HomeHome Metamath Proof Explorer
Theorem List (p. 230 of 311)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-31058)
 

Theorem List for Metamath Proof Explorer - 22901-23000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremerdszelem9 22901* Lemma for erdsze 22904. (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 22902* Lemma for erdsze 22904. (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 22903* Lemma for erdsze 22904. (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 22904* 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 22905* Lemma for erdsze2 22907. (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 22906* Lemma for erdsze2 22907. (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 22907* Generalize the statement of the Erdős-Szekeres theorem erdsze 22904 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 ) ) ) ) )
 
16.3.6  The Kuratowski closure-complement theorem
 
Theoremkur14lem1 22908 Lemma for kur14 22918. (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 22909 Lemma for kur14 22918. 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 22910 Lemma for kur14 22918. 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 22911 Lemma for kur14 22918. 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 22912 Lemma for kur14 22918. 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 22913 Lemma for kur14 22918. 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 22914 Lemma for kur14 22918: 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 22913. (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 22915 Lemma for kur14 22918. 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 22916* Lemma for kur14 22918. 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 22917* Lemma for kur14 22918. 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 22918* 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 ) )
 
16.3.7  Retracts and sections
 
Syntaxcretr 22919 Extend class notation with the retract relation.
 class Retr
 
Definitiondf-retr 22920* 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 ) )  =/=  (/) } )
 
16.3.8  Path-connected and simply connected spaces
 
Syntaxcpcon 22921 Extend class notation with the class of path-connected topologies.
 class PCon
 
Syntaxcscon 22922 Extend class notation with the class of simply connected topologies.
 class SCon
 
Definitiondf-pcon 22923* 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 22924* 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 22925* 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 22926* 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 22927 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( J  e. PCon  ->  J  e.  Top )
 
Theoremisscon 22928* 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 22929 A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( J  e. SCon  ->  J  e. PCon )
 
Theoremscontop 22930 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( J  e. SCon  ->  J  e.  Top )
 
Theoremsconpht 22931 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 22932 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 22933 A path-connected space is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( J  e. PCon  ->  J  e.  Con )
 
Theoremtxpcon 22934 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 22935 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 22936 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 22937 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 22938 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 22939 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 22940 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 22941 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 22942 Lemma for txscon 22943. (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 22943 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 22944* 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 22945* 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 22946 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 22947 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 22948 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 22949 An open interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  (
 ( topGen `  ran  (,) )t  ( A (,) B ) )  e. SCon
 
Theoremiccscon 22950 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 22951 The real numbers are simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  ( topGen `
  ran  (,) )  e. SCon
 
Theoremiccllyscon 22952 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 22953 The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  ( topGen `
  ran  (,) )  e. Locally SCon
 
Theoremiiscon 22954 The unit interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  II  e. SCon
 
Theoremiillyscon 22955 The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  II  e. Locally SCon
 
Theoremiinllycon 22956 The unit interval is locally connected. (Contributed by Mario Carneiro, 6-Jul-2015.)
 |-  II  e. 𝑛Locally  Con
 
16.3.9  Covering maps
 
Syntaxccvm 22957 Extend class notation with the class of covering maps.
 class CovMap
 
Definitiondf-cvm 22958* 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 22959 Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.)
 |- CovMap  Fn  ( Top  X.  Top )
 
Theoremcvmscbv 22960* 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 22961* 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 22962 Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( F  e.  ( C CovMap  J )  ->  C  e.  Top )
 
Theoremcvmtop2 22963 Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
 |-  ( F  e.  ( C CovMap  J )  ->  J  e.  Top )
 
Theoremcvmcn 22964 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 22965* 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 22966* 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 22967* One direction of cvmsval 22968. (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 22968* 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 22969* 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 22970* 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 22971* 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 ) )
 
Theoremcvmsdisj 22972* An even covering of  U is a disjoint union. (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 ) 
 /\  A  e.  T  /\  B  e.  T ) 
 ->  ( A  =  B  \/  ( A  i^i  B )  =  (/) ) )
 
Theoremcvmshmeo 22973* Every element of an even covering of  U is homeomorphic to  U via  F. (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 ) 
 /\  A  e.  T )  ->  ( F  |`  A )  e.  ( ( Ct  A )  Homeo  ( Jt  U ) ) )
 
