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Theorem List for Metamath Proof Explorer - 23701-23800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremderangsn 23701* 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 23702* One half of derangen 23703. (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 23703* 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 23704* The subfactorial is defined as the number of derangements (see derangval 23698) 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 23705* 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 23706* 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 23707* 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 23708* 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 23709* 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 23710* Lemma for subfacp1 23717. 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 23711* Lemma for subfacp1 23717. 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 23712* Lemma for subfacp1 23717. 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 23713* Lemma for subfacp1 23717. In subfacp1lem6 23716 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 23714* Lemma for subfacp1 23717. 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 23715* Lemma for subfacp1 23717. In subfacp1lem6 23716 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 23716* Lemma for subfacp1 23717. 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 23713), while the subset with  ( f `  M
)  =/=  1 has size  S
( N ) (by subfacp1lem5 23715). 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 23717* 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 23708, subfac1 23709. (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 23718* 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 23719* 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 23720* 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 23721* 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
 ) ) ) )
 
18.4.5  The Erdős-Szekeres theorem
 
Theoremerdszelem1 23722* Lemma for erdsze 23733. (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 23723* Lemma for erdsze 23733. (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 23724* Lemma for erdsze 23733. (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 23725* Lemma for erdsze 23733. (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 23726* Lemma for erdsze 23733. (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 23727* Lemma for erdsze 23733. (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 23728* Lemma for erdsze 23733. (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 23729* Lemma for erdsze 23733. (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 23730* Lemma for erdsze 23733. (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 23731* Lemma for erdsze 23733. (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 23732* Lemma for erdsze 23733. (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 23733* 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 23734* Lemma for erdsze2 23736. (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 23735* Lemma for erdsze2 23736. (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 23736* Generalize the statement of the Erdős-Szekeres theorem erdsze 23733 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 ) ) ) ) )
 
18.4.6  The Kuratowski closure-complement theorem
 
Theoremkur14lem1 23737 Lemma for kur14 23747. (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 23738 Lemma for kur14 23747. 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 23739 Lemma for kur14 23747. 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 23740 Lemma for kur14 23747. 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 23741 Lemma for kur14 23747. 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 23742 Lemma for kur14 23747. 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 23743 Lemma for kur14 23747: 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 23742. (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 23744 Lemma for kur14 23747. 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 23745* Lemma for kur14 23747. 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 )
 
Theoremkur14lem10 23746* Lemma for kur14 23747. Discharge the set  T. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  J  e.  Top   &    |-  X  =  U. J   &    |-  K  =  ( cls `  J )   &    |-  S  =  |^| { x  e.  ~P ~P X  |  ( A  e.  x  /\  A. y  e.  x  { ( X  \  y ) ,  ( K `  y ) }  C_  x ) }   &    |-  A  C_  X   =>    |-  ( S  e.  Fin  /\  ( # `  S )  <_ ; 1 4 )
 
Theoremkur14 23747* 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.)
 |-  X  =  U. J   &    |-  K  =  ( cls `  J )   &    |-  S  =  |^| { x  e. 
 ~P ~P X  |  ( A  e.  x  /\  A. y  e.  x  {
 ( X  \  y
 ) ,  ( K `
  y ) }  C_  x ) }   =>    |-  ( ( J  e.  Top  /\  A  C_  X )  ->  ( S  e.  Fin  /\  ( # `  S )  <_ ; 1 4 ) )
 
18.4.7  Retracts and sections
 
Syntaxcretr 23748 Extend class notation with the retract relation.
 class Retr
 
Definitiondf-retr 23749* Define the set of retractions on two topological spaces. We say that  R is a retraction from  J to  K. or  R  e.  ( J Retr  K ) iff there is an  S such that  R : J --> K ,  S : K
--> J are continuous functions called the retraction and section respectively, and their composite  R  o.  S is homotopic to the identity map. If a retraction exists, we say  J is a retract of  K. (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  =  ( j  e.  Top ,  k  e.  Top  |->  { r  e.  ( j  Cn  k
 )  |  E. s  e.  ( k  Cn  j
 ) ( ( r  o.  s ) ( j Htpy  j ) (  _I  |`  U. j ) )  =/=  (/) } )
 
18.4.8  Path-connected and simply connected spaces
 
Syntaxcpcon 23750 Extend class notation with the class of path-connected topologies.
 class PCon
 
Syntaxcscon 23751 Extend class notation with the class of simply connected topologies.
 class SCon
 
Definitiondf-pcon 23752* 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  x to  y for any points  x ,  y in the space. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |- PCon  =  {
 j  e.  Top  |  A. x  e.  U. j A. y  e.  U. j E. f  e.  ( II  Cn  j ) ( ( f `  0
 )  =  x  /\  ( f `  1
 )  =  y ) }
 
