29415. (Contributed by NM, 25-Nov-2012.)

Theorem4atexlem7 29415* Whenever there are at least 4 atoms under (specifically, , , , and ), there are also at least 4 atoms under . This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p q/0 and hence p s/0 contains at least four atoms..." Note that by cvlsupr2 28684, our is a shorter way to express . With a longer proof, the condition could be eliminated (see 4atex 29416), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.)

Theorem4atex 29416* Whenever there are at least 4 atoms under (specifically, , , , and ), there are also at least 4 atoms under . This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p q/0 and hence p s/0 contains at least four atoms..." Note that by cvlsupr2 28684, our is a shorter way to express . (Contributed by NM, 27-May-2013.)

Theorem4atex2 29417* More general version of 4atex 29416 for a line not necessarily connected to . (Contributed by NM, 27-May-2013.)

Theorem4atex2-0aOLDN 29418* Same as 4atex2 29417 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex2-0bOLDN 29419* Same as 4atex2 29417 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex2-0cOLDN 29420* Same as 4atex2 29417 except that and are zero. TODO: do we need this one or 4atex2-0aOLDN 29418 or 4atex2-0bOLDN 29419? (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex3 29421* More general version of 4atex 29416 for a line not necessarily connected to . (Contributed by NM, 29-May-2013.)

Theoremlautset 29422* The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)

Theoremislaut 29423* The predictate "is a lattice automorphism." (Contributed by NM, 11-May-2012.)

Theoremlautle 29424 Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)

Theoremlaut1o 29425 A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012.)

Theoremlaut11 29426 One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautcl 29427 A lattice automorphism value belongs to the base set. (Contributed by NM, 20-May-2012.)

TheoremlautcnvclN 29428 Reverse closure of a lattice automorphism. (Contributed by NM, 25-May-2012.) (New usage is discouraged.)

Theoremlautcnvle 29429 Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.)

Theoremlautcnv 29430 The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)

Theoremlautlt 29431 Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautcvr 29432 Covering property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautj 29433 Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.)

Theoremlautm 29434 Meet property of a lattice automorphism. (Contributed by NM, 19-May-2012.)

Theoremlauteq 29435* A lattice automorphism argument is equal to its value if all atoms are equal to their values. (Contributed by NM, 24-May-2012.)

Theoremidlaut 29436 The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)

Theoremlautco 29437 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)

TheorempautsetN 29438* The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

TheoremispautN 29439* The predictate "is a projective automorphism." (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

Syntaxcldil 29440 Extend class notation with set of all lattice dilations.

Syntaxcltrn 29441 Extend class notation with set of all lattice translations.

SyntaxcdilN 29442 Extend class notation with set of all dilations.

SyntaxctrnN 29443 Extend class notation with set of all translations.

Definitiondf-ldil 29444* Define set of all lattice dilations. Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Definitiondf-ltrn 29445* Define set of all lattice translations. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Definitiondf-dilN 29446* Define set of all dilations. Definition of dilation in [Crawley] p. 111. (Contributed by NM, 30-Jan-2012.)

Definitiondf-trnN 29447* Define set of all translations. Definition of translation in [Crawley] p. 111. (Contributed by NM, 4-Feb-2012.)

Theoremldilfset 29448* The mapping from fiducial co-atom to its set of lattice dilations. (Contributed by NM, 11-May-2012.)

Theoremldilset 29449* The set of lattice dilations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)

Theoremisldil 29450* The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Theoremldillaut 29451 A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012.)

Theoremldil1o 29452 A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013.)

Theoremldilval 29453 Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)

Theoremidldil 29454 The identity function is a lattice dilation. (Contributed by NM, 18-May-2012.)

Theoremldilcnv 29455 The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013.)

Theoremldilco 29456 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)

Theoremltrnfset 29457* The set of all lattice translations for a lattice . (Contributed by NM, 11-May-2012.)

Theoremltrnset 29458* The set of lattice translations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)

Theoremisltrn 29459* The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Theoremisltrn2N 29460* The predicate "is a lattice translation". Version of isltrn 29459 that considers only different and . TODO: Can this eliminate some separate proofs for the case? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)

Theoremltrnu 29461 Uniqueness property of a lattice translation value for atoms not under the fiducial co-atom . Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 20-May-2012.)

