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Theorem List for Metamath Proof Explorer - 24801-24900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlgamgulmlem6 24801* The series is uniformly convergent on the compact region , which describes a circle of radius with holes of size around the poles of the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremlgamgulm 24802* The series is uniformly convergent on the compact region , which describes a circle of radius with holes of size around the poles of the gamma function. (Contributed by Mario Carneiro, 3-Jul-2017.)

Theoremlgamgulm2 24803* Rewrite the limit of the sequence in terms of the log-Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)

Theoremlgambdd 24804* The log-Gamma function is bounded on the region . (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremlgamucov 24805* The regions used in the proof of lgamgulm 24802 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
fld

Theoremlgamucov2 24806* The regions used in the proof of lgamgulm 24802 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.)

Theoremlgamcvglem 24807* Lemma for lgamf 24809 and lgamcvg 24821. (Contributed by Mario Carneiro, 8-Jul-2017.)

Theoremlgamcl 24808 The log-Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 8-Jul-2017.)

Theoremlgamf 24809 The log-Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 6-Jul-2017.)

Theoremgamf 24810 The Gamma function is a complex function defined on the whole complex plane except for the negative integers. (Contributed by Mario Carneiro, 6-Jul-2017.)

Theoremgamcl 24811 The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)

Theoremeflgam 24812 The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)

Theoremgamne0 24813 The Gamma function is never zero. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremigamval 24814 Value of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/

Theoremigamz 24815 Value of the inverse Gamma function on nonpositive integers. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/

Theoremigamgam 24816 Value of the inverse Gamma function in terms of the Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/

Theoremigamlgam 24817 Value of the inverse Gamma function in terms of the log-Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/

Theoremigamf 24818 Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/

Theoremigamcl 24819 Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/

Theoremgamigam 24820 The Gamma function is the inverse of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
1/

Theoremlgamcvg 24821* The series converges to . (Contributed by Mario Carneiro, 6-Jul-2017.)

Theoremlgamcvg2 24822* The series converges to . (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremgamcvg 24823* The pointwise exponential of the series converges to . (Contributed by Mario Carneiro, 6-Jul-2017.)

Theoremlgamp1 24824 The functional equation of the (log) Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremgamp1 24825 The functional equation of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremgamcvg2lem 24826* Lemma for gamcvg2 24827. (Contributed by Mario Carneiro, 10-Jul-2017.)

Theoremgamcvg2 24827* An infinite product expression for the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremregamcl 24828 The Gamma function is real for real input. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremrelgamcl 24829 The log-Gamma function is real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremrpgamcl 24830 The log-Gamma function is positive real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremlgam1 24831 The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremgam1 24832 The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremfacgam 24833 The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremgamfac 24834 The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)

19.4.4  Derangements and the Subfactorial

Theoremderanglem 24835* Lemma for derangements. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremderangval 24836* Define the derangement function, which counts the number of bijections from a set to itself such that no element is mapped to itself. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremderangf 24837* The derangement number is a function from finite sets to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremderang0 24838* The derangement number of the empty set. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremderangsn 24839* The derangement number of a singleton. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremderangenlem 24840* One half of derangen 24841. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremderangen 24841* The derangement number is a cardinal invariant, i.e. it only depends on the size of a set and not on its contents. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremsubfacval 24842* The subfactorial is defined as the number of derangements (see derangval 24836) of the set . (Contributed by Mario Carneiro, 21-Jan-2015.)

Theoremderangen2 24843* Write the derangement number in terms of the subfactorial. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremsubfacf 24844* The subfactorial is a function from nonnegative integers to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfaclefac 24845* The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfac0 24846* The subfactorial at zero. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfac1 24847* The subfactorial at one. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfacp1lem1 24848* Lemma for subfacp1 24855. The set together with partitions the set . (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacp1lem2a 24849* Lemma for subfacp1 24855. Properties of a bijection on augmented with the two-element flip to get a bijection on . (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacp1lem2b 24850* Lemma for subfacp1 24855. Properties of a bijection on augmented with the two-element flip to get a bijection on . (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacp1lem3 24851* Lemma for subfacp1 24855. In subfacp1lem6 24854 we cut up the set of all derangements on first according to the value at , and then by whether or not . In this lemma, we show that the subset of all derangements that satisfy this for fixed is in bijection with derangements, by simply dropping the and points from the function to get a derangement on . (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacp1lem4 24852* Lemma for subfacp1 24855. The function , which swaps with and leaves all other elements alone, is a bijection of order , i.e. it is its own inverse. (Contributed by Mario Carneiro, 19-Jan-2015.)

Theoremsubfacp1lem5 24853* Lemma for subfacp1 24855. In subfacp1lem6 24854 we cut up the set of all derangements on first according to the value at , and then by whether or not . In this lemma, we show that the subset of all derangements with for fixed is in bijection with derangements of , because pre-composing with the function swaps and and turns the function into a bijection with and for all other , so dropping the point at yields a derangement on the remaining points. (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacp1lem6 24854* Lemma for subfacp1 24855. By induction, we cut up the set of all derangements on according to the possible values of (since ), and for each set for fixed , the subset of derangements with has size (by subfacp1lem3 24851), while the subset with has size (by subfacp1lem5 24853). Adding it all up yields the desired equation for the number of derangements on . (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremsubfacp1 24855* A two-term recurrence for the subfactorial. This theorem allows us to forget the combinatorial definition of the derangement number in favor of the recursive definition provided by this theorem and subfac0 24846, subfac1 24847. (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacval2 24856* A closed-form expression for the subfactorial. (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfaclim 24857* The subfactorial converges rapidly to . (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremsubfacval3 24858* Another closed form expression for the subfactorial. The expression is a way of saying "rounded to the nearest integer". (Contributed by Mario Carneiro, 23-Jan-2015.)

