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Theorem List for Metamath Proof Explorer - 24801-24900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcgam 24801 The Gamma function.
 class  _G
 
Syntaxcigam 24802 The inverse Gamma function.
 class 1/ _G
 
Definitiondf-lgam 24803* Define the log-Gamma function. We can work with this form of the gamma function a bit easier than the equivalent expression for the gamma function itself, and moreover this function is not actually equal to  log ( _G ( x ) ) because the branch cuts are placed differently (we do have  exp ( log  _G ( x ) )  =  _G ( x ), though). This definition is attributed to Euler, and unlike the usual integral definition is defined on the entire complex plane except the nonpositive integers  ZZ  \  NN, where the function has simple poles. (Contributed by Mario Carneiro, 12-Jul-2014.)
 |-  log  _G  =  ( z  e.  ( CC  \  ( ZZ  \  NN ) ) 
 |->  ( sum_ m  e.  NN  ( ( z  x.  ( log `  (
 ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( z 
 /  m )  +  1 ) ) )  -  ( log `  z
 ) ) )
 
Definitiondf-gam 24804 Define the Gamma function. See df-lgam 24803 for more information about the reason for this definition in terms of the log-gamma function. (Contributed by Mario Carneiro, 12-Jul-2014.)
 |-  _G  =  ( exp  o.  log  _G )
 
Definitiondf-igam 24805 Define the inverse Gamma function, which is defined everywhere, unlike the Gamma function itself. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |- 1/ _G  =  ( x  e.  CC  |->  if ( x  e.  ( ZZ  \  NN ) ,  0 ,  ( 1 
 /  ( _G `  x ) ) ) )
 
Theoremeldmgm 24806 Elementhood in the set of non-nonpositive integers. (Contributed by Mario Carneiro, 12-Jul-2014.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  <->  ( A  e.  CC  /\  -.  -u A  e.  NN0 ) )
 
Theoremdmgmaddn0 24807 If  A is not a nonpositive integer, then  A  +  N is nonzero for any nonnegative integer  N. (Contributed by Mario Carneiro, 12-Jul-2014.)
 |-  (
 ( A  e.  ( CC  \  ( ZZ  \  NN ) )  /\  N  e.  NN0 )  ->  ( A  +  N )  =/=  0 )
 
Theoremdmlogdmgm 24808 If  A is in the continuous domain of the logarithm, then it is in the domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.)
 |-  ( A  e.  ( CC  \  (  -oo (,] 0
 ) )  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )
 
Theoremrpdmgm 24809 A positive real number is in the domain of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  RR+  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )
 
Theoremdmgmn0 24810 If  A is not a nonpositive integer, then  A is nonzero. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremdmgmaddnn0 24811 If  A is not a nonpositive integer and  N is a nonnegative integer, then  A  +  N is also not a nonpositive integer. (Contributed by Mario Carneiro, 6-Jul-2017.)
 |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A  +  N )  e.  ( CC  \  ( ZZ  \  NN ) ) )
 
Theoremdmgmdivn0 24812 Lemma for lgamf 24826. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   &    |-  ( ph  ->  M  e.  NN )   =>    |-  ( ph  ->  (
 ( A  /  M )  +  1 )  =/=  0 )
 
Theoremlgamgulmlem1 24813* Lemma for lgamgulm 24819. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1  /  R )  <_  ( abs `  ( x  +  k
 ) ) ) }   =>    |-  ( ph  ->  U  C_  ( CC  \  ( ZZ  \  NN ) ) )
 
Theoremlgamgulmlem2 24814* Lemma for lgamgulm 24819. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1  /  R )  <_  ( abs `  ( x  +  k
 ) ) ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  ( 2  x.  R ) 
 <_  N )   =>    |-  ( ph  ->  ( abs `  ( ( A 
 /  N )  -  ( log `  ( ( A  /  N )  +  1 ) ) ) )  <_  ( R  x.  ( ( 1  /  ( N  -  R ) )  -  (
 1  /  N )
 ) ) )
 
Theoremlgamgulmlem3 24815* Lemma for lgamgulm 24819. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1  /  R )  <_  ( abs `  ( x  +  k
 ) ) ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  ( 2  x.  R ) 
 <_  N )   =>    |-  ( ph  ->  ( abs `  ( ( A  x.  ( log `  (
 ( N  +  1 )  /  N ) ) )  -  ( log `  ( ( A 
 /  N )  +  1 ) ) ) )  <_  ( R  x.  ( ( 2  x.  ( R  +  1 ) )  /  ( N ^ 2 ) ) ) )
 
Theoremlgamgulmlem4 24816* Lemma for lgamgulm 24819. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1  /  R )  <_  ( abs `  ( x  +  k
 ) ) ) }   &    |-  G  =  ( m  e.  NN  |->  ( z  e.  U  |->  ( ( z  x.  ( log `  (
 ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( z 
 /  m )  +  1 ) ) ) ) )   &    |-  T  =  ( m  e.  NN  |->  if ( ( 2  x.  R )  <_  m ,  ( R  x.  (
 ( 2  x.  ( R  +  1 )
 )  /  ( m ^ 2 ) ) ) ,  ( ( R  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  +  ( ( log `  (
 ( R  +  1 )  x.  m ) )  +  pi ) ) ) )   =>    |-  ( ph  ->  seq  1 (  +  ,  T )  e.  dom  ~~>  )
 
Theoremlgamgulmlem5 24817* Lemma for lgamgulm 24819. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1  /  R )  <_  ( abs `  ( x  +  k
 ) ) ) }   &    |-  G  =  ( m  e.  NN  |->  ( z  e.  U  |->  ( ( z  x.  ( log `  (
 ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( z 
 /  m )  +  1 ) ) ) ) )   &    |-  T  =  ( m  e.  NN  |->  if ( ( 2  x.  R )  <_  m ,  ( R  x.  (
 ( 2  x.  ( R  +  1 )
 )  /  ( m ^ 2 ) ) ) ,  ( ( R  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  +  ( ( log `  (
 ( R  +  1 )  x.  m ) )  +  pi ) ) ) )   =>    |-  ( ( ph  /\  ( n  e.  NN  /\  y  e.  U ) )  ->  ( abs `  ( ( G `  n ) `  y
 ) )  <_  ( T `  n ) )
 
