Home Metamath Proof ExplorerTheorem List (p. 254 of 328) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22421) Hilbert Space Explorer (22422-23944) Users' Mathboxes (23945-32762)

Theorem List for Metamath Proof Explorer - 25301-25400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfprodrev 25301* Reversal of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)

Theoremfprodconst 25302* The product of constant terms ( is not free in .) (Contributed by Scott Fenton, 12-Jan-2018.)

Theoremfprodn0 25303* A finite product of non-zero terms is non-zero. (Contributed by Scott Fenton, 15-Jan-2018.)

Theoremfprod2dlem 25304* Lemma for fprod2d 25305- induction step. (Contributed by Scott Fenton, 30-Jan-2018.)

Theoremfprod2d 25305* Write a double product as a product over a two dimensional region. Compare fsum2d 12555. (Contributed by Scott Fenton, 30-Jan-2018.)

Theoremfprodxp 25306* Combine two products into a single product over the cartesian product. (Contributed by Scott Fenton, 1-Feb-2018.)

Theoremfprodcnv 25307* Transform a product region using the converse operation. (Contributed by Scott Fenton, 1-Feb-2018.)

Theoremfprodcom2 25308* Interchange order of multiplication. Note that and are not necessarily constant expressions. (Contributed by Scott Fenton, 1-Feb-2018.)

Theoremfprodcom 25309* Interchange product order. (Contributed by Scott Fenton, 2-Feb-2018.)

Theoremfprod0diag 25310* Two ways to express "the product of over the the triangular region , , . Compare fsum0diag 12561. (Contributed by Scott Fenton, 2-Feb-2018.)

19.7.10  Infinite products

Theoremiprodclim 25311* An infinite product equals the value its sequence converges to. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremiprodclim2 25312* A converging product converges to its infinite product. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremiprodclim3 25313* The sequence of partial finite product of a converging infinite product converge to the infinite product of the series. Note that must not occur in . (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremiprodcl 25314* The product of a non-trivially converging infinite sequence is a complex number. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremiprodrecl 25315* The product of a non-trivially converging infinite real sequence is a real number. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremiprodmul 25316* Multiplication of infinite sums. (Contributed by Scott Fenton, 18-Dec-2017.)

Theoremiprodefisumlem 25317 Lemma for iprodefisum 25318. (Contributed by Scott Fenton, 11-Feb-2018.)

Theoremiprodefisum 25318* Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.)

Theoremiprodgam 25319* An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.)

19.7.11  Falling and Rising Factorial

Syntaxcfallfac 25320 Declare the syntax for the falling factorial.
FallFac

Syntaxcrisefac 25321 Declare the syntax for the rising factorial.
RiseFac

Definitiondf-risefac 25322* Define the rising factorial function. This is the function for complex and non-negative integers . (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Definitiondf-fallfac 25323* Define the falling factorial function. This is the function for complex and non-negative integers . (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremrisefacval 25324* The value of the rising factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremfallfacval 25325* The value of the falling factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremrisefacval2 25326* One-based value of rising factorial. (Contributed by Scott Fenton, 15-Jan-2018.)
RiseFac

Theoremfallfacval2 25327* One-based value of falling factorial. (Contributed by Scott Fenton, 15-Jan-2018.)
FallFac

Theoremfallfacval3 25328* A product representation of falling factorial when is a non-negative integer. (Contributed by Scott Fenton, 20-Mar-2018.)
FallFac

Theoremrisefaccllem 25329* Lemma for rising factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremfallfaccllem 25330* Lemma for falling factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremrisefaccl 25331 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremfallfaccl 25332 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremrerisefaccl 25333 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremrefallfaccl 25334 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremnnrisefaccl 25335 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremzrisefaccl 25336 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremzfallfaccl 25337 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremnn0risefaccl 25338 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremrprisefaccl 25339 Closure law for rising factorial. (Contributed by Scott Fenton, 9-Jan-2018.)
RiseFac

Theoremrisefallfac 25340 A relationship between rising and falling factorials. (Contributed by Scott Fenton, 15-Jan-2018.)
RiseFac FallFac

Theoremfallrisefac 25341 A relationship between falling and rising factorials. (Contributed by Scott Fenton, 17-Jan-2018.)
FallFac RiseFac

Theoremrisefall0lem 25342 Lemma for risefac0 25343 and fallfac0 25344. Show a particular set of finite integers is empty. (Contributed by Scott Fenton, 5-Jan-2018.)

