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Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcvmsrcl 23201* 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 23202* One direction of cvmsval 23203. (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 23203* 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 23204* 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 23205* 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 23206* 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 23207* 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 23208* 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 23209*  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 23210* 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 23211* 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 23212* 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 23213* 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 23214* 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 23215* Lemma for cvmopn 23217. (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 23216* Lemma for cvmfo 23237. (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 23217 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 23218* Lemma for cvmliftmo 23221. (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 23219* Lemma for cvmliftmo 23221. (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 23220 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 23221* 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.) (Revised by NM, 17-Jun-2017.)
 |-  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  e.  ( K  Cn  C ) ( ( F  o.  f
 )  =  G  /\  ( f `  O )  =  P )
 )
 
Theoremcvmliftlem1 23222* Lemma for cvmlift 23236. In cvmliftlem15 23235, 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 23223* Lemma for cvmlift 23236. 
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 23224* Lemma for cvmlift 23236. 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 23225* Lemma for cvmlift 23236. 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 23226* Lemma for cvmlift 23236. 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 23227* Lemma for cvmlift 23236. Induction step for cvmliftlem7 23228. 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 23228* Lemma for cvmlift 23236. Prove by induction that every  Q function is well-defined (we can immediately follow this theorem with cvmliftlem6 23227 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 23229* Lemma for cvmlift 23236. 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 23230* Lemma for cvmlift 23236. 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 23231* Lemma for cvmlift 23236. 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 23227, cvmliftlem7 23228 (to show it is a function and a lift), cvmliftlem8 23229 (to show it is continuous), and cvmliftlem9 23230 (to show that different 
Q functions agree on the intersection of their domains, so that the pasting lemma paste 17017 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 23232* Lemma for cvmlift 23236. (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 23233* Lemma for cvmlift 23236. 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 23234* Lemma for cvmlift 23236. Putting the results of cvmliftlem11 23232, cvmliftlem13 23233 and cvmliftmo 23221 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 23235* Lemma for cvmlift 23236. Discharge the assumptions of cvmliftlem14 23234. 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 18459, 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 7096 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 23234. (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 ) )
 
Theoremcvmlift 23236* One of the important properties of covering maps is that any path  G in the base space "lifts" to a path  f in the covering space such that  F  o.  f  =  G, and given a starting point  P in the covering space this lift is unique. The proof is contained in cvmliftlem1 23222 thru cvmliftlem15 23235. (Contributed by Mario Carneiro, 16-Feb-2015.)
 |-  B  =  U. C   =>    |-  ( ( ( F  e.  ( C CovMap  J )  /\  G  e.  ( II  Cn  J ) ) 
 /\  ( P  e.  B  /\  ( F `  P )  =  ( G `  0 ) ) )  ->  E! f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `
  0 )  =  P ) )
 
Theoremcvmfo 23237 A covering map is an onto function. (Contributed by Mario Carneiro, 13-Feb-2015.)
 |-  B  =  U. C   &    |-  X  =  U. J   =>    |-  ( F  e.  ( C CovMap  J )  ->  F : B -onto-> X )
 
Theoremcvmliftiota 23238* Write out a function  H that is the unique lift of  F. (Contributed by Mario Carneiro, 16-Feb-2015.)
 |-  B  =  U. C   &    |-  H  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  G  /\  ( f `  0
 )  =  P ) )   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0
 ) )   =>    |-  ( ph  ->  ( H  e.  ( II  Cn  C )  /\  ( F  o.  H )  =  G  /\  ( H `
  0 )  =  P ) )
 
Theoremcvmlift2lem1 23239* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 1-Jun-2015.)
 |-  ( A. y  e.  (
 0 [,] 1 ) E. u  e.  ( ( nei `  II ) `  { y } )
 ( ( u  X.  { x } )  C_  M 
 <->  ( u  X.  {
 t } )  C_  M )  ->  ( ( ( 0 [,] 1
 )  X.  { x } )  C_  M  ->  ( ( 0 [,] 1
 )  X.  { t } )  C_  M ) )
 