Theoremcvmsf1o 22974*  F, localized to an element of an even covering of  U, is a bijection. (Contributed by Mario Carneiro, 14-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 ) ) ) ) }
 )   =>    |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U )  /\  A  e.  T )  ->  ( F  |`  A ) : A -1-1-onto-> U )
 
Theoremcvmscld 22975* The sets of an even covering are clopen in the subspace topology on  T. (Contributed by Mario Carneiro, 14-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 ) ) ) ) }
 )   =>    |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U )  /\  A  e.  T )  ->  A  e.  ( Clsd `  ( Ct  ( `' F " U ) ) ) )
 
Theoremcvmsss2 22976* An open subset of an evenly covered set is evenly covered. (Contributed by Mario Carneiro, 7-Jul-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 ) ) ) ) }
 )   =>    |-  ( ( F  e.  ( C CovMap  J )  /\  V  e.  J  /\  V  C_  U )  ->  ( ( S `  U )  =/=  (/)  ->  ( S `  V )  =/=  (/) ) )
 
Theoremcvmcov2 22977* The covering map property can be restricted to an open subset. (Contributed by Mario Carneiro, 7-Jul-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 ) ) ) ) }
 )   =>    |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  E. x  e.  ~P  U ( P  e.  x  /\  ( S `  x )  =/=  (/) ) )
 
Theoremcvmseu 22978* Every element in  U. T is a member of a unique element of  T. (Contributed by Mario Carneiro, 14-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 ) ) ) ) }
 )   &    |-  B  =  U. C   =>    |-  (
 ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `
  A )  e.  U ) )  ->  E! x  e.  T  A  e.  x )
 
Theoremcvmsiota 22979* Identify the unique element of  T containing  A. (Contributed by Mario Carneiro, 14-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  W  =  ( iota_ x  e.  T A  e.  x )   =>    |-  (
 ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `
  A )  e.  U ) )  ->  ( W  e.  T  /\  A  e.  W ) )
 
Theoremcvmopnlem 22980* Lemma for cvmopn 22982. (Contributed by Mario Carneiro, 7-May-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 ) ) ) ) }
 )   &    |-  B  =  U. C   =>    |-  (
 ( F  e.  ( C CovMap  J )  /\  A  e.  C )  ->  ( F " A )  e.  J )
 
Theoremcvmfolem 22981* Lemma for cvmfo 23002. (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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   =>    |-  ( F  e.  ( C CovMap  J )  ->  F : B -onto-> X )
 
Theoremcvmopn 22982 A covering map is an open map. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  (
 ( F  e.  ( C CovMap  J )  /\  A  e.  C )  ->  ( F " A )  e.  J )
 
Theoremcvmliftmolem1 22983* Lemma for cvmliftmo 22986. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e.  Con )   &    |-  ( ph  ->  K  e. 𝑛Locally  Con )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  M  e.  ( K  Cn  C ) )   &    |-  ( ph  ->  N  e.  ( K  Cn  C ) )   &    |-  ( ph  ->  ( F  o.  M )  =  ( F  o.  N ) )   &    |-  ( ph  ->  ( M `  O )  =  ( N `  O ) )   &    |-  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 ) ) ) ) }
 )   &    |-  ( ( ph  /\  ps )  ->  T  e.  ( S `  U ) )   &    |-  ( ( ph  /\  ps )  ->  W  e.  T )   &    |-  ( ( ph  /\  ps )  ->  I  C_  ( `' M " W ) )   &    |-  ( ( ph  /\ 
 ps )  ->  ( Kt  I )  e.  Con )   &    |-  ( ( ph  /\  ps )  ->  X  e.  I
 )   &    |-  ( ( ph  /\  ps )  ->  Q  e.  I
 )   &    |-  ( ( ph  /\  ps )  ->  R  e.  I
 )   &    |-  ( ( ph  /\  ps )  ->  ( F `  ( M `  X ) )  e.  U )   =>    |-  ( ( ph  /\  ps )  ->  ( Q  e.  dom  (  M  i^i  N )  ->  R  e.  dom  (  M  i^i  N ) ) )
 