Theoremrisefac0 25343 The value of the rising factorial when . (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremfallfac0 25344 The value of the falling factorial when . (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremrisefacp1 25345 The value of the rising factorial at a successor. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac RiseFac

Theoremfallfacp1 25346 The value of the falling factorial at a successor. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac FallFac

Theoremrisefacp1d 25347 The value of the rising factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
RiseFac RiseFac

Theoremfallfacp1d 25348 The value of the falling factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
FallFac FallFac

Theoremrisefac1 25349 The value of rising factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremfallfac1 25350 The value of falling factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremrisefacfac 25351 Relate rising factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremfallfacfwd 25352 The forward difference of a falling factorial. (Contributed by Scott Fenton, 21-Jan-2018.)
FallFac FallFac FallFac

Theorem0fallfac 25353 The value of the zero falling factorial at natural . (Contributed by Scott Fenton, 17-Feb-2018.)
FallFac

Theorem0risefac 25354 The value of the zero rising factorial at natural . (Contributed by Scott Fenton, 17-Feb-2018.)
RiseFac

Theorembinomfallfaclem1 25355 Lemma for binomfallfac 25357. Closure law. (Contributed by Scott Fenton, 13-Mar-2018.)
FallFac FallFac

Theorembinomfallfaclem2 25356* Lemma for binomfallfac 25357. Inductive step. (Contributed by Scott Fenton, 13-Mar-2018.)
FallFac FallFac FallFac        FallFac FallFac FallFac

Theorembinomfallfac 25357* A version of the binomial theorem using falling factorials instead of exponentials. (Contributed by Scott Fenton, 13-Mar-2018.)
FallFac FallFac FallFac

Theorembinomrisefac 25358* A version of the binomial theorem using rising factorials instead of exponentials. (Contributed by Scott Fenton, 16-Mar-2018.)
RiseFac RiseFac RiseFac

Theoremfallfacval4 25359 Represent the falling factorial via factorials when the first argument is a natural. (Contributed by Scott Fenton, 20-Mar-2018.)
FallFac

Theorembcfallfac 25360 Binomial coefficient in terms of falling factorials. (Contributed by Scott Fenton, 20-Mar-2018.)
FallFac

Theoremfallfacfac 25361 Relate falling factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

19.7.12  Factorial limits

Theoremfaclimlem1 25362* Lemma for faclim 25365. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.)

Theoremfaclimlem2 25363* Lemma for faclim 25365. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.)

Theoremfaclimlem3 25364 Lemma for faclim 25365. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.)

Theoremfaclim 25365* An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.)

Theoremiprodfac 25366* An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.)

Theoremfaclim2 25367* Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.)

19.7.13  Greatest common divisor and divisibility

Theorempdivsq 25368 Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdvdspw 25369 Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcd32 25370 Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdabsorb 25371 Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

19.7.14  Properties of relationships

Theorembrtp 25372 A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)

Theoremdftr6 25373 A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)

Theoremcoep 25374* Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremcoepr 25375* Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremdffr5 25376 A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)

Theoremdfso2 25377 Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)

Theoremdfpo2 25378 Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.)

Theorembr8 25379* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Theorembr6 25380* Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Theorembr4 25381* Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)

Theoremdfres3 25382 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcnvco1 25383 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)

Theoremcnvco2 25384 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)

Theoremeldm3 25385 Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)

Theoremelrn3 25386 Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)

Theorempocnv 25387 The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)

Theoremsocnv 25388 The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.)

19.7.15  Properties of functions and mappings

Theoremfunpsstri 25389 A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)

Theoremfundmpss 25390 If a class is a proper subset of a function , then . (Contributed by Scott Fenton, 20-Apr-2011.)

Theoremfvresval 25391 The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011.)

Theoremmptrel 25392 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfunsseq 25393 Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Theoremfununiq 25394 The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremfunbreq 25395 An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremmpteq12d 25396 An equality inference for the maps to notation. Compare mpteq12dv 4287. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremfprb 25397* A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)

Theorembr1steq 25398 Uniqueness condition for binary relationship over the relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Theorembr2ndeq 25399 Uniqueness condition for binary relationship over the relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Theoremdfdm5 25400 Definition of domain in terms of and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

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 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32762
 Copyright terms: Public domain < Previous  Next >