Theoremlgamgulmlem6 24818* The series  G is uniformly convergent on the compact region  U, which describes a circle of radius  R with holes of size  1  /  R around the poles of the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1  /  R )  <_  ( abs `  ( x  +  k
 ) ) ) }   &    |-  G  =  ( m  e.  NN  |->  ( z  e.  U  |->  ( ( z  x.  ( log `  (
 ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( z 
 /  m )  +  1 ) ) ) ) )   &    |-  T  =  ( m  e.  NN  |->  if ( ( 2  x.  R )  <_  m ,  ( R  x.  (
 ( 2  x.  ( R  +  1 )
 )  /  ( m ^ 2 ) ) ) ,  ( ( R  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  +  ( ( log `  (
 ( R  +  1 )  x.  m ) )  +  pi ) ) ) )   =>    |-  ( ph  ->  ( 
 seq  1 (  o F  +  ,  G )  e.  dom  ( ~~> u `  U )  /\  (  seq  1 (  o F  +  ,  G )
 ( ~~> u `  U ) ( z  e.  U  |->  O )  ->  E. r  e.  RR  A. z  e.  U  ( abs `  O )  <_  r ) ) )
 
Theoremlgamgulm 24819* The series  G is uniformly convergent on the compact region  U, which describes a circle of radius  R with holes of size 
1  /  R around the poles of the gamma function. (Contributed by Mario Carneiro, 3-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1  /  R )  <_  ( abs `  ( x  +  k
 ) ) ) }   &    |-  G  =  ( m  e.  NN  |->  ( z  e.  U  |->  ( ( z  x.  ( log `  (
 ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( z 
 /  m )  +  1 ) ) ) ) )   =>    |-  ( ph  ->  seq  1
 (  o F  +  ,  G )  e.  dom  (
 ~~> u `  U ) )
 
Theoremlgamgulm2 24820* Rewrite the limit of the sequence 
G in terms of the log-Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1  /  R )  <_  ( abs `  ( x  +  k
 ) ) ) }   &    |-  G  =  ( m  e.  NN  |->  ( z  e.  U  |->  ( ( z  x.  ( log `  (
 ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( z 
 /  m )  +  1 ) ) ) ) )   =>    |-  ( ph  ->  ( A. z  e.  U  ( log  _G `  z )  e.  CC  /\  seq  1 (  o F  +  ,  G )
 ( ~~> u `  U ) ( z  e.  U  |->  ( ( log  _G `  z )  +  ( log `  z )
 ) ) ) )
 
Theoremlgambdd 24821* The log-Gamma function is bounded on the region  U. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( ph  ->  R  e.  NN )   &    |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  R  /\  A. k  e.  NN0  ( 1  /  R )  <_  ( abs `  ( x  +  k
 ) ) ) }   &    |-  G  =  ( m  e.  NN  |->  ( z  e.  U  |->  ( ( z  x.  ( log `  (
 ( m  +  1 )  /  m ) ) )  -  ( log `  ( ( z 
 /  m )  +  1 ) ) ) ) )   =>    |-  ( ph  ->  E. r  e.  RR  A. z  e.  U  ( abs `  ( log  _G `  z ) )  <_  r )
 
Theoremlgamucov 24822* The  U regions used in the proof of lgamgulm 24819 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 6-Jul-2017.)
 |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  r  /\  A. k  e.  NN0  ( 1 
 /  r )  <_  ( abs `  ( x  +  k ) ) ) }   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN )
 ) )   &    |-  J  =  (
 TopOpen ` fld )   =>    |-  ( ph  ->  E. r  e.  NN  A  e.  (
 ( int `  J ) `  U ) )
 
Theoremlgamucov2 24823* The  U regions used in the proof of lgamgulm 24819 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.)
 |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  r  /\  A. k  e.  NN0  ( 1 
 /  r )  <_  ( abs `  ( x  +  k ) ) ) }   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN )
 ) )   =>    |-  ( ph  ->  E. r  e.  NN  A  e.  U )
 
Theoremlgamcvglem 24824* Lemma for lgamf 24826 and lgamcvg 24838. (Contributed by Mario Carneiro, 8-Jul-2017.)
 |-  U  =  { x  e.  CC  |  ( ( abs `  x )  <_  r  /\  A. k  e.  NN0  ( 1 
 /  r )  <_  ( abs `  ( x  +  k ) ) ) }   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN )
 ) )   &    |-  G  =  ( m  e.  NN  |->  ( ( A  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  (
 ( A  /  m )  +  1 )
 ) ) )   =>    |-  ( ph  ->  ( ( log  _G `  A )  e.  CC  /\  seq  1 (  +  ,  G ) 
 ~~>  ( ( log  _G `  A )  +  ( log `  A ) ) ) )
 
Theoremlgamcl 24825 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.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( log  _G `  A )  e. 
 CC )
 
Theoremlgamf 24826 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.)
 |-  log  _G : ( CC  \  ( ZZ  \  NN )
 ) --> CC
 
Theoremgamf 24827 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.)
 |-  _G :
 ( CC  \  ( ZZ  \  NN ) ) --> CC
 
Theoremgamcl 24828 The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( _G `  A )  e.  CC )
 
Theoremeflgam 24829 The exponential of the log-Gamma function is the Gamma function (by definition). (Contributed by Mario Carneiro, 8-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( exp `  ( log  _G `  A ) )  =  ( _G `  A ) )
 
Theoremgamne0 24830 The Gamma function is never zero. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( _G `  A )  =/=  0
 )
 
Theoremigamval 24831 Value of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |-  ( A  e.  CC  ->  (1/
 _G `  A )  =  if ( A  e.  ( ZZ  \  NN ) ,  0 ,  (
 1  /  ( _G `  A ) ) ) )
 
Theoremigamz 24832 Value of the inverse Gamma function on nonpositive integers. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |-  ( A  e.  ( ZZ  \  NN )  ->  (1/ _G
 `  A )  =  0 )
 
Theoremigamgam 24833 Value of the inverse Gamma function in terms of the Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  (1/ _G `  A )  =  ( 1  /  ( _G `  A ) ) )
 
