Home Metamath Proof ExplorerTheorem List (p. 113 of 309) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21328) Hilbert Space Explorer (21329-22851) Users' Mathboxes (22852-30843)

Theorem List for Metamath Proof Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembcn0 11201 choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcnn 11202 choose is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcn1 11203 Binomial coefficient: choose . (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcnp1n 11204 Binomial coefficient: choose . (Contributed by NM, 20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcm1k 11205 The proportion of one binomial coefficient to another with decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembcp1n 11206 The proportion of one binomial coefficient to another with increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembcp1nk 11207 The proportion of one binomial coefficient to another with and increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theorembcval5 11208 Write out the top and bottom parts of the binomial coefficient explicitly. In this form, it is valid even for , although it is no longer valid for non-positive . (Contributed by Mario Carneiro, 22-May-2014.)

Theorembcn2 11209 Binomial coefficient: choose . (Contributed by Mario Carneiro, 22-May-2014.)

Theorembcp1m1 11210 Compute the binomial coefficent of over (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.)

Theorembcpasc 11211 Pascal's rule for the binomial coefficient, generalized to all integers . Equation 2 of [Gleason] p. 295. (Contributed by NM, 13-Jul-2005.) (Revised by Mario Carneiro, 10-Mar-2014.)

Theorembccl 11212 A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.)

Theorembccl2 11213 A binomial coefficient, in its standard domain, is a natural number. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.)

Theorempermnn 11214 The number of permutations of objects from a collection of objects is a natural number. (Contributed by Jason Orendorff, 24-Jan-2007.)

5.6.8  The ` # ` (finite set size) function

Syntaxchash 11215 Extend the definition of a class to include the size function.

Definitiondf-hash 11216 Define the function, which gives the cardinality of a finite set as a member of , and assigns all infinite sets the value . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremhashkf 11217 The finite part of the size function maps all finite sets to their cardinality, as members of . (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremhashgval 11218* The value of the function in terms of the mapping from to . The proof avoids the use of ax-ac 7969. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremhashginv 11219* maps the size function's value to . (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashinf 11220 The value of the function on an infinite set. (Contributed by Mario Carneiro, 13-Jul-2014.)

Theoremhashbnd 11221 If has size bounded by an integer , then is finite. (Contributed by Mario Carneiro, 14-Jun-2015.)

Theoremhashf 11222 The size function maps all finite sets to their cardinality, as members of , and infinite sets to . (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 13-Jul-2014.)

Theoremhashfz1 11223 The set has elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashen 11224 Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhasheni 11225 Equinumerous sets have the same number of elements (even if they are not finite). (Contributed by Mario Carneiro, 15-Apr-2015.)

Theoremfz1eqb 11226 Two possibly-empty 1-based finite sets of sequential integers are equal iff their endpoints are equal. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 29-Mar-2014.)

Theoremhashcard 11227 The size function of the cardinality function. (Contributed by Mario Carneiro, 19-Sep-2013.) (Revised by Mario Carneiro, 4-Nov-2013.)

Theoremhashcl 11228 Closure of the function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)

Theoremhashxrcl 11229 Extended real closure of the function. (Contributed by Mario Carneiro, 22-Apr-2015.)

Theoremhashclb 11230 Reverse closure of the function. (Contributed by Mario Carneiro, 15-Jan-2015.)

Theoremhasheq0 11231 Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.)

Theoremhashnncl 11232 Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theoremhash0 11233 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)

Theoremhashsng 11234 The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)

Theoremhashfn 11235 A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.)

Theoremfseq1hash 11236 The value of the size function on a finite 1-based sequence. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)

Theoremhashgadd 11237 maps ordinal addition to integer addition. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashgval2 11238 A short expression for the function of hashgf1o 10911. (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremhashdom 11239 Dominance relation for the size function. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 22-Apr-2015.)

Theoremhashdomi 11240 Non-strict order relation of the function on the full cardinal poset. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Theoremhashsdom 11241 Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.)

Theoremhashun 11242 The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashun2 11243 The size of the union of finite sets is less than or equal to the sum of their sizes. (Contributed by Mario Carneiro, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 27-Jul-2014.)

Theoremhashunsng 11244 The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.)

Theoremhashprg 11245 The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.)

Theoremhashp1i 11246 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash1 11247 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash2 11248 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash3 11249 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash4 11250 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhashssdif 11251 The size of the difference of a finite set and a subset is the set's size minus the subset's. (Contributed by Steve Rodriguez, 24-Oct-2015.)

Theoremhashdif 11252 The size of the difference of a finite set and another set is the first set's size minus that of the intersection of both. (Contributed by Steve Rodriguez, 24-Oct-2015.)

