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

Theoremrisefacfac 25301 Relate rising factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
RiseFac

Theoremfallfacfac 25302 Relate falling factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
FallFac

Theoremfallfacfwd 25303 The forward difference of a falling factorial. (Contributed by Scott Fenton, 21-Jan-2018.)
FallFac FallFac FallFac

Theorem0fallfac 25304 The value of the zero falling factorial at natural . (Contributed by Scott Fenton, 17-Feb-2018.)
FallFac

Theorem0risefac 25305 The value of the zero rising factorial at natural . (Contributed by Scott Fenton, 17-Feb-2018.)
RiseFac

Theorembinomfallfaclem1 25306 Lemma for binomfallfac 25308. Closure law. (Contributed by Scott Fenton, 13-Mar-2018.)
FallFac FallFac

Theorembinomfallfaclem2 25307* Lemma for binomfallfac 25308. Inductive step. (Contributed by Scott Fenton, 13-Mar-2018.)
FallFac FallFac FallFac        FallFac FallFac FallFac

Theorembinomfallfac 25308* A version of the binomial theorem using falling factorials instead of exponentials. (Contributed by Scott Fenton, 13-Mar-2018.)
FallFac FallFac FallFac

Theorembinomrisefac 25309* A version of the binomial theorem using rising factorials instead of exponentials. (Contributed by Scott Fenton, 16-Mar-2018.)
RiseFac RiseFac RiseFac

19.7.12  Factorial limits

Theoremfaclimlem1 25310* Lemma for faclim 25313. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.)

Theoremfaclimlem2 25311* Lemma for faclim 25313. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.)

Theoremfaclimlem3 25312 Lemma for faclim 25313. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.)

Theoremfaclim 25313* An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.)

Theoremiprodfac 25314* An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.)

Theoremfaclim2 25315* Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.)

19.7.13  Greatest common divisor and divisibility

Theorempdivsq 25316 Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdvdspw 25317 Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcd32 25318 Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremgcdabsorb 25319 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 25320 A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)

Theoremdftr6 25321 A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)

Theoremcoep 25322* Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremcoepr 25323* Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremdffr5 25324 A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)

Theoremdfso2 25325 Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)

Theoremdfpo2 25326 Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.)

Theorembr8 25327* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Theorembr6 25328* Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)

Theorembr4 25329* Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)

Theoremdfres3 25330 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremcnvco1 25331 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)

Theoremcnvco2 25332 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)

Theoremeldm3 25333 Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)

Theoremelrn3 25334 Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)

19.7.15  Properties of functions and mappings

Theoremfunpsstri 25335 A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)

Theoremfundmpss 25336 If a class is a proper subset of a function , then . (Contributed by Scott Fenton, 20-Apr-2011.)

Theoremfvresval 25337 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 25338 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfunsseq 25339 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 25340 The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremfunbreq 25341 An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremmpteq12d 25342 An equality inference for the maps to notation. Compare mpteq12dv 4247. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremfprb 25343* A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)

Theorembr1steq 25344 Uniqueness condition for binary relationship over the relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Theorembr2ndeq 25345 Uniqueness condition for binary relationship over the relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Theoremdfdm5 25346 Definition of domain in terms of and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdfrn5 25347 Definition of range in terms of and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

19.7.16  Epsilon induction

Theoremsetinds 25348* Principle of induction (set induction). If a property passes from all elements of to itself, then it holds for all . (Contributed by Scott Fenton, 10-Mar-2011.)

Theoremsetinds2f 25349* induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremsetinds2 25350* induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.)

19.7.17  Ordinal numbers

Theoremelpotr 25351* A class of transitive sets is partially ordered by . (Contributed by Scott Fenton, 15-Oct-2010.)

Theoremdford5reg 25352 Given ax-reg 7516, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.)

Theoremdfon2lem1 25353 Lemma for dfon2 25362. (Contributed by Scott Fenton, 28-Feb-2011.)

Theoremdfon2lem2 25354* Lemma for dfon2 25362 (Contributed by Scott Fenton, 28-Feb-2011.)