Theoremcvmliftmolem2 22984* Lemma for cvmliftmo 22986. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e.  Con )   &    |-  ( ph  ->  K  e. 𝑛Locally  Con )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  M  e.  ( K  Cn  C ) )   &    |-  ( ph  ->  N  e.  ( K  Cn  C ) )   &    |-  ( ph  ->  ( F  o.  M )  =  ( F  o.  N ) )   &    |-  ( ph  ->  ( M `  O )  =  ( N `  O ) )   &    |-  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 ) ) ) ) }
 )   =>    |-  ( ph  ->  M  =  N )
 
Theoremcvmliftmoi 22985 A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e.  Con )   &    |-  ( ph  ->  K  e. 𝑛Locally  Con )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  M  e.  ( K  Cn  C ) )   &    |-  ( ph  ->  N  e.  ( K  Cn  C ) )   &    |-  ( ph  ->  ( F  o.  M )  =  ( F  o.  N ) )   &    |-  ( ph  ->  ( M `  O )  =  ( N `  O ) )   =>    |-  ( ph  ->  M  =  N )
 
Theoremcvmliftmo 22986* A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e.  Con )   &    |-  ( ph  ->  K  e. 𝑛Locally  Con )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  G  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  O ) )   =>    |-  ( ph  ->  E* f ( f  e.  ( K  Cn  C )  /\  ( ( F  o.  f )  =  G  /\  ( f `
  O )  =  P ) ) )
 
Theoremcvmliftlem1 22987* Lemma for cvmlift 23001. In cvmliftlem15 23000, we picked an  N large enough so that the sections  ( G " [ ( k  -  1 )  /  N ,  k  /  N ] ) are all contained in an even covering, and the function  T enumerates these even coverings. So  1st `  ( T `  M
) is a neighborhood of  ( G " [
( M  -  1 )  /  N ,  M  /  N ] ), and  2nd `  ( T `  M ) is an even covering of  1st `  ( T `  M ), which is to say a disjoint union of open sets in  C whose image is  1st `  ( T `
 M ). (Contributed by Mario Carneiro, 14-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T : ( 1 ... N ) -->
 U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )   &    |-  ( ph  ->  A. k  e.  ( 1
 ... N ) ( G " ( ( ( k  -  1
 )  /  N ) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k ) ) )   &    |-  L  =  ( topGen `  ran  (,) )   &    |-  ( ( ph  /\ 
 ps )  ->  M  e.  ( 1 ... N ) )   =>    |-  ( ( ph  /\  ps )  ->  ( 2nd `  ( T `  M ) )  e.  ( S `  ( 1st `  ( T `  M ) ) ) )
 
Theoremcvmliftlem2 22988* Lemma for cvmlift 23001. 
W  =  [ ( k  -  1 )  /  N ,  k  /  N ] is a subset of  [ 0 ,  1 ] for each  M  e.  ( 1 ... N
). (Contributed by Mario Carneiro, 16-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T : ( 1 ... N ) -->
 U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )   &    |-  ( ph  ->  A. k  e.  ( 1
 ... N ) ( G " ( ( ( k  -  1
 )  /  N ) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k ) ) )   &    |-  L  =  ( topGen `  ran  (,) )   &    |-  ( ( ph  /\ 
 ps )  ->  M  e.  ( 1 ... N ) )   &    |-  W  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  W  C_  ( 0 [,] 1
 ) )
 
Theoremcvmliftlem3 22989* Lemma for cvmlift 23001. Since  1st `  ( T `  M
) is a neighborhood of  ( G " W ), every element  A  e.  W satisfies  ( G `  A )  e.  ( 1st `  ( T `
 M ) ). (Contributed by Mario Carneiro, 16-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T : ( 1 ... N ) -->
 U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )   &    |-  ( ph  ->  A. k  e.  ( 1
 ... N ) ( G " ( ( ( k  -  1
 )  /  N ) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k ) ) )   &    |-  L  =  ( topGen `  ran  (,) )   &    |-  ( ( ph  /\ 
 ps )  ->  M  e.  ( 1 ... N ) )   &    |-  W  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )   &    |-  (
 ( ph  /\  ps )  ->  A  e.  W )   =>    |-  ( ( ph  /\  ps )  ->  ( G `  A )  e.  ( 1st `  ( T `  M ) ) )
 