Definitiondf-scon 23753* 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  =  {
 j  e. PCon  |  A. f  e.  ( II  Cn  j
 ) ( ( f `
  0 )  =  ( f `  1
 )  ->  f (  ~=ph  `  j ) ( ( 0 [,] 1 )  X.  { ( f `
  0 ) }
 ) ) }
 
Theoremispcon 23754* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  X  =  U. J   =>    |-  ( J  e. PCon  <->  ( J  e.  Top  /\  A. x  e.  X  A. y  e.  X  E. f  e.  ( II  Cn  J ) ( ( f `  0 )  =  x  /\  (
 f `  1 )  =  y ) ) )
 
Theorempconcn 23755* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  X  =  U. J   =>    |-  ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  ->  E. f  e.  ( II  Cn  J ) ( ( f `  0
 )  =  A  /\  ( f `  1
 )  =  B ) )
 
Theorempcontop 23756 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( J  e. PCon  ->  J  e.  Top )
 
Theoremisscon 23757* The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( J  e. SCon  <->  ( J  e. PCon  /\ 
 A. f  e.  ( II  Cn  J ) ( ( f `  0
 )  =  ( f `
  1 )  ->  f (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( f `  0
 ) } ) ) ) )
 
Theoremsconpcon 23758 A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( J  e. SCon  ->  J  e. PCon )
 
Theoremscontop 23759 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( J  e. SCon  ->  J  e.  Top )
 
Theoremsconpht 23760 A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  (
 ( J  e. SCon  /\  F  e.  ( II  Cn  J )  /\  ( F `  0 )  =  ( F `  1 ) ) 
 ->  F (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  { ( F `  0
 ) } ) )
 
Theoremcnpcon 23761 An image of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  Y  =  U. K   =>    |-  ( ( J  e. PCon  /\  F : X -onto-> Y  /\  F  e.  ( J  Cn  K ) ) 
 ->  K  e. PCon )
 
Theorempconcon 23762 A path-connected space is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( J  e. PCon  ->  J  e.  Con )
 
Theoremtxpcon 23763 The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  (
 ( R  e. PCon  /\  S  e. PCon )  ->  ( R  tX  S )  e. PCon )
 
Theoremptpcon 23764 The topological product of a collection of path-connected spaces is path-connected. The proof uses the axiom of choice. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  (
 ( A  e.  V  /\  F : A -->PCon )  ->  ( Xt_ `  F )  e. PCon )
 
Theoremindispcon 23765 The indiscrete topology (or trivial topology) on any set is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  { (/) ,  A }  e. PCon
 
Theoremconpcon 23766 A connected and locally path-connected space is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.)
 |-  (
 ( J  e.  Con  /\  J  e. 𝑛Locally PCon )  ->  J  e. PCon )
 
Theoremqtoppcon 23767 A quotient of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
 |-  X  =  U. J   =>    |-  ( ( J  e. PCon  /\  F  Fn  X ) 
 ->  ( J qTop  F )  e. PCon )
 
Theorempconpi1 23768 All fundamental groups in a path-connected space are isomorphic. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  X  =  U. J   &    |-  P  =  ( J  pi 1  A )   &    |-  Q  =  ( J  pi 1  B )   &    |-  S  =  ( Base `  P )   &    |-  T  =  (
 Base `  Q )   =>    |-  ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  ->  P  ~=ph𝑔 
 Q )
 
Theoremsconpht2 23769 Any two paths in a simply connected space with the same start and end point are path-homotopic. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  ( ph  ->  J  e. SCon )   &    |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  ( F `  0 )  =  ( G `  0 ) )   &    |-  ( ph  ->  ( F `  1 )  =  ( G `  1 ) )   =>    |-  ( ph  ->  F (  ~=ph  `  J ) G )
 
Theoremsconpi1 23770 A path-connected topological space is simply connected iff its fundamental group is trivial. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  X  =  U. J   =>    |-  ( ( J  e. PCon  /\  Y  e.  X ) 
 ->  ( J  e. SCon  <->  ( Base `  ( J  pi 1  Y ) )  ~~  1o )
 )
 