Theoremcvmlift2lem9a 23240* Lemma for cvmlift2 23253 and cvmlift3 23265. (Contributed by Mario Carneiro, 9-Jul-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. c  e.  s  ( A. d  e.  (
 s  \  { c } ) ( c  i^i  d )  =  (/)  /\  ( F  |`  c )  e.  ( ( Ct  c )  Homeo  ( Jt  k ) ) ) ) }
 )   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  H : Y --> B )   &    |-  ( ph  ->  ( F  o.  H )  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  K  e.  Top )   &    |-  ( ph  ->  X  e.  Y )   &    |-  ( ph  ->  T  e.  ( S `  A ) )   &    |-  ( ph  ->  ( W  e.  T  /\  ( H `
  X )  e.  W ) )   &    |-  ( ph  ->  M  C_  Y )   &    |-  ( ph  ->  ( H " M )  C_  W )   =>    |-  ( ph  ->  ( H  |`  M )  e.  ( ( Kt  M )  Cn  C ) )
 
Theoremcvmlift2lem2 23241* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  B  =  U. C   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  (
 0 G 0 ) )   &    |-  H  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) 
 /\  ( f `  0 )  =  P ) )   =>    |-  ( ph  ->  ( H  e.  ( II  Cn  C )  /\  ( F  o.  H )  =  ( z  e.  (
 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( H `
  0 )  =  P ) )
 
Theoremcvmlift2lem3 23242* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  B  =  U. C   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  (
 0 G 0 ) )   &    |-  H  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) 
 /\  ( f `  0 )  =  P ) )   &    |-  K  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) ) 
 /\  ( f `  0 )  =  ( H `  X ) ) )   =>    |-  ( ( ph  /\  X  e.  ( 0 [,] 1
 ) )  ->  ( K  e.  ( II  Cn  C )  /\  ( F  o.  K )  =  ( z  e.  (
 0 [,] 1 )  |->  ( X G z ) )  /\  ( K `
  0 )  =  ( H `  X ) ) )
 
Theoremcvmlift2lem4 23243* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 1-Jun-2015.)
 |-  B  =  U. C   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  (
 0 G 0 ) )   &    |-  H  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) 
 /\  ( f `  0 )  =  P ) )   &    |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) 
 /\  ( f `  0 )  =  ( H `  x ) ) ) `  y ) )   =>    |-  ( ( X  e.  ( 0 [,] 1
 )  /\  Y  e.  ( 0 [,] 1
 ) )  ->  ( X K Y )  =  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  (
 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
  0 )  =  ( H `  X ) ) ) `  Y ) )
 
Theoremcvmlift2lem5 23244* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  B  =  U. C   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  (
 0 G 0 ) )   &    |-  H  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) 
 /\  ( f `  0 )  =  P ) )   &    |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) 
 /\  ( f `  0 )  =  ( H `  x ) ) ) `  y ) )   =>    |-  ( ph  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
 ) ) --> B )
 
Theoremcvmlift2lem6 23245* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  B  =  U. C   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  (
 0 G 0 ) )   &    |-  H  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) 
 /\  ( f `  0 )  =  P ) )   &    |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) 
 /\  ( f `  0 )  =  ( H `  x ) ) ) `  y ) )   =>    |-  ( ( ph  /\  X  e.  ( 0 [,] 1
 ) )  ->  ( K  |`  ( { X }  X.  ( 0 [,] 1 ) ) )  e.  ( ( ( II  tX  II )t  ( { X }  X.  (
 0 [,] 1 ) ) )  Cn  C ) )
 
Theoremcvmlift2lem7 23246* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  B  =  U. C   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  (
 0 G 0 ) )   &    |-  H  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) 
 /\  ( f `  0 )  =  P ) )   &    |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) 
 /\  ( f `  0 )  =  ( H `  x ) ) ) `  y ) )   =>    |-  ( ph  ->  ( F  o.  K )  =  G )
 
Theoremcvmlift2lem8 23247* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  B  =  U. C   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  (
 0 G 0 ) )   &    |-  H  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) 
 /\  ( f `  0 )  =  P ) )   &    |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) 
 /\  ( f `  0 )  =  ( H `  x ) ) ) `  y ) )   =>    |-  ( ( ph  /\  X  e.  ( 0 [,] 1
 ) )  ->  ( X K 0 )  =  ( H `  X ) )
 