Theoremigamlgam 24834 Value of the inverse Gamma function in terms of the log-Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  (1/ _G `  A )  =  ( exp `  -u ( log  _G `  A ) ) )
 
Theoremigamf 24835 Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |- 1/ _G : CC
 --> CC
 
Theoremigamcl 24836 Closure of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |-  ( A  e.  CC  ->  (1/
 _G `  A )  e.  CC )
 
Theoremgamigam 24837 The Gamma function is the inverse of the inverse Gamma function. (Contributed by Mario Carneiro, 16-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( _G `  A )  =  ( 1  /  (1/ _G `  A ) ) )
 
Theoremlgamcvg 24838* The series  G converges to  log  _G ( A )  +  log ( A ). (Contributed by Mario Carneiro, 6-Jul-2017.)
 |-  G  =  ( m  e.  NN  |->  ( ( A  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  (
 ( A  /  m )  +  1 )
 ) ) )   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   =>    |-  ( ph  ->  seq  1 (  +  ,  G )  ~~>  ( ( log  _G `  A )  +  ( log `  A )
 ) )
 
Theoremlgamcvg2 24839* The series  G converges to  log  _G ( A  +  1 ). (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  G  =  ( m  e.  NN  |->  ( ( A  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  (
 ( A  /  m )  +  1 )
 ) ) )   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   =>    |-  ( ph  ->  seq  1 (  +  ,  G )  ~~>  ( log  _G `  ( A  +  1 )
 ) )
 
Theoremgamcvg 24840* The pointwise exponential of the series  G converges to  _G ( A )  x.  A. (Contributed by Mario Carneiro, 6-Jul-2017.)
 |-  G  =  ( m  e.  NN  |->  ( ( A  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  (
 ( A  /  m )  +  1 )
 ) ) )   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   =>    |-  ( ph  ->  ( exp  o.  seq  1
 (  +  ,  G ) )  ~~>  ( ( _G `  A )  x.  A ) )
 
Theoremlgamp1 24841 The functional equation of the (log) Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( log  _G `  ( A  +  1 ) )  =  ( ( log  _G `  A )  +  ( log `  A ) ) )
 
Theoremgamp1 24842 The functional equation of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  ( _G `  ( A  +  1 ) )  =  ( ( _G `  A )  x.  A ) )
 
Theoremgamcvg2lem 24843* Lemma for gamcvg2 24844. (Contributed by Mario Carneiro, 10-Jul-2017.)
 |-  F  =  ( m  e.  NN  |->  ( ( ( ( m  +  1 ) 
 /  m )  ^ c  A )  /  (
 ( A  /  m )  +  1 )
 ) )   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN )
 ) )   &    |-  G  =  ( m  e.  NN  |->  ( ( A  x.  ( log `  ( ( m  +  1 )  /  m ) ) )  -  ( log `  (
 ( A  /  m )  +  1 )
 ) ) )   =>    |-  ( ph  ->  ( exp  o.  seq  1
 (  +  ,  G ) )  =  seq  1 (  x.  ,  F ) )
 
Theoremgamcvg2 24844* An infinite product expression for the gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  F  =  ( m  e.  NN  |->  ( ( ( ( m  +  1 ) 
 /  m )  ^ c  A )  /  (
 ( A  /  m )  +  1 )
 ) )   &    |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN )
 ) )   =>    |-  ( ph  ->  seq  1
 (  x.  ,  F ) 
 ~~>  ( ( _G `  A )  x.  A ) )
 
Theoremregamcl 24845 The Gamma function is real for real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  ( RR  \  ( ZZ  \  NN ) )  ->  ( _G `  A )  e.  RR )
 
Theoremrelgamcl 24846 The log-Gamma function is real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  RR+  ->  ( log  _G `  A )  e.  RR )
 
Theoremrpgamcl 24847 The log-Gamma function is positive real for positive real input. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( A  e.  RR+  ->  ( _G `  A )  e.  RR+ )
 
Theoremlgam1 24848 The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( log  _G `  1 )  =  0
 
Theoremgam1 24849 The log-Gamma function at one. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( _G `  1 )  =  1
 
Theoremfacgam 24850 The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( N  e.  NN0  ->  ( ! `  N )  =  ( _G `  ( N  +  1 )
 ) )
 
Theoremgamfac 24851 The Gamma function generalizes the factorial. (Contributed by Mario Carneiro, 9-Jul-2017.)
 |-  ( N  e.  NN  ->  (
 _G `  N )  =  ( ! `  ( N  -  1 ) ) )
 
19.4.4  Derangements and the Subfactorial
 
Theoremderanglem 24852* Lemma for derangements. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  ( A  e.  Fin  ->  { f  |  ( f : A -1-1-onto-> A  /\  ph ) }  e.  Fin )
 
Theoremderangval 24853* 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.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   =>    |-  ( A  e.  Fin 
 ->  ( D `  A )  =  ( # `  { f  |  ( f : A -1-1-onto-> A  /\  A. y  e.  A  ( f `  y
 )  =/=  y ) } ) )
 
Theoremderangf 24854* The derangement number is a function from finite sets to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   =>    |-  D : Fin --> NN0
 
Theoremderang0 24855* The derangement number of the empty set. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   =>    |-  ( D `  (/) )  =  1
 
Theoremderangsn 24856* The derangement number of a singleton. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   =>    |-  ( A  e.  V  ->  ( D `  { A } )  =  0 )
 
Theoremderangenlem 24857* One half of derangen 24858. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   =>    |-  ( ( A 
 ~~  B  /\  B  e.  Fin )  ->  ( D `  A )  <_  ( D `  B ) )
 
Theoremderangen 24858* 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.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   =>    |-  ( ( A 
 ~~  B  /\  B  e.  Fin )  ->  ( D `  A )  =  ( D `  B ) )
 
Theoremsubfacval 24859* The subfactorial is defined as the number of derangements (see derangval 24853) of the set  ( 1 ... N ). (Contributed by Mario Carneiro, 21-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   =>    |-  ( N  e.  NN0  ->  ( S `  N )  =  ( D `  ( 1 ... N ) ) )
 