Theoremcvmliftlem4 22990* Lemma for cvmlift 23001. The function  Q will be our lifted path, defined piecewise on each section  [ ( M  -  1 )  /  N ,  M  /  N ] for  M  e.  ( 1 ... N ). For 
M  =  0, it is a "seed" value which makes the rest of the recursion work, a singleton function mapping  0 to  P. (Contributed by Mario Carneiro, 15-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T : ( 1 ... N ) -->
 U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )   &    |-  ( ph  ->  A. k  e.  ( 1
 ... N ) ( G " ( ( ( k  -  1
 )  /  N ) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k ) ) )   &    |-  L  =  ( topGen `  ran  (,) )   &    |-  Q  =  seq  0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N ) ) 
 |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m ) ) ( x `  ( ( m  -  1 ) 
 /  N ) )  e.  b ) ) `
  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
 >. } ) )   =>    |-  ( Q `  0 )  =  { <. 0 ,  P >. }
 
Theoremcvmliftlem5 22991* Lemma for cvmlift 23001. Definition of  Q at a successor. This is a function defined on  W as  `' ( T  |`  I )  o.  G where  I is the unique covering set of  2nd `  ( T `  M ) that contains  Q ( M  -  1 ) evaluated at the last defined point, namely  ( M  - 
1 )  /  N (note that for  M  =  1 this is using the seed value  Q ( 0 ) ( 0 )  =  P). (Contributed by Mario Carneiro, 15-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T : ( 1 ... N ) -->
 U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )   &    |-  ( ph  ->  A. k  e.  ( 1
 ... N ) ( G " ( ( ( k  -  1
 )  /  N ) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k ) ) )   &    |-  L  =  ( topGen `  ran  (,) )   &    |-  Q  =  seq  0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N ) ) 
 |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m ) ) ( x `  ( ( m  -  1 ) 
 /  N ) )  e.  b ) ) `
  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
 >. } ) )   &    |-  W  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )   =>    |-  ( ( ph  /\  M  e.  NN )  ->  ( Q `  M )  =  ( z  e.  W  |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  M ) ) ( ( Q `  ( M  -  1
 ) ) `  (
 ( M  -  1
 )  /  N )
 )  e.  b ) ) `  ( G `
  z ) ) ) )
 
Theoremcvmliftlem6 22992* Lemma for cvmlift 23001. Induction step for cvmliftlem7 22993. Assuming that  Q ( M  - 
1 ) is defined at  ( M  -  1 )  /  N and is a preimage of  G ( ( M  -  1 )  /  N ), the next segment  Q ( M ) is also defined and is a function on  W which is a lift  G for this segment. This follows explicitly from the definition  Q ( M )  =  `' ( F  |`  I )  o.  G since  G is in  1st `  ( F `  M ) for the entire interval so that  `' ( F  |`  I ) maps this into  I and  F  o.  Q maps back to  G. (Contributed by Mario Carneiro, 16-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T : ( 1 ... N ) -->
 U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )   &    |-  ( ph  ->  A. k  e.  ( 1
 ... N ) ( G " ( ( ( k  -  1
 )  /  N ) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k ) ) )   &    |-  L  =  ( topGen `  ran  (,) )   &    |-  Q  =  seq  0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N ) ) 
 |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m ) ) ( x `  ( ( m  -  1 ) 
 /  N ) )  e.  b ) ) `
  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
 >. } ) )   &    |-  W  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )   &    |-  ( ( ph  /\  ps )  ->  M  e.  (
 1 ... N ) )   &    |-  ( ( ph  /\  ps )  ->  ( ( Q `
  ( M  -  1 ) ) `  ( ( M  -  1 )  /  N ) )  e.  ( `' F " { ( G `  ( ( M  -  1 )  /  N ) ) }
 ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( Q `
  M ) : W --> B  /\  ( F  o.  ( Q `  M ) )  =  ( G  |`  W ) ) )
 