Theoremcvmlift2lem9 23248* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 1-Jun-2015.)
 |-  B  =  U. C   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  (
 0 G 0 ) )   &    |-  H  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) 
 /\  ( f `  0 )  =  P ) )   &    |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) 
 /\  ( f `  0 )  =  ( H `  x ) ) ) `  y ) )   &    |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. c  e.  s  ( A. d  e.  (
 s  \  { c } ) ( c  i^i  d )  =  (/)  /\  ( F  |`  c )  e.  ( ( Ct  c )  Homeo  ( Jt  k ) ) ) ) }
 )   &    |-  ( ph  ->  ( X G Y )  e.  M )   &    |-  ( ph  ->  T  e.  ( S `  M ) )   &    |-  ( ph  ->  U  e.  II )   &    |-  ( ph  ->  V  e.  II )   &    |-  ( ph  ->  ( IIt  U )  e.  Con )   &    |-  ( ph  ->  ( IIt  V )  e.  Con )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( U  X.  V ) 
 C_  ( `' G " M ) )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( K  |`  ( U  X.  { Z } ) )  e.  ( ( ( II  tX  II )t  ( U  X.  { Z }
 ) )  Cn  C ) )   &    |-  W  =  (
 iota_ b  e.  T ( X K Y )  e.  b )   =>    |-  ( ph  ->  ( K  |`  ( U  X.  V ) )  e.  ( ( ( II  tX  II )t  ( U  X.  V ) )  Cn  C ) )
 
Theoremcvmlift2lem10 23249* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 1-Jun-2015.)
 |-  B  =  U. C   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  (
 0 G 0 ) )   &    |-  H  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) 
 /\  ( f `  0 )  =  P ) )   &    |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) 
 /\  ( f `  0 )  =  ( H `  x ) ) ) `  y ) )   &    |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. c  e.  s  ( A. d  e.  (
 s  \  { c } ) ( c  i^i  d )  =  (/)  /\  ( F  |`  c )  e.  ( ( Ct  c )  Homeo  ( Jt  k ) ) ) ) }
 )   &    |-  ( ph  ->  X  e.  ( 0 [,] 1
 ) )   &    |-  ( ph  ->  Y  e.  ( 0 [,] 1 ) )   =>    |-  ( ph  ->  E. u  e.  II  E. v  e.  II  ( X  e.  u  /\  Y  e.  v  /\  ( E. w  e.  v  ( K  |`  ( u  X.  { w }
 ) )  e.  (
 ( ( II  tX  II )t  ( u  X.  { w } ) )  Cn  C )  ->  ( K  |`  ( u  X.  v
 ) )  e.  (
 ( ( II  tX  II )t  ( u  X.  v
 ) )  Cn  C ) ) ) )
 
Theoremcvmlift2lem11 23250* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 1-Jun-2015.)
 |-  B  =  U. C   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  (
 0 G 0 ) )   &    |-  H  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) 
 /\  ( f `  0 )  =  P ) )   &    |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) 
 /\  ( f `  0 )  =  ( H `  x ) ) ) `  y ) )   &    |-  M  =  {
 z  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  |  K  e.  ( ( ( II  tX  II )  CnP  C ) `  z ) }   &    |-  ( ph  ->  U  e.  II )   &    |-  ( ph  ->  V  e.  II )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( E. w  e.  V  ( K  |`  ( U  X.  { w }
 ) )  e.  (
 ( ( II  tX  II )t  ( U  X.  { w } ) )  Cn  C )  ->  ( K  |`  ( U  X.  V ) )  e.  (
 ( ( II  tX  II )t  ( U  X.  V ) )  Cn  C ) ) )   =>    |-  ( ph  ->  (
 ( U  X.  { Y } )  C_  M  ->  ( U  X.  { Z } )  C_  M ) )
 