Theoremderangen2 24860* Write the derangement number in terms of the subfactorial. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   =>    |-  ( A  e.  Fin  ->  ( D `  A )  =  ( S `  ( # `  A ) ) )
 
Theoremsubfacf 24861* The subfactorial is a function from nonnegative integers to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   =>    |-  S : NN0 --> NN0
 
Theoremsubfaclefac 24862* The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   =>    |-  ( N  e.  NN0  ->  ( S `  N ) 
 <_  ( ! `  N ) )
 
Theoremsubfac0 24863* The subfactorial at zero. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   =>    |-  ( S `  0
 )  =  1
 
Theoremsubfac1 24864* The subfactorial at one. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   =>    |-  ( S `  1
 )  =  0
 
Theoremsubfacp1lem1 24865* Lemma for subfacp1 24872. The set  K together with  { 1 ,  M } partitions the set  1 ... ( N  +  1 ). (Contributed by Mario Carneiro, 23-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   &    |-  A  =  {
 f  |  ( f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 )
 )  /\  A. y  e.  ( 1 ... ( N  +  1 )
 ) ( f `  y )  =/=  y
 ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  (
 2 ... ( N  +  1 ) ) )   &    |-  M  e.  _V   &    |-  K  =  ( ( 2 ... ( N  +  1 )
 )  \  { M } )   =>    |-  ( ph  ->  (
 ( K  i^i  {
 1 ,  M }
 )  =  (/)  /\  ( K  u.  { 1 ,  M } )  =  ( 1 ... ( N  +  1 )
 )  /\  ( # `  K )  =  ( N  -  1 ) ) )
 
Theoremsubfacp1lem2a 24866* Lemma for subfacp1 24872. Properties of a bijection on  K augmented with the two-element flip to get a bijection on  K  u.  {
1 ,  M }. (Contributed by Mario Carneiro, 23-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   &    |-  A  =  {
 f  |  ( f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 )
 )  /\  A. y  e.  ( 1 ... ( N  +  1 )
 ) ( f `  y )  =/=  y
 ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  (
 2 ... ( N  +  1 ) ) )   &    |-  M  e.  _V   &    |-  K  =  ( ( 2 ... ( N  +  1 )
 )  \  { M } )   &    |-  F  =  ( G  u.  { <. 1 ,  M >. ,  <. M ,  1 >. } )   &    |-  ( ph  ->  G : K -1-1-onto-> K )   =>    |-  ( ph  ->  ( F : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 )
 )  /\  ( F `  1 )  =  M  /\  ( F `  M )  =  1 )
 )
 
Theoremsubfacp1lem2b 24867* Lemma for subfacp1 24872. Properties of a bijection on  K augmented with the two-element flip to get a bijection on  K  u.  {
1 ,  M }. (Contributed by Mario Carneiro, 23-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   &    |-  A  =  {
 f  |  ( f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 )
 )  /\  A. y  e.  ( 1 ... ( N  +  1 )
 ) ( f `  y )  =/=  y
 ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  (
 2 ... ( N  +  1 ) ) )   &    |-  M  e.  _V   &    |-  K  =  ( ( 2 ... ( N  +  1 )
 )  \  { M } )   &    |-  F  =  ( G  u.  { <. 1 ,  M >. ,  <. M ,  1 >. } )   &    |-  ( ph  ->  G : K -1-1-onto-> K )   =>    |-  ( ( ph  /\  X  e.  K )  ->  ( F `  X )  =  ( G `  X ) )
 
Theoremsubfacp1lem3 24868* Lemma for subfacp1 24872. In subfacp1lem6 24871 we cut up the set of all derangements on  1 ... ( N  +  1 ) first according to the value at  1, and then by whether or not  ( f `  ( f `  1
) )  =  1. In this lemma, we show that the subset of all  N  +  1 derangements that satisfy this for fixed  M  =  ( f `  1 ) is in bijection with  N  -  1 derangements, by simply dropping the  x  =  1 and  x  =  M points from the function to get a derangement on  K  =  ( 1 ... ( N  -  1 ) ) 
\  { 1 ,  M }. (Contributed by Mario Carneiro, 23-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   &    |-  A  =  {
 f  |  ( f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 )
 )  /\  A. y  e.  ( 1 ... ( N  +  1 )
 ) ( f `  y )  =/=  y
 ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  (
 2 ... ( N  +  1 ) ) )   &    |-  M  e.  _V   &    |-  K  =  ( ( 2 ... ( N  +  1 )
 )  \  { M } )   &    |-  B  =  {
 g  e.  A  |  ( ( g `  1 )  =  M  /\  ( g `  M )  =  1 ) }   &    |-  C  =  { f  |  ( f : K -1-1-onto-> K  /\  A. y  e.  K  ( f `  y
 )  =/=  y ) }   =>    |-  ( ph  ->  ( # `
  B )  =  ( S `  ( N  -  1 ) ) )
 
Theoremsubfacp1lem4 24869* Lemma for subfacp1 24872. The function  F, which swaps  1 with  M and leaves all other elements alone, is a bijection of order  2, i.e. it is its own inverse. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   &    |-  A  =  {
 f  |  ( f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 )
 )  /\  A. y  e.  ( 1 ... ( N  +  1 )
 ) ( f `  y )  =/=  y
 ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  (
 2 ... ( N  +  1 ) ) )   &    |-  M  e.  _V   &    |-  K  =  ( ( 2 ... ( N  +  1 )
 )  \  { M } )   &    |-  B  =  {
 g  e.  A  |  ( ( g `  1 )  =  M  /\  ( g `  M )  =/=  1 ) }   &    |-  F  =  ( (  _I  |`  K )  u.  { <. 1 ,  M >. ,  <. M , 
 1 >. } )   =>    |-  ( ph  ->  `' F  =  F )
 