Theoremdfon2lem3 25355* Lemma for dfon2 25362. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem4 25356* Lemma for dfon2 25362. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem5 25357* Lemma for dfon2 25362. Two sets satisfying the new definition also satisfy trichotomy with respect to (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem6 25358* Lemma for dfon2 25362. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem7 25359* Lemma for dfon2 25362. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremdfon2lem8 25360* Lemma for dfon2 25362. The intersection of a non-empty class of new ordinals is itself a new ordinal and is contained within (Contributed by Scott Fenton, 26-Feb-2011.)

Theoremdfon2lem9 25361* Lemma for dfon2 25362. A class of new ordinals is well-founded by . (Contributed by Scott Fenton, 3-Mar-2011.)

Theoremdfon2 25362* consists of all sets that contain all its transitive proper subsets. This definition comes from J. R. Isbell, "A Definition of Ordinal Numbers," American Mathematical Monthly, vol 67 (1960), pp. 51-52. (Contributed by Scott Fenton, 20-Feb-2011.)

Theoremdomep 25363 The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)

Theoremrdgprc0 25364 The value of the recursive definition generator at when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremrdgprc 25365 The value of the recursive definition generator when is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdfrdg2 25366* Alternate definition of the recursive function generator when is a set. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremdfrdg3 25367* Generalization of dfrdg2 25366 to remove sethood requirement. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

19.7.18  Defined equality axioms

Theoremaxextdfeq 25368 A version of ax-ext 2385 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)

Theoremax13dfeq 25369 A version of ax-13 1723 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)

Theoremaxextdist 25370 ax-ext 2385 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremaxext4dist 25371 axext4 2388 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)

Theorem19.12b 25372* 19.12vv 1917 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremexnel 25373 There is always a set not in . (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremdistel 25374 Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4341 and elirrv 7521.) (Contributed by Scott Fenton, 15-Dec-2010.)

Theoremaxextndbi 25375 axextnd 8422 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)

19.7.19  Hypothesis builders

Theoremhbntg 25376 A more general form of hbnt 1795. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbimtg 25377 A more general and closed form of hbim 1832. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbaltg 25378 A more general and closed form of hbal 1747. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbng 25379 A more general form of hbn 1797. (Contributed by Scott Fenton, 13-Dec-2010.)

Theoremhbimg 25380 A more general form of hbim 1832. (Contributed by Scott Fenton, 13-Dec-2010.)

19.7.20  The Predecessor Class

Syntaxcpred 25381 The predecessors symbol.

Definitiondf-pred 25382 Define the predecessor class of a relationship. This is the class of all elements of such that (see elpred 25391) . (Contributed by Scott Fenton, 29-Jan-2011.)

Theorempredeq1 25383 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredeq2 25384 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredeq3 25385 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredpredss 25386 If is a subset of , then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredss 25387 The predecessor class of is a subset of (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremsspred 25388 Another subset/predecessor class relationship. (Contributed by Scott Fenton, 6-Feb-2011.)

Theoremdfpred2 25389* An alternate definition of predecessor class when is a set. (Contributed by Scott Fenton, 8-Feb-2011.)

Theoremelpredim 25390 Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)

Theoremelpred 25391 Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.)

Theoremelpredg 25392 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)

Theorempredreseq 25393* Equality of restriction to predecessor classes. (Contributed by Scott Fenton, 8-Feb-2011.)

Theorempredasetex 25394 The predecessor class exists when does. (Contributed by Scott Fenton, 8-Feb-2011.)

Theoremcbvsetlike 25395* Change the bound variable in the statement stating that is set-like. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremdffr4 25396* Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.)

Theorempredel 25397 Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.)

Theorempredpo 25398 Property of the precessor class for partial orderings. (Contributed by Scott Fenton, 28-Apr-2012.)

Theorempredso 25399 Property of the predecessor class for strict orderings. (Contributed by Scott Fenton, 11-Feb-2011.)

Theorempredbrg 25400 Closed form of elpredim 25390. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)

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