Theoremcvmlift2lem12 23251* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 1-Jun-2015.)
 |-  B  =  U. C   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  (
 0 G 0 ) )   &    |-  H  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) 
 /\  ( f `  0 )  =  P ) )   &    |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) 
 /\  ( f `  0 )  =  ( H `  x ) ) ) `  y ) )   &    |-  M  =  {
 z  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  |  K  e.  ( ( ( II  tX  II )  CnP  C ) `  z ) }   &    |-  A  =  { a  e.  (
 0 [,] 1 )  |  ( ( 0 [,] 1 )  X.  {
 a } )  C_  M }   &    |-  S  =  { <. r ,  t >.  |  ( t  e.  (
 0 [,] 1 )  /\  E. u  e.  ( ( nei `  II ) `  { r } )
 ( ( u  X.  { a } )  C_  M 
 <->  ( u  X.  {
 t } )  C_  M ) ) }   =>    |-  ( ph  ->  K  e.  (
 ( II  tX  II )  Cn  C ) )
 
Theoremcvmlift2lem13 23252* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  B  =  U. C   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  (
 0 G 0 ) )   &    |-  H  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) ) 
 /\  ( f `  0 )  =  P ) )   &    |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) ) 
 /\  ( f `  0 )  =  ( H `  x ) ) ) `  y ) )   =>    |-  ( ph  ->  E! g  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  g )  =  G  /\  (
 0 g 0 )  =  P ) )
 
Theoremcvmlift2 23253* A two-dimensional version of cvmlift 23236. There is a unique lift of functions on the unit square 
II  tX  II which commutes with the covering map. (Contributed by Mario Carneiro, 1-Jun-2015.)
 |-  B  =  U. C   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  (
 0 G 0 ) )   =>    |-  ( ph  ->  E! f  e.  ( ( II  tX  II )  Cn  C ) ( ( F  o.  f )  =  G  /\  (
 0 f 0 )  =  P ) )
 
Theoremcvmliftphtlem 23254* Lemma for cvmliftpht 23255. (Contributed by Mario Carneiro, 6-Jul-2015.)
 |-  B  =  U. C   &    |-  M  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  G  /\  ( f `  0
 )  =  P ) )   &    |-  N  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  H  /\  ( f `  0
 )  =  P ) )   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0 ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  H  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  K  e.  ( G ( PHtpy `  J ) H ) )   &    |-  ( ph  ->  A  e.  (
 ( II  tX  II )  Cn  C ) )   &    |-  ( ph  ->  ( F  o.  A )  =  K )   &    |-  ( ph  ->  (
 0 A 0 )  =  P )   =>    |-  ( ph  ->  A  e.  ( M (
 PHtpy `  C ) N ) )
 
Theoremcvmliftpht 23255* If  G and  H are path-homotopic, then their lifts  M and  N are also path-homotopic. (Contributed by Mario Carneiro, 6-Jul-2015.)
 |-  B  =  U. C   &    |-  M  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  G  /\  ( f `  0
 )  =  P ) )   &    |-  N  =  (
 iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f
 )  =  H  /\  ( f `  0
 )  =  P ) )   &    |-  ( ph  ->  F  e.  ( C CovMap  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  0 ) )   &    |-  ( ph  ->  G (  ~=ph  `  J ) H )   =>    |-  ( ph  ->  M (  ~=ph  `  C ) N )
 
Theoremcvmlift3lem1 23256* Lemma for cvmlift3 23265. (Contributed by Mario Carneiro, 6-Jul-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e. SCon )   &    |-  ( ph  ->  K  e. 𝑛Locally PCon )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  G  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  O ) )   &    |-  ( ph  ->  M  e.  ( II  Cn  K ) )   &    |-  ( ph  ->  ( M `  0 )  =  O )   &    |-  ( ph  ->  N  e.  ( II  Cn  K ) )   &    |-  ( ph  ->  ( N `  0 )  =  O )   &    |-  ( ph  ->  ( M `  1 )  =  ( N `  1 ) )   =>    |-  ( ph  ->  ( ( iota_
 g  e.  ( II 
 Cn  C ) ( ( F  o.  g
 )  =  ( G  o.  M )  /\  ( g `  0
 )  =  P ) ) `  1 )  =  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  N )  /\  ( g `
  0 )  =  P ) ) `  1 ) )
 
Theoremcvmlift3lem2 23257* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 6-Jul-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e. SCon )   &    |-  ( ph  ->  K  e. 𝑛Locally PCon )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  G  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  O ) )   =>    |-  ( ( ph  /\  X  e.  Y ) 
 ->  E! z  e.  B  E. f  e.  ( II  Cn  K ) ( ( f `  0
 )  =  O  /\  ( f `  1
 )  =  X  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f
 )  /\  ( g `  0 )  =  P ) ) `  1
 )  =  z ) )
 