Theoremsubfacp1lem5 24870* Lemma for subfacp1 24872. In subfacp1lem6 24871 we cut up the set of all derangements on  1 ... ( N  +  1 ) first according to the value at  1, and then by whether or not  ( f `  ( f `  1
) )  =  1. In this lemma, we show that the subset of all  N  +  1 derangements with  ( f `  ( f `  1
) )  =/=  1 for fixed  M  =  ( f ` 
1 ) is in bijection with derangements of  2 ... ( N  + 
1 ), because pre-composing with the function  F swaps  1 and  M and turns the function into a bijection with  ( f `  1 )  =  1 and  ( f `  x )  =/=  x for all other  x, so dropping the point at  1 yields a derangement on the  N remaining points. (Contributed by Mario Carneiro, 23-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   &    |-  A  =  {
 f  |  ( f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 )
 )  /\  A. y  e.  ( 1 ... ( N  +  1 )
 ) ( f `  y )  =/=  y
 ) }   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  M  e.  (
 2 ... ( N  +  1 ) ) )   &    |-  M  e.  _V   &    |-  K  =  ( ( 2 ... ( N  +  1 )
 )  \  { M } )   &    |-  B  =  {
 g  e.  A  |  ( ( g `  1 )  =  M  /\  ( g `  M )  =/=  1 ) }   &    |-  F  =  ( (  _I  |`  K )  u.  { <. 1 ,  M >. ,  <. M , 
 1 >. } )   &    |-  C  =  { f  |  ( f : ( 2
 ... ( N  +  1 ) ) -1-1-onto-> ( 2
 ... ( N  +  1 ) )  /\  A. y  e.  ( 2
 ... ( N  +  1 ) ) ( f `  y )  =/=  y ) }   =>    |-  ( ph  ->  ( # `  B )  =  ( S `  N ) )
 
Theoremsubfacp1lem6 24871* Lemma for subfacp1 24872. By induction, we cut up the set of all derangements on  N  +  1 according to the  N possible values of  ( f ` 
1 ) (since  ( f `  1 )  =/=  1), and for each set for fixed  M  =  ( f `  1 ), the subset of derangements with  ( f `  M )  =  1 has size  S ( N  - 
1 ) (by subfacp1lem3 24868), while the subset with  ( f `  M
)  =/=  1 has size  S
( N ) (by subfacp1lem5 24870). Adding it all up yields the desired equation  N ( S ( N )  +  S ( N  - 
1 ) ) for the number of derangements on  N  +  1. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   &    |-  A  =  {
 f  |  ( f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 )
 )  /\  A. y  e.  ( 1 ... ( N  +  1 )
 ) ( f `  y )  =/=  y
 ) }   =>    |-  ( N  e.  NN  ->  ( S `  ( N  +  1 )
 )  =  ( N  x.  ( ( S `
  N )  +  ( S `  ( N  -  1 ) ) ) ) )
 
Theoremsubfacp1 24872* 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 24863, subfac1 24864. (Contributed by Mario Carneiro, 23-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   =>    |-  ( N  e.  NN  ->  ( S `  ( N  +  1 )
 )  =  ( N  x.  ( ( S `
  N )  +  ( S `  ( N  -  1 ) ) ) ) )
 
Theoremsubfacval2 24873* A closed-form expression for the subfactorial. (Contributed by Mario Carneiro, 23-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   =>    |-  ( N  e.  NN0  ->  ( S `  N )  =  ( ( ! `
  N )  x. 
 sum_ k  e.  (
 0 ... N ) ( ( -u 1 ^ k
 )  /  ( ! `  k ) ) ) )
 
Theoremsubfaclim 24874* The subfactorial converges rapidly to  N !  /  _e. (Contributed by Mario Carneiro, 23-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   =>    |-  ( N  e.  NN  ->  ( abs `  (
 ( ( ! `  N )  /  _e )  -  ( S `  N ) ) )  <  ( 1  /  N ) )
 
Theoremsubfacval3 24875* Another closed form expression for the subfactorial. The expression  |_ `  (
x  +  1  / 
2 ) is a way of saying "rounded to the nearest integer". (Contributed by Mario Carneiro, 23-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   &    |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1
 ... n ) ) )   =>    |-  ( N  e.  NN  ->  ( S `  N )  =  ( |_ `  ( ( ( ! `
  N )  /  _e )  +  (
 1  /  2 )
 ) ) )
 
Theoremderangfmla 24876* The derangements formula, which expresses the number of derangements of a finite nonempty set in terms of the factorial. The expression  |_ `  (
x  +  1  / 
2 ) is a way of saying "rounded to the nearest integer". (Contributed by Mario Carneiro, 23-Jan-2015.)
 |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y
 )  =/=  y ) } ) )   =>    |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  ->  ( D `  A )  =  ( |_ `  ( ( ( ! `  ( # `
  A ) ) 
 /  _e )  +  ( 1  /  2
 ) ) ) )
 
19.4.5  The Erdős-Szekeres theorem
 
Theoremerdszelem1 24877* Lemma for erdsze 24888. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  S  =  { y  e.  ~P ( 1 ... A )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
 ) )  /\  A  e.  y ) }   =>    |-  ( X  e.  S 
 <->  ( X  C_  (
 1 ... A )  /\  ( F  |`  X ) 
 Isom  <  ,  O  ( X ,  ( F
 " X ) ) 
 /\  A  e.  X ) )
 
Theoremerdszelem2 24878* Lemma for erdsze 24888. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  S  =  { y  e.  ~P ( 1 ... A )  |  ( ( F  |`  y )  Isom  <  ,  O  ( y ,  ( F " y
 ) )  /\  A  e.  y ) }   =>    |-  ( ( # " S )  e.  Fin  /\  ( # " S )  C_  NN )
 
Theoremerdszelem3 24879* Lemma for erdsze 24888. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 N ) -1-1-> RR )   &    |-  K  =  ( x  e.  (
 1 ... N )  |->  sup ( ( # " {
 y  e.  ~P (
 1 ... x )  |  ( ( F  |`  y ) 
 Isom  <  ,  O  ( y ,  ( F
 " y ) ) 
 /\  x  e.  y
 ) } ) ,  RR ,  <  )
 )   =>    |-  ( A  e.  (
 1 ... N )  ->  ( K `  A )  =  sup ( ( # " { y  e. 
 ~P ( 1 ...
 A )  |  ( ( F  |`  y ) 
 Isom  <  ,  O  ( y ,  ( F
 " y ) ) 
 /\  A  e.  y
 ) } ) ,  RR ,  <  )
 )
 