Theoremcvmliftlem7 22993* Lemma for cvmlift 23001. Prove by induction that every  Q function is well-defined (we can immediately follow this theorem with cvmliftlem6 22992 to show functionality and lifting of  Q). (Contributed by Mario Carneiro, 14-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T : ( 1 ... N ) -->
 U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )   &    |-  ( ph  ->  A. k  e.  ( 1
 ... N ) ( G " ( ( ( k  -  1
 )  /  N ) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k ) ) )   &    |-  L  =  ( topGen `  ran  (,) )   &    |-  Q  =  seq  0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N ) ) 
 |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m ) ) ( x `  ( ( m  -  1 ) 
 /  N ) )  e.  b ) ) `
  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
 >. } ) )   &    |-  W  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )   =>    |-  ( ( ph  /\  M  e.  ( 1 ... N ) )  ->  ( ( Q `  ( M  -  1 ) ) `
  ( ( M  -  1 )  /  N ) )  e.  ( `' F " { ( G `  ( ( M  -  1 )  /  N ) ) } ) )
 
Theoremcvmliftlem8 22994* Lemma for cvmlift 23001. The functions  Q are continuous functions because they are defined as  `' ( F  |`  I )  o.  G where  G is continuous and  ( F  |`  I ) is a homeomorphism. (Contributed by Mario Carneiro, 16-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T : ( 1 ... N ) -->
 U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )   &    |-  ( ph  ->  A. k  e.  ( 1
 ... N ) ( G " ( ( ( k  -  1
 )  /  N ) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k ) ) )   &    |-  L  =  ( topGen `  ran  (,) )   &    |-  Q  =  seq  0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N ) ) 
 |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m ) ) ( x `  ( ( m  -  1 ) 
 /  N ) )  e.  b ) ) `
  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
 >. } ) )   &    |-  W  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )   =>    |-  ( ( ph  /\  M  e.  ( 1 ... N ) )  ->  ( Q `
  M )  e.  ( ( Lt  W )  Cn  C ) )
 
Theoremcvmliftlem9 22995* Lemma for cvmlift 23001. The  Q ( M ) functions are defined on almost disjoint intervals, but they overlap at the edges. Here we show that at these points the  Q functions agree on their common domain. (Contributed by Mario Carneiro, 14-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T : ( 1 ... N ) -->
 U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )   &    |-  ( ph  ->  A. k  e.  ( 1
 ... N ) ( G " ( ( ( k  -  1
 )  /  N ) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k ) ) )   &    |-  L  =  ( topGen `  ran  (,) )   &    |-  Q  =  seq  0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N ) ) 
 |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m ) ) ( x `  ( ( m  -  1 ) 
 /  N ) )  e.  b ) ) `
  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
 >. } ) )   =>    |-  ( ( ph  /\  M  e.  ( 1
 ... N ) ) 
 ->  ( ( Q `  M ) `  (
 ( M  -  1
 )  /  N )
 )  =  ( ( Q `  ( M  -  1 ) ) `
  ( ( M  -  1 )  /  N ) ) )
 
Theoremcvmliftlem10 22996* Lemma for cvmlift 23001. The function  K is going to be our complete lifted path, formed by unioning together all the  Q functions (each of which is defined on one segment  [ ( M  -  1 )  /  N ,  M  /  N ] of the interval). Here we prove by induction that  K is a continuous function and a lift of  G by applying cvmliftlem6 22992, cvmliftlem7 22993 (to show it is a function and a lift), cvmliftlem8 22994 (to show it is continuous), and cvmliftlem9 22995 (to show that different 
Q functions agree on the intersection of their domains, so that the pasting lemma paste 16854 gives that  K is well-defined and continuous). (Contributed by Mario Carneiro, 14-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T : ( 1 ... N ) -->
 U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )   &    |-  ( ph  ->  A. k  e.  ( 1
 ... N ) ( G " ( ( ( k  -  1
 )  /  N ) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k ) ) )   &    |-  L  =  ( topGen `  ran  (,) )   &    |-  Q  =  seq  0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N ) ) 
 |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m ) ) ( x `  ( ( m  -  1 ) 
 /  N ) )  e.  b ) ) `
  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
 >. } ) )   &    |-  K  =  U_ k  e.  (
 1 ... N ) ( Q `  k )   &    |-  ( ch  <->  ( ( n  e.  NN  /\  ( n  +  1 )  e.  ( 1 ... N ) )  /\  ( U_ k  e.  ( 1 ... n ) ( Q `
  k )  e.  ( ( Lt  ( 0 [,] ( n  /  N ) ) )  Cn  C )  /\  ( F  o.  U_ k  e.  ( 1 ... n ) ( Q `  k ) )  =  ( G  |`  ( 0 [,] ( n  /  N ) ) ) ) ) )   =>    |-  ( ph  ->  ( K  e.  ( ( Lt  ( 0 [,] ( N  /  N ) ) )  Cn  C ) 
 /\  ( F  o.  K )  =  ( G  |`  ( 0 [,] ( N  /  N ) ) ) ) )
 