Theoremcvmlift3lem3 23258* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 6-Jul-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e. SCon )   &    |-  ( ph  ->  K  e. 𝑛Locally PCon )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  G  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  O ) )   &    |-  H  =  ( x  e.  Y  |->  ( iota_ z  e.  B E. f  e.  ( II  Cn  K ) ( ( f `  0
 )  =  O  /\  ( f `  1
 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f
 )  /\  ( g `  0 )  =  P ) ) `  1
 )  =  z ) ) )   =>    |-  ( ph  ->  H : Y --> B )
 
Theoremcvmlift3lem4 23259* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 6-Jul-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e. SCon )   &    |-  ( ph  ->  K  e. 𝑛Locally PCon )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  G  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  O ) )   &    |-  H  =  ( x  e.  Y  |->  ( iota_ z  e.  B E. f  e.  ( II  Cn  K ) ( ( f `  0
 )  =  O  /\  ( f `  1
 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f
 )  /\  ( g `  0 )  =  P ) ) `  1
 )  =  z ) ) )   =>    |-  ( ( ph  /\  X  e.  Y )  ->  (
 ( H `  X )  =  A  <->  E. f  e.  ( II  Cn  K ) ( ( f `  0
 )  =  O  /\  ( f `  1
 )  =  X  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f
 )  /\  ( g `  0 )  =  P ) ) `  1
 )  =  A ) ) )
 
Theoremcvmlift3lem5 23260* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 6-Jul-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e. SCon )   &    |-  ( ph  ->  K  e. 𝑛Locally PCon )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  G  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  O ) )   &    |-  H  =  ( x  e.  Y  |->  ( iota_ z  e.  B E. f  e.  ( II  Cn  K ) ( ( f `  0
 )  =  O  /\  ( f `  1
 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f
 )  /\  ( g `  0 )  =  P ) ) `  1
 )  =  z ) ) )   =>    |-  ( ph  ->  ( F  o.  H )  =  G )
 
Theoremcvmlift3lem6 23261* Lemma for cvmlift3 23265. (Contributed by Mario Carneiro, 9-Jul-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e. SCon )   &    |-  ( ph  ->  K  e. 𝑛Locally PCon )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  G  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  O ) )   &    |-  H  =  ( x  e.  Y  |->  ( iota_ z  e.  B E. f  e.  ( II  Cn  K ) ( ( f `  0
 )  =  O  /\  ( f `  1
 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f
 )  /\  ( g `  0 )  =  P ) ) `  1
 )  =  z ) ) )   &    |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. c  e.  s  ( A. d  e.  (
 s  \  { c } ) ( c  i^i  d )  =  (/)  /\  ( F  |`  c )  e.  ( ( Ct  c )  Homeo  ( Jt  k ) ) ) ) }
 )   &    |-  ( ph  ->  ( G `  X )  e.  A )   &    |-  ( ph  ->  T  e.  ( S `  A ) )   &    |-  ( ph  ->  M  C_  ( `' G " A ) )   &    |-  W  =  (
 iota_ b  e.  T ( H `  X )  e.  b )   &    |-  ( ph  ->  X  e.  M )   &    |-  ( ph  ->  Z  e.  M )   &    |-  ( ph  ->  Q  e.  ( II  Cn  K ) )   &    |-  R  =  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g
 )  =  ( G  o.  Q )  /\  ( g `  0
 )  =  P ) )   &    |-  ( ph  ->  ( ( Q `  0
 )  =  O  /\  ( Q `  1 )  =  X  /\  ( R `  1 )  =  ( H `  X ) ) )   &    |-  ( ph  ->  N  e.  ( II  Cn  ( Kt  M ) ) )   &    |-  ( ph  ->  ( ( N `  0
 )  =  X  /\  ( N `  1 )  =  Z ) )   &    |-  I  =  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  N )  /\  ( g `  0 )  =  ( H `  X ) ) )   =>    |-  ( ph  ->  ( H `  Z )  e.  W )
 