Theoremerdszelem4 24880* Lemma for erdsze 24888. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 N ) -1-1-> RR )   &    |-  K  =  ( x  e.  (
 1 ... N )  |->  sup ( ( # " {
 y  e.  ~P (
 1 ... x )  |  ( ( F  |`  y ) 
 Isom  <  ,  O  ( y ,  ( F
 " y ) ) 
 /\  x  e.  y
 ) } ) ,  RR ,  <  )
 )   &    |-  O  Or  RR   =>    |-  ( ( ph  /\  A  e.  ( 1
 ... N ) ) 
 ->  { A }  e.  { y  e.  ~P (
 1 ... A )  |  ( ( F  |`  y ) 
 Isom  <  ,  O  ( y ,  ( F
 " y ) ) 
 /\  A  e.  y
 ) } )
 
Theoremerdszelem5 24881* Lemma for erdsze 24888. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 N ) -1-1-> RR )   &    |-  K  =  ( x  e.  (
 1 ... N )  |->  sup ( ( # " {
 y  e.  ~P (
 1 ... x )  |  ( ( F  |`  y ) 
 Isom  <  ,  O  ( y ,  ( F
 " y ) ) 
 /\  x  e.  y
 ) } ) ,  RR ,  <  )
 )   &    |-  O  Or  RR   =>    |-  ( ( ph  /\  A  e.  ( 1
 ... N ) ) 
 ->  ( K `  A )  e.  ( # " {
 y  e.  ~P (
 1 ... A )  |  ( ( F  |`  y ) 
 Isom  <  ,  O  ( y ,  ( F
 " y ) ) 
 /\  A  e.  y
 ) } ) )
 
Theoremerdszelem6 24882* Lemma for erdsze 24888. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 N ) -1-1-> RR )   &    |-  K  =  ( x  e.  (
 1 ... N )  |->  sup ( ( # " {
 y  e.  ~P (
 1 ... x )  |  ( ( F  |`  y ) 
 Isom  <  ,  O  ( y ,  ( F
 " y ) ) 
 /\  x  e.  y
 ) } ) ,  RR ,  <  )
 )   &    |-  O  Or  RR   =>    |-  ( ph  ->  K : ( 1 ...
 N ) --> NN )
 
Theoremerdszelem7 24883* Lemma for erdsze 24888. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 N ) -1-1-> RR )   &    |-  K  =  ( x  e.  (
 1 ... N )  |->  sup ( ( # " {
 y  e.  ~P (
 1 ... x )  |  ( ( F  |`  y ) 
 Isom  <  ,  O  ( y ,  ( F
 " y ) ) 
 /\  x  e.  y
 ) } ) ,  RR ,  <  )
 )   &    |-  O  Or  RR   &    |-  ( ph  ->  A  e.  (
 1 ... N ) )   &    |-  ( ph  ->  R  e.  NN )   &    |-  ( ph  ->  -.  ( K `  A )  e.  ( 1 ... ( R  -  1
 ) ) )   =>    |-  ( ph  ->  E. s  e.  ~P  (
 1 ... N ) ( R  <_  ( # `  s
 )  /\  ( F  |`  s )  Isom  <  ,  O  ( s ,  ( F " s
 ) ) ) )
 
Theoremerdszelem8 24884* Lemma for erdsze 24888. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 N ) -1-1-> RR )   &    |-  K  =  ( x  e.  (
 1 ... N )  |->  sup ( ( # " {
 y  e.  ~P (
 1 ... x )  |  ( ( F  |`  y ) 
 Isom  <  ,  O  ( y ,  ( F
 " y ) ) 
 /\  x  e.  y
 ) } ) ,  RR ,  <  )
 )   &    |-  O  Or  RR   &    |-  ( ph  ->  A  e.  (
 1 ... N ) )   &    |-  ( ph  ->  B  e.  ( 1 ... N ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( ( K `  A )  =  ( K `  B )  ->  -.  ( F `  A ) O ( F `  B ) ) )
 
Theoremerdszelem9 24885* Lemma for erdsze 24888. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 N ) -1-1-> RR )   &    |-  I  =  ( x  e.  (
 1 ... N )  |->  sup ( ( # " {
 y  e.  ~P (
 1 ... x )  |  ( ( F  |`  y ) 
 Isom  <  ,  <  (
 y ,  ( F
 " y ) ) 
 /\  x  e.  y
 ) } ) ,  RR ,  <  )
 )   &    |-  J  =  ( x  e.  ( 1 ...
 N )  |->  sup (
 ( # " { y  e.  ~P ( 1 ... x )  |  ( ( F  |`  y ) 
 Isom  <  ,  `'  <  ( y ,  ( F
 " y ) ) 
 /\  x  e.  y
 ) } ) ,  RR ,  <  )
 )   &    |-  T  =  ( n  e.  ( 1 ...
 N )  |->  <. ( I `
  n ) ,  ( J `  n ) >. )   =>    |-  ( ph  ->  T : ( 1 ...
 N ) -1-1-> ( NN 
 X.  NN ) )
 
Theoremerdszelem10 24886* Lemma for erdsze 24888. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 N ) -1-1-> RR )   &    |-  I  =  ( x  e.  (
 1 ... N )  |->  sup ( ( # " {
 y  e.  ~P (
 1 ... x )  |  ( ( F  |`  y ) 
 Isom  <  ,  <  (
 y ,  ( F
 " y ) ) 
 /\  x  e.  y
 ) } ) ,  RR ,  <  )
 )   &    |-  J  =  ( x  e.  ( 1 ...
 N )  |->  sup (
 ( # " { y  e.  ~P ( 1 ... x )  |  ( ( F  |`  y ) 
 Isom  <  ,  `'  <  ( y ,  ( F
 " y ) ) 
 /\  x  e.  y
 ) } ) ,  RR ,  <  )
 )   &    |-  T  =  ( n  e.  ( 1 ...
 N )  |->  <. ( I `
  n ) ,  ( J `  n ) >. )   &    |-  ( ph  ->  R  e.  NN )   &    |-  ( ph  ->  S  e.  NN )   &    |-  ( ph  ->  (
 ( R  -  1
 )  x.  ( S  -  1 ) )  <  N )   =>    |-  ( ph  ->  E. m  e.  ( 1
 ... N ) ( -.  ( I `  m )  e.  (
 1 ... ( R  -  1 ) )  \/ 
 -.  ( J `  m )  e.  (
 1 ... ( S  -  1 ) ) ) )
 