Theoremltrnldil 29462 A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)

Theoremltrnlaut 29463 A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremltrn1o 29464 A lattice translation is a one-to-one onto function. (Contributed by NM, 20-May-2012.)

Theoremltrncl 29465 Closure of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrn11 29466 One-to-one property of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrncnvnid 29467 If a translation is different from the identity, so is its converse. (Contributed by NM, 17-Jun-2013.)

TheoremltrncoidN 29468 Two translations are equal if the composition of one with the converse of the other is the zero translation. This is an analog of vector subtraction. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremltrnle 29469 Less-than or equal property of a lattice translation. (Contributed by NM, 20-May-2012.)

TheoremltrncnvleN 29470 Less-than or equal property of lattice translation converse. (Contributed by NM, 10-May-2013.) (New usage is discouraged.)

Theoremltrnm 29471 Lattice translation of a meet. (Contributed by NM, 20-May-2012.)

Theoremltrnj 29472 Lattice translation of a meet. TODO: change antecedent to (Contributed by NM, 25-May-2012.)

Theoremltrncvr 29473 Covering property of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrnval1 29474 Value of a lattice translation under its co-atom. (Contributed by NM, 20-May-2012.)

Theoremltrnid 29475* A lattice translation is the identity function iff all atoms not under the fiducial co-atom are equal to their values. (Contributed by NM, 24-May-2012.)

Theoremltrnnid 29476* If a lattice translation is not the identity, then there is an atom not under the fiducial co-atom and not equal to its translation. (Contributed by NM, 24-May-2012.)

Theoremltrnatb 29477 The lattice translation of an atom is an atom. (Contributed by NM, 20-May-2012.)

Theoremltrncnvatb 29478 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)

Theoremltrnel 29479 The lattice translation of an atom not under the fiducial co-atom is also an atom not under the fiducial co-atom. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 22-May-2012.)

Theoremltrnat 29480 The lattice translation of an atom is also an atom. TODO: See if this can shorten some ltrnel 29479 uses. (Contributed by NM, 25-May-2012.)

Theoremltrncnvat 29481 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 9-May-2013.)

Theoremltrncnvel 29482 The converse of the lattice translation of an atom not under the fiducial co-atom. (Contributed by NM, 10-May-2013.)

TheoremltrncoelN 29483 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 29479 uses. (Contributed by NM, 1-May-2013.) (New usage is discouraged.)

Theoremltrncoat 29484 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 29479, ltrnat 29480 uses. (Contributed by NM, 1-May-2013.)

Theoremltrncoval 29485 Two ways to express value of translation composition. (Contributed by NM, 31-May-2013.)

Theoremltrncnv 29486 The converse of a lattice translation is a lattice translation. (Contributed by NM, 10-May-2013.)

Theoremltrn11at 29487 Frequently used one-to-one property of lattice translation atoms. (Contributed by NM, 5-May-2013.)

Theoremltrneq2 29488* The equality of two translations is determined by their equality at atoms. (Contributed by NM, 2-Mar-2014.)

Theoremltrneq 29489* The equality of two translations is determined by their equality at atoms not under co-atom . (Contributed by NM, 20-Jun-2013.)

Theoremidltrn 29490 The identity function is a lattice translation. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 18-May-2012.)

Theoremltrnmw 29491 Property of lattice translation value. Remark below Lemma B in [Crawley] p. 112. TODO: Can this be used in more places? (Contributed by NM, 20-May-2012.)

TheoremdilfsetN 29492* The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-2012.) (New usage is discouraged.)

TheoremdilsetN 29493* The set of dilations for a fiducial atom . (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

TheoremisdilN 29494* The predicate "is a dilation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

TheoremtrnfsetN 29495* The mapping from fiducial atom to set of translations. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

TheoremtrnsetN 29496* The set of translations for a fiducial atom . (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

TheoremistrnN 29497* The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

Syntaxctrl 29498 Extend class notation with set of all traces of lattice translations.

Definitiondf-trl 29499* Define trace of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremtrlfset 29500* The set of all traces of lattice translations for a lattice . (Contributed by NM, 20-May-2012.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31305
 Copyright terms: Public domain < Previous  Next >