Theoremhashsnlei 11253 Get an upper bound on a concretely specified finite set. Base case: singleton set. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhashunlei 11254 Get an upper bound on a concretely specified finite set. Induction step: union of two finite bounded sets. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhashsslei 11255 Get an upper bound on a concretely specified finite set. Transfer boundedness to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhashprlei 11256 An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhashtplei 11257 An unordered triple has at most three elements. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremhashfz 11258 Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)

Theoremfzsdom2 11259 Condition for finite ranges to have a strict dominance relation. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2015.)

Theoremhashfzo 11260 Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremhashfzo0 11261 Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremhashxplem 11262 Lemma for hashxp 11263. (Contributed by Paul Chapman, 30-Nov-2012.)

Theoremhashxp 11263 The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)

Theoremhashmap 11264 The size of the set exponential of two finite sets is the exponential of their sizes. (This is the original motivation behind the notation for set exponentiation.) (Contributed by Mario Carneiro, 5-Aug-2014.)

Theoremhashpw 11265 The size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.) (Proof shortened by Mario Carneiro, 5-Aug-2014.)

Theoremhashfun 11266 A finite set is a function iff it is equinumerous to its domain. (Contributed by Mario Carneiro, 26-Sep-2013.) (Revised by Mario Carneiro, 12-Mar-2015.)

Theoremhashbclem 11267* Lemma for hashbc 11268: inductive step. (Contributed by Mario Carneiro, 13-Jul-2014.)

Theoremhashbc 11268* The binomial coefficient counts the number of subsets of a finite set of a given size. (Contributed by Mario Carneiro, 13-Jul-2014.)

Theoremhashfacen 11269* The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)

Theoremhashf1lem1 11270* Lemma for hashf1 11272. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremhashf1lem2 11271* Lemma for hashf1 11272. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremhashf1 11272* The permutation number counts the number of injections from to . (Contributed by Mario Carneiro, 21-Jan-2015.)

Theoremhashfac 11273* A factorial counts the number of bijections on a finite set. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof shortened by Mario Carneiro, 17-Apr-2015.)

Theoremleiso 11274 Two ways to write a strictly increasing function on the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremleisorel 11275 Version of isorel 5675 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)

Theoremfz1isolem 11276* Lemma for fz1iso 11277. (Contributed by Mario Carneiro, 2-Apr-2014.)
OrdIso

Theoremfz1iso 11277* Any finite ordered set has an order isometry to a one-based finite sequence. (Contributed by Mario Carneiro, 2-Apr-2014.)

Theoremseqcoll 11278* The function contains a sparse set of non-zero values to be summed. The function is an order isomorphism from the set of non-zero values of to a 1-based finite sequence, and collects these non-zero values together. Under these conditions, the sum over the values in yields the same result as the sum over the original set . (Contributed by Mario Carneiro, 2-Apr-2014.)

Theoremseqcoll2 11279* The function contains a sparse set of non-zero values to be summed. The function is an order isomorphism from the set of non-zero values of to a 1-based finite sequence, and collects these non-zero values together. Under these conditions, the sum over the values in yields the same result as the sum over the original set . (Contributed by Mario Carneiro, 13-Dec-2014.)

5.6.9  Words over a set

Syntaxcword 11280 Syntax for the Word operator.
Word

Syntaxcconcat 11281 Syntax for the concatenation operator.
concat

Syntaxcs1 11282 Syntax for the singleton word constructor.

Syntaxcsubstr 11283 Syntax for the word slicing operator.
substr

Syntaxcsplice 11284 Syntax for the word splicing operator.
splice

Syntaxcreverse 11285 Syntax for the word reverse operator.
reverse

Definitiondf-word 11286* Define the class of words over a set. A word is an finite sequence of symbols from a set. The domain is forced so that two words with the same symbols in the same order will be the same. This is sometimes denoted with the Kleene star, although properly speaking that is an operator on languages. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 14-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word ..^

Definitiondf-concat 11287* Define the concatenation operator which combines two words. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 15-Aug-2015.)
concat ..^ ..^

Definitiondf-s1 11288 Define the canonical injection from symbols to words. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Definitiondf-substr 11289* Define an operation which extracts portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
substr ..^ ..^

Definitiondf-splice 11290* Define an operation which replaces portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
splice substr concat concat substr

Definitiondf-reverse 11291* Define an operation which reverses the order of symbols in a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
reverse ..^

Theoremiswrd 11292* Property of being a word over a set with a quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word ..^

Theoremwrdval 11293* Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word ..^

Theoremiswrdi 11294 A one-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
..^ Word

Theoremwrd0 11295 The empty set is a word (frequently denoted ε in this context). (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word

Theoremwrdf 11296 A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word ..^

Theoremwrdfin 11297 A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.)
Word

Theoremlencl 11298 The length of a word is a nonnegative integer. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Word

Theoremlennncl 11299 The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.)
Word

Theoremsswrd 11300 The set of words respects ordering on the base set. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word Word

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 30801-30843
 Copyright terms: Public domain < Previous  Next >