Theoremtxsconlem 23771 Lemma for txscon 23772. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  ( ph  ->  R  e.  Top )   &    |-  ( ph  ->  S  e.  Top )   &    |-  ( ph  ->  F  e.  ( II  Cn  ( R  tX  S ) ) )   &    |-  A  =  ( ( 1st  |`  ( U. R  X.  U. S ) )  o.  F )   &    |-  B  =  ( ( 2nd  |`  ( U. R  X.  U. S ) )  o.  F )   &    |-  ( ph  ->  G  e.  ( A ( PHtpy `  R ) ( ( 0 [,] 1 )  X.  { ( A `  0
 ) } ) ) )   &    |-  ( ph  ->  H  e.  ( B (
 PHtpy `  S ) ( ( 0 [,] 1
 )  X.  { ( B `  0 ) }
 ) ) )   =>    |-  ( ph  ->  F (  ~=ph  `  ( R  tX  S ) ) ( ( 0 [,] 1
 )  X.  { ( F `  0 ) }
 ) )
 
Theoremtxscon 23772 The topological product of two simply connected spaces is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  (
 ( R  e. SCon  /\  S  e. SCon )  ->  ( R  tX  S )  e. SCon )
 
Theoremcvxpcon 23773* A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  ( ph  ->  S  C_  CC )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  t  e.  ( 0 [,] 1 ) ) ) 
 ->  ( ( t  x.  x )  +  (
 ( 1  -  t
 )  x.  y ) )  e.  S )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  S )   =>    |-  ( ph  ->  K  e. PCon )
 
Theoremcvxscon 23774* A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  ( ph  ->  S  C_  CC )   &    |-  ( ( ph  /\  ( x  e.  S  /\  y  e.  S  /\  t  e.  ( 0 [,] 1 ) ) ) 
 ->  ( ( t  x.  x )  +  (
 ( 1  -  t
 )  x.  y ) )  e.  S )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  S )   =>    |-  ( ph  ->  K  e. SCon )
 
Theoremblscon 23775 An open ball in the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  J  =  ( TopOpen ` fld )   &    |-  S  =  ( P ( ball `  ( abs  o.  -  ) ) R )   &    |-  K  =  ( Jt  S )   =>    |-  ( ( P  e.  CC  /\  R  e.  RR* )  ->  K  e. SCon )
 
Theoremcnllyscon 23776 The topology of the complex numbers is locally simply connected. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e. Locally SCon
 
Theoremrescon 23777 A subset of  RR is simply connected iff it is connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  J  =  ( ( topGen `  ran  (,) )t  A )   =>    |-  ( A  C_  RR  ->  ( J  e. SCon  <->  J  e.  Con ) )
 
Theoremiooscon 23778 An open interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  (
 ( topGen `  ran  (,) )t  ( A (,) B ) )  e. SCon
 
Theoremiccscon 23779 An closed interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B ) )  e. SCon )
 
Theoremretopscon 23780 The real numbers are simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  ( topGen `
  ran  (,) )  e. SCon
 
Theoremiccllyscon 23781 An closed interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A [,] B ) )  e. Locally SCon )
 
Theoremrellyscon 23782 The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  ( topGen `
  ran  (,) )  e. Locally SCon
 
Theoremiiscon 23783 The unit interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  II  e. SCon
 
Theoremiillyscon 23784 The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  II  e. Locally SCon
 
Theoremiinllycon 23785 The unit interval is locally connected. (Contributed by Mario Carneiro, 6-Jul-2015.)
 |-  II  e. 𝑛Locally  Con
 
18.4.9  Covering maps
 
Syntaxccvm 23786 Extend class notation with the class of covering maps.
 class CovMap
 
Definitiondf-cvm 23787* Define the class of covering maps on two topological spaces. A function  f : c --> j is a covering map if it is continuous and for every point  x in the target space there is a neighborhood 
k of  x and a decomposition  s of the preimage of  k as a disjoint union such that  f is a homeomorphism of each set  u  e.  s onto  k. (Contributed by Mario Carneiro, 13-Feb-2015.)
 |- CovMap  =  ( c  e.  Top ,  j  e.  Top  |->  { f  e.  ( c  Cn  j
 )  |  A. x  e.  U. j E. k  e.  j  ( x  e.  k  /\  E. s  e.  ( ~P c  \  { (/) } ) (
 U. s  =  ( `' f " k ) 
 /\  A. u  e.  s  ( A. v  e.  (
 s  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( f  |`  u )  e.  (
 ( ct  u )  Homeo  ( jt  k ) ) ) ) ) } )
 
Theoremfncvm 23788 Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.)
 |- CovMap  Fn  ( Top  X.  Top )
 