Theoremcvmliftlem11 22997* Lemma for cvmlift 23001. (Contributed by Mario Carneiro, 14-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T : ( 1 ... N ) -->
 U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )   &    |-  ( ph  ->  A. k  e.  ( 1
 ... N ) ( G " ( ( ( k  -  1
 )  /  N ) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k ) ) )   &    |-  L  =  ( topGen `  ran  (,) )   &    |-  Q  =  seq  0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N ) ) 
 |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m ) ) ( x `  ( ( m  -  1 ) 
 /  N ) )  e.  b ) ) `
  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
 >. } ) )   &    |-  K  =  U_ k  e.  (
 1 ... N ) ( Q `  k )   =>    |-  ( ph  ->  ( K  e.  ( II  Cn  C )  /\  ( F  o.  K )  =  G ) )
 
Theoremcvmliftlem13 22998* Lemma for cvmlift 23001. The initial value of  K is  P because  Q ( 1 ) is a subset of  K which takes value  P at  0. (Contributed by Mario Carneiro, 16-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T : ( 1 ... N ) -->
 U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )   &    |-  ( ph  ->  A. k  e.  ( 1
 ... N ) ( G " ( ( ( k  -  1
 )  /  N ) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k ) ) )   &    |-  L  =  ( topGen `  ran  (,) )   &    |-  Q  =  seq  0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N ) ) 
 |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m ) ) ( x `  ( ( m  -  1 ) 
 /  N ) )  e.  b ) ) `
  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
 >. } ) )   &    |-  K  =  U_ k  e.  (
 1 ... N ) ( Q `  k )   =>    |-  ( ph  ->  ( K `  0 )  =  P )
 
Theoremcvmliftlem14 22999* Lemma for cvmlift 23001. Putting the results of cvmliftlem11 22997, cvmliftlem13 22998 and cvmliftmo 22986 together, we have that  K is a continuous function, satisfies  F  o.  K  =  G and  K ( 0 )  =  P, and is equal to any other function which also has these properties, so it follows that  K is the unique lift of  G. (Contributed by Mario Carneiro, 16-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T : ( 1 ... N ) -->
 U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )   &    |-  ( ph  ->  A. k  e.  ( 1
 ... N ) ( G " ( ( ( k  -  1
 )  /  N ) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k ) ) )   &    |-  L  =  ( topGen `  ran  (,) )   &    |-  Q  =  seq  0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N ) ) 
 |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m ) ) ( x `  ( ( m  -  1 ) 
 /  N ) )  e.  b ) ) `
  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. }
 >. } ) )   &    |-  K  =  U_ k  e.  (
 1 ... N ) ( Q `  k )   =>    |-  ( ph  ->  E! f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `
  0 )  =  P ) )
 
Theoremcvmliftlem15 23000* Lemma for cvmlift 23001. Discharge the assumptions of cvmliftlem14 22999. The set of all open subsets 
u of the unit interval such that  G " u is contained in an even covering of some open set in  J is a cover of  II by the definition of a covering map, so by the Lebesgue number lemma lebnumii 18296, there is a subdivision of the unit interval into  N equal parts such that each part is entirely contained within one such open set of  J. Then using finite choice ac6sfi 6986 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 22999. (Contributed by Mario Carneiro, 14-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 ) ) ) ) }
 )   &    |-  B  =  U. C   &    |-  X  =  U. J   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   =>    |-  ( ph  ->  E! f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  (
 f `  0 )  =  P ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31058
  Copyright terms: Public domain < Previous  Next >