Theoremerdszelem11 24887* Lemma for erdsze 24888. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 N ) -1-1-> RR )   &    |-  I  =  ( x  e.  (
 1 ... N )  |->  sup ( ( # " {
 y  e.  ~P (
 1 ... x )  |  ( ( F  |`  y ) 
 Isom  <  ,  <  (
 y ,  ( F
 " y ) ) 
 /\  x  e.  y
 ) } ) ,  RR ,  <  )
 )   &    |-  J  =  ( x  e.  ( 1 ...
 N )  |->  sup (
 ( # " { y  e.  ~P ( 1 ... x )  |  ( ( F  |`  y ) 
 Isom  <  ,  `'  <  ( y ,  ( F
 " y ) ) 
 /\  x  e.  y
 ) } ) ,  RR ,  <  )
 )   &    |-  T  =  ( n  e.  ( 1 ...
 N )  |->  <. ( I `
  n ) ,  ( J `  n ) >. )   &    |-  ( ph  ->  R  e.  NN )   &    |-  ( ph  ->  S  e.  NN )   &    |-  ( ph  ->  (
 ( R  -  1
 )  x.  ( S  -  1 ) )  <  N )   =>    |-  ( ph  ->  E. s  e.  ~P  (
 1 ... N ) ( ( R  <_  ( # `
  s )  /\  ( F  |`  s ) 
 Isom  <  ,  <  (
 s ,  ( F
 " s ) ) )  \/  ( S 
 <_  ( # `  s
 )  /\  ( F  |`  s )  Isom  <  ,  `'  <  ( s ,  ( F " s
 ) ) ) ) )
 
Theoremerdsze 24888* The Erdős-Szekeres theorem. For any injective sequence  F on the reals of length at least 
( R  -  1 )  x.  ( S  -  1 )  +  1, there is either a subsequence of length at least  R on which  F is increasing (i.e. a  <  ,  < order isomorphism) or a subsequence of length at least  S on which  F is decreasing (i.e. a  <  ,  `'  < order isomorphism, recalling that  `'  < is the greater-than relation). (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 N ) -1-1-> RR )   &    |-  ( ph  ->  R  e.  NN )   &    |-  ( ph  ->  S  e.  NN )   &    |-  ( ph  ->  ( ( R  -  1
 )  x.  ( S  -  1 ) )  <  N )   =>    |-  ( ph  ->  E. s  e.  ~P  (
 1 ... N ) ( ( R  <_  ( # `
  s )  /\  ( F  |`  s ) 
 Isom  <  ,  <  (
 s ,  ( F
 " s ) ) )  \/  ( S 
 <_  ( # `  s
 )  /\  ( F  |`  s )  Isom  <  ,  `'  <  ( s ,  ( F " s
 ) ) ) ) )
 
Theoremerdsze2lem1 24889* Lemma for erdsze2 24891. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ph  ->  R  e.  NN )   &    |-  ( ph  ->  S  e.  NN )   &    |-  ( ph  ->  F : A -1-1-> RR )   &    |-  ( ph  ->  A  C_  RR )   &    |-  N  =  ( ( R  -  1 )  x.  ( S  -  1 ) )   &    |-  ( ph  ->  N  <  ( # `
  A ) )   =>    |-  ( ph  ->  E. f
 ( f : ( 1 ... ( N  +  1 ) )
 -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1
 ... ( N  +  1 ) ) , 
 ran  f ) ) )
 
Theoremerdsze2lem2 24890* Lemma for erdsze2 24891. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ph  ->  R  e.  NN )   &    |-  ( ph  ->  S  e.  NN )   &    |-  ( ph  ->  F : A -1-1-> RR )   &    |-  ( ph  ->  A  C_  RR )   &    |-  N  =  ( ( R  -  1 )  x.  ( S  -  1 ) )   &    |-  ( ph  ->  N  <  ( # `
  A ) )   &    |-  ( ph  ->  G :
 ( 1 ... ( N  +  1 )
 ) -1-1-> A )   &    |-  ( ph  ->  G 
 Isom  <  ,  <  (
 ( 1 ... ( N  +  1 )
 ) ,  ran  G ) )   =>    |-  ( ph  ->  E. s  e.  ~P  A ( ( R  <_  ( # `  s
 )  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
 ) ) )  \/  ( S  <_  ( # `
  s )  /\  ( F  |`  s ) 
 Isom  <  ,  `'  <  ( s ,  ( F
 " s ) ) ) ) )
 
Theoremerdsze2 24891* Generalize the statement of the Erdős-Szekeres theorem erdsze 24888 to "sequences" indexed by an arbitrary subset of  RR, which can be infinite. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  ( ph  ->  R  e.  NN )   &    |-  ( ph  ->  S  e.  NN )   &    |-  ( ph  ->  F : A -1-1-> RR )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  (
 ( R  -  1
 )  x.  ( S  -  1 ) )  <  ( # `  A ) )   =>    |-  ( ph  ->  E. s  e.  ~P  A ( ( R  <_  ( # `  s
 )  /\  ( F  |`  s )  Isom  <  ,  <  ( s ,  ( F " s
 ) ) )  \/  ( S  <_  ( # `
  s )  /\  ( F  |`  s ) 
 Isom  <  ,  `'  <  ( s ,  ( F
 " s ) ) ) ) )
 
19.4.6  The Kuratowski closure-complement theorem
 
Theoremkur14lem1 24892 Lemma for kur14 24902. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  A  C_  X   &    |-  ( X  \  A )  e.  T   &    |-  ( K `  A )  e.  T   =>    |-  ( N  =  A  ->  ( N  C_  X  /\  { ( X  \  N ) ,  ( K `  N ) }  C_  T ) )
 
Theoremkur14lem2 24893 Lemma for kur14 24902. Write interior in terms of closure and complement:  i A  =  c k c A where 
c is complement and  k is closure. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  J  e.  Top   &    |-  X  =  U. J   &    |-  K  =  ( cls `  J )   &    |-  I  =  ( int `  J )   &    |-  A  C_  X   =>    |-  ( I `  A )  =  ( X  \  ( K `  ( X  \  A ) ) )
 