Theoremcvmscbv 23789* Change bound variables in the set of even coverings. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. u  e.  s  ( A. v  e.  (
 s  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )  Homeo  ( Jt  k ) ) ) ) }
 )   =>    |-  S  =  ( a  e.  J  |->  { b  e.  ( ~P C  \  { (/) } )  |  ( U. b  =  ( `' F "
 a )  /\  A. c  e.  b  ( A. d  e.  (
 b  \  { c } ) ( c  i^i  d )  =  (/)  /\  ( F  |`  c )  e.  ( ( Ct  c )  Homeo  ( Jt  a ) ) ) ) }
 )
 
Theoremiscvm 23790* The property of being a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
 |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. u  e.  s  ( A. v  e.  (
 s  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )  Homeo  ( Jt  k ) ) ) ) }
 )   &    |-  X  =  U. J   =>    |-  ( F  e.  ( C CovMap  J )  <->  ( ( C  e.  Top  /\  J  e.  Top  /\  F  e.  ( C  Cn  J ) ) 
 /\  A. x  e.  X  E. k  e.  J  ( x  e.  k  /\  ( S `  k
 )  =/=  (/) ) ) )
 
Theoremcvmtop1 23791 Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( F  e.  ( C CovMap  J )  ->  C  e.  Top )
 
Theoremcvmtop2 23792 Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
 |-  ( F  e.  ( C CovMap  J )  ->  J  e.  Top )
 
Theoremcvmcn 23793 A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015.)
 |-  ( F  e.  ( C CovMap  J )  ->  F  e.  ( C  Cn  J ) )
 
Theoremcvmcov 23794* Property of a covering map. In order to make the covering property more manageable, we define here the set  S ( k ) of all even coverings of an open set  k in the range. Then the covering property states that every point has a neighborhood which has an even covering. (Contributed by Mario Carneiro, 13-Feb-2015.)
 |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. u  e.  s  ( A. v  e.  (
 s  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )  Homeo  ( Jt  k ) ) ) ) }
 )   &    |-  X  =  U. J   =>    |-  (
 ( F  e.  ( C CovMap  J )  /\  P  e.  X )  ->  E. x  e.  J  ( P  e.  x  /\  ( S `  x )  =/=  (/) ) )
 
Theoremcvmsrcl 23795* Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. u  e.  s  ( A. v  e.  (
 s  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )  Homeo  ( Jt  k ) ) ) ) }
 )   =>    |-  ( T  e.  ( S `  U )  ->  U  e.  J )
 
Theoremcvmsi 23796* One direction of cvmsval 23797. (Contributed by Mario Carneiro, 13-Feb-2015.)
 |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. u  e.  s  ( A. v  e.  (
 s  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )  Homeo  ( Jt  k ) ) ) ) }
 )   =>    |-  ( T  e.  ( S `  U )  ->  ( U  e.  J  /\  ( T  C_  C  /\  T  =/=  (/) )  /\  ( U. T  =  ( `' F " U ) 
 /\  A. u  e.  T  ( A. v  e.  ( T  \  { u }
 ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u ) 
 Homeo  ( Jt  U ) ) ) ) ) )
 
Theoremcvmsval 23797* Elementhood in the set  S of all even coverings of an open set in  J.  S is an even covering of  U if it is a nonempty collection of disjoint open sets in  C whose union is the preimage of  U, such that each set  u  e.  S is homeomorphic under  F to  U. (Contributed by Mario Carneiro, 13-Feb-2015.)
 |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. u  e.  s  ( A. v  e.  (
 s  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )  Homeo  ( Jt  k ) ) ) ) }
 )   =>    |-  ( C  e.  V  ->  ( T  e.  ( S `  U )  <->  ( U  e.  J  /\  ( T  C_  C  /\  T  =/=  (/) )  /\  ( U. T  =  ( `' F " U ) 
 /\  A. u  e.  T  ( A. v  e.  ( T  \  { u }
 ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u ) 
 Homeo  ( Jt  U ) ) ) ) ) ) )
 
Theoremcvmsss 23798* An even covering is a subset of the topology of the domain (i.e. a collection of open sets). (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. u  e.  s  ( A. v  e.  (
 s  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )  Homeo  ( Jt  k ) ) ) ) }
 )   =>    |-  ( T  e.  ( S `  U )  ->  T  C_  C )
 
Theoremcvmsn0 23799* An even covering is nonempty. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. u  e.  s  ( A. v  e.  (
 s  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )  Homeo  ( Jt  k ) ) ) ) }
 )   =>    |-  ( T  e.  ( S `  U )  ->  T  =/=  (/) )
 
Theoremcvmsuni 23800* An even covering of  U has union equal to the preimage of 
U by  F. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. u  e.  s  ( A. v  e.  (
 s  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )  Homeo  ( Jt  k ) ) ) ) }
 )   =>    |-  ( T  e.  ( S `  U )  ->  U. T  =  ( `' F " U ) )
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