Theoremderangfmla 24859* The derangements formula, which expresses the number of derangements of a finite nonempty set in terms of the factorial. The expression is a way of saying "rounded to the nearest integer". (Contributed by Mario Carneiro, 23-Jan-2015.)

19.4.5  The Erdős-Szekeres theorem

Theoremerdszelem1 24860* Lemma for erdsze 24871. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem2 24861* Lemma for erdsze 24871. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem3 24862* Lemma for erdsze 24871. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem4 24863* Lemma for erdsze 24871. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem5 24864* Lemma for erdsze 24871. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem6 24865* Lemma for erdsze 24871. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem7 24866* Lemma for erdsze 24871. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem8 24867* Lemma for erdsze 24871. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem9 24868* Lemma for erdsze 24871. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem10 24869* Lemma for erdsze 24871. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdszelem11 24870* Lemma for erdsze 24871. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze 24871* The Erdős-Szekeres theorem. For any injective sequence on the reals of length at least , there is either a subsequence of length at least on which is increasing (i.e. a order isomorphism) or a subsequence of length at least on which is decreasing (i.e. a order isomorphism, recalling that is the greater-than relation). (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze2lem1 24872* Lemma for erdsze2 24874. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze2lem2 24873* Lemma for erdsze2 24874. (Contributed by Mario Carneiro, 22-Jan-2015.)

Theoremerdsze2 24874* Generalize the statement of the Erdős-Szekeres theorem erdsze 24871 to "sequences" indexed by an arbitrary subset of , which can be infinite. (Contributed by Mario Carneiro, 22-Jan-2015.)

19.4.6  The Kuratowski closure-complement theorem

Theoremkur14lem1 24875 Lemma for kur14 24885. (Contributed by Mario Carneiro, 17-Feb-2015.)

Theoremkur14lem2 24876 Lemma for kur14 24885. Write interior in terms of closure and complement: where is complement and is closure. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem3 24877 Lemma for kur14 24885. A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem4 24878 Lemma for kur14 24885. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem5 24879 Lemma for kur14 24885. Closure is an idempotent operation in the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem6 24880 Lemma for kur14 24885. If is the complementation operator and is the closure operator, this expresses the identity for any subset of the topological space. This is the key result that lets us cut down long enough sequences of that arise when applying closure and complement repeatedly to , and explains why we end up with a number as large as , yet no larger. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem7 24881 Lemma for kur14 24885: main proof. The set here contains all the distinct combinations of and that can arise, and we prove here that applying or to any element of yields another elemnt of . In operator shorthand, we have . From the identities and , we can reduce any operator combination containing two adjacent identical operators, which is why the list only contains alternating sequences. The reason the sequences don't keep going after a certain point is due to the identity , proved in kur14lem6 24880. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremkur14lem8 24882 Lemma for kur14 24885. Show that the set contains at most elements. (It could be less if some of the operators take the same value for a given set, but Kuratowski showed that this upper bound of is tight in the sense that there exist topological spaces and subsets of these spaces for which all generated sets are distinct, and indeed the real numbers form such a topological space.) (Contributed by Mario Carneiro, 11-Feb-2015.)
;

Theoremkur14lem9 24883* Lemma for kur14 24885. Since the set is closed under closure and complement, it contains the minimal set as a subset, so also has at most elements. (Indeed , and it's not hard to prove this, but we don't need it for this proof.) (Contributed by Mario Carneiro, 11-Feb-2015.)
;

Theoremkur14lem10 24884* Lemma for kur14 24885. Discharge the set . (Contributed by Mario Carneiro, 11-Feb-2015.)
;

Theoremkur14 24885* Kuratowski's closure-complement theorem. There are at most 14 sets which can be obtained by the application of the closure and complement operations to a set in a topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
;

19.4.7  Retracts and sections

Syntaxcretr 24886 Extend class notation with the retract relation.
Retr

Definitiondf-retr 24887* Define the set of retractions on two topological spaces. We say that is a retraction from to . or Retr iff there is an such that are continuous functions called the retraction and section respectively, and their composite is homotopic to the identity map. If a retraction exists, we say is a retract of . (This terminology is borrowed from HoTT and appears to be nonstandard, although it has similaries to the concept of retract in the category of topological spaces and to a deformation retract in general topology.) Two topological spaces that are retracts of each other are called homotopy equivalent. (Contributed by Mario Carneiro, 11-Feb-2015.)
Retr Htpy

Theoremm1expevenALT 24888 Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)

19.4.8  Path-connected and simply connected spaces

Syntaxcpcon 24889 Extend class notation with the class of path-connected topologies.
PCon

Syntaxcscon 24890 Extend class notation with the class of simply connected topologies.
SCon

Definitiondf-pcon 24891* Define the class of path-connected topologies. A topology is path-connected if there is a path (a continuous function from the unit interval) that goes from to for any points in the space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Definitiondf-scon 24892* Define the class of simply connected topologies. A topology is simply connected if it is path-connected and every loop (continuous path with identical start and endpoint) is contractible to a point (path-homotopic to a constant function). (Contributed by Mario Carneiro, 11-Feb-2015.) (New usage is discouraged.)
SCon PCon

Theoremispcon 24893* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theorempconcn 24894* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theorempcontop 24895 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon

Theoremisscon 24896* The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon PCon

Theoremsconpcon 24897 A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon PCon

Theoremscontop 24898 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon

Theoremsconpht 24899 A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.)
SCon

Theoremcnpcon 24900 An image of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
PCon PCon

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