Theoremkur14lem3 24894 Lemma for kur14 24902. A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  J  e.  Top   &    |-  X  =  U. J   &    |-  K  =  ( cls `  J )   &    |-  I  =  ( int `  J )   &    |-  A  C_  X   =>    |-  ( K `  A )  C_  X
 
Theoremkur14lem4 24895 Lemma for kur14 24902. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  J  e.  Top   &    |-  X  =  U. J   &    |-  K  =  ( cls `  J )   &    |-  I  =  ( int `  J )   &    |-  A  C_  X   =>    |-  ( X  \  ( X  \  A ) )  =  A
 
Theoremkur14lem5 24896 Lemma for kur14 24902. Closure is an idempotent operation in the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  J  e.  Top   &    |-  X  =  U. J   &    |-  K  =  ( cls `  J )   &    |-  I  =  ( int `  J )   &    |-  A  C_  X   =>    |-  ( K `  ( K `  A ) )  =  ( K `  A )
 
Theoremkur14lem6 24897 Lemma for kur14 24902. If  k is the complementation operator and  k is the closure operator, this expresses the identity  k c
k A  =  k c k c k c k A for any subset  A of the topological space. This is the key result that lets us cut down long enough sequences of  c k c k ... that arise when applying closure and complement repeatedly to  A, and explains why we end up with a number as large as  1 4, yet no larger. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  J  e.  Top   &    |-  X  =  U. J   &    |-  K  =  ( cls `  J )   &    |-  I  =  ( int `  J )   &    |-  A  C_  X   &    |-  B  =  ( X  \  ( K `  A ) )   =>    |-  ( K `  ( I `
  ( K `  B ) ) )  =  ( K `  B )
 
Theoremkur14lem7 24898 Lemma for kur14 24902: main proof. The set  T here contains all the distinct combinations of  k and  c that can arise, and we prove here that applying  k or  c to any element of  T yields another elemnt of  T. In operator shorthand, we have  T  =  { A ,  c A ,  k A  ,  c k A ,  k c A ,  c k c A ,  k c k A , 
c k c k A ,  k c k c A ,  c k
c k c A ,  k c k c k A , 
c k c k c k A , 
k c k c k c A ,  c k
c k c k c A }. From the identities  c c A  =  A and  k k A  =  k A, 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  k c k A  =  k c k c k c k A, proved in kur14lem6 24897. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  J  e.  Top   &    |-  X  =  U. J   &    |-  K  =  ( cls `  J )   &    |-  I  =  ( int `  J )   &    |-  A  C_  X   &    |-  B  =  ( X  \  ( K `  A ) )   &    |-  C  =  ( K `  ( X  \  A ) )   &    |-  D  =  ( I `  ( K `
  A ) )   &    |-  T  =  ( (
 ( { A ,  ( X  \  A ) ,  ( K `  A ) }  u.  { B ,  C ,  ( I `  A ) } )  u.  {
 ( K `  B ) ,  D ,  ( K `  ( I `
  A ) ) } )  u.  ( { ( I `  C ) ,  ( K `  D ) ,  ( I `  ( K `  B ) ) }  u.  { ( K `  ( I `  C ) ) ,  ( I `  ( K `  ( I `  A ) ) ) } ) )   =>    |-  ( N  e.  T  ->  ( N  C_  X  /\  { ( X 
 \  N ) ,  ( K `  N ) }  C_  T ) )
 
Theoremkur14lem8 24899 Lemma for kur14 24902. Show that the set  T contains at most  1
4 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  1 4 is tight in the sense that there exist topological spaces and subsets of these spaces for which all  1 4 generated sets are distinct, and indeed the real numbers form such a topological space.) (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  J  e.  Top   &    |-  X  =  U. J   &    |-  K  =  ( cls `  J )   &    |-  I  =  ( int `  J )   &    |-  A  C_  X   &    |-  B  =  ( X  \  ( K `  A ) )   &    |-  C  =  ( K `  ( X  \  A ) )   &    |-  D  =  ( I `  ( K `
  A ) )   &    |-  T  =  ( (
 ( { A ,  ( X  \  A ) ,  ( K `  A ) }  u.  { B ,  C ,  ( I `  A ) } )  u.  {
 ( K `  B ) ,  D ,  ( K `  ( I `
  A ) ) } )  u.  ( { ( I `  C ) ,  ( K `  D ) ,  ( I `  ( K `  B ) ) }  u.  { ( K `  ( I `  C ) ) ,  ( I `  ( K `  ( I `  A ) ) ) } ) )   =>    |-  ( T  e.  Fin  /\  ( # `  T )  <_ ; 1 4 )
 
Theoremkur14lem9 24900* Lemma for kur14 24902. Since the set  T is closed under closure and complement, it contains the minimal set  S as a subset, so  S also has at most  1 4 elements. (Indeed  S  =  T, and it's not hard to prove this, but we don't need it for this proof.) (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  J  e.  Top   &    |-  X  =  U. J   &    |-  K  =  ( cls `  J )   &    |-  I  =  ( int `  J )   &    |-  A  C_  X   &    |-  B  =  ( X  \  ( K `  A ) )   &    |-  C  =  ( K `  ( X  \  A ) )   &    |-  D  =  ( I `  ( K `
  A ) )   &    |-  T  =  ( (
 ( { A ,  ( X  \  A ) ,  ( K `  A ) }  u.  { B ,  C ,  ( I `  A ) } )  u.  {
 ( K `  B ) ,  D ,  ( K `  ( I `
  A ) ) } )  u.  ( { ( I `  C ) ,  ( K `  D ) ,  ( I `  ( K `  B ) ) }  u.  { ( K `  ( I `  C ) ) ,  ( I `  ( K `  ( I `  A ) ) ) } ) )   &    |-  S  =  |^| { x  e. 
 ~P ~P X  |  ( A  e.  x  /\  A. y  e.  x  {
 ( X  \  y
 ) ,  ( K `
  y ) }  C_  x ) }   =>    |-  ( S  e.  Fin  /\  ( # `  S )  <_ ; 1 4 )
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