Theoremcvmlift3lem7 23262* Lemma for cvmlift3 23265. (Contributed by Mario Carneiro, 9-Jul-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e. SCon )   &    |-  ( ph  ->  K  e. 𝑛Locally PCon )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  G  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  O ) )   &    |-  H  =  ( x  e.  Y  |->  ( iota_ z  e.  B E. f  e.  ( II  Cn  K ) ( ( f `  0
 )  =  O  /\  ( f `  1
 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f
 )  /\  ( g `  0 )  =  P ) ) `  1
 )  =  z ) ) )   &    |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. c  e.  s  ( A. d  e.  (
 s  \  { c } ) ( c  i^i  d )  =  (/)  /\  ( F  |`  c )  e.  ( ( Ct  c )  Homeo  ( Jt  k ) ) ) ) }
 )   &    |-  ( ph  ->  ( G `  X )  e.  A )   &    |-  ( ph  ->  T  e.  ( S `  A ) )   &    |-  ( ph  ->  M  C_  ( `' G " A ) )   &    |-  W  =  (
 iota_ b  e.  T ( H `  X )  e.  b )   &    |-  ( ph  ->  ( Kt  M )  e. PCon )   &    |-  ( ph  ->  V  e.  K )   &    |-  ( ph  ->  V  C_  M )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  H  e.  ( ( K  CnP  C ) `  X ) )
 
Theoremcvmlift3lem8 23263* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 6-Jul-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e. SCon )   &    |-  ( ph  ->  K  e. 𝑛Locally PCon )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  G  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  O ) )   &    |-  H  =  ( x  e.  Y  |->  ( iota_ z  e.  B E. f  e.  ( II  Cn  K ) ( ( f `  0
 )  =  O  /\  ( f `  1
 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f
 )  /\  ( g `  0 )  =  P ) ) `  1
 )  =  z ) ) )   &    |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. c  e.  s  ( A. d  e.  (
 s  \  { c } ) ( c  i^i  d )  =  (/)  /\  ( F  |`  c )  e.  ( ( Ct  c )  Homeo  ( Jt  k ) ) ) ) }
 )   =>    |-  ( ph  ->  H  e.  ( K  Cn  C ) )
 
Theoremcvmlift3lem9 23264* Lemma for cvmlift2 23253. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e. SCon )   &    |-  ( ph  ->  K  e. 𝑛Locally PCon )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  G  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  O ) )   &    |-  H  =  ( x  e.  Y  |->  ( iota_ z  e.  B E. f  e.  ( II  Cn  K ) ( ( f `  0
 )  =  O  /\  ( f `  1
 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f
 )  /\  ( g `  0 )  =  P ) ) `  1
 )  =  z ) ) )   &    |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
 k )  /\  A. c  e.  s  ( A. d  e.  (
 s  \  { c } ) ( c  i^i  d )  =  (/)  /\  ( F  |`  c )  e.  ( ( Ct  c )  Homeo  ( Jt  k ) ) ) ) }
 )   =>    |-  ( ph  ->  E. f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `
  O )  =  P ) )
 
Theoremcvmlift3 23265* A general version of cvmlift 23236. If  K is simply connected and weakly locally path-connected, then there is a unique lift of functions on  K which commutes with the covering map. (Contributed by Mario Carneiro, 9-Jul-2015.)
 |-  B  =  U. C   &    |-  Y  =  U. K   &    |-  ( ph  ->  F  e.  ( C CovMap  J )
 )   &    |-  ( ph  ->  K  e. SCon )   &    |-  ( ph  ->  K  e. 𝑛Locally PCon )   &    |-  ( ph  ->  O  e.  Y )   &    |-  ( ph  ->  G  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  ( F `  P )  =  ( G `  O ) )   =>    |-  ( ph  ->  E! f  e.  ( K  Cn  C ) ( ( F  o.  f
 )  =  G  /\  ( f `  O )  =  P )
 )
 
18.4.10  Undirected multigraphs
 
Syntaxcumg 23266 Extend class notation with undirected multigraphs.
 class UMGrph
 
Syntaxceup 23267 Extend class notation with Eulerian paths.
 class EulPaths
 
Syntaxcvdg 23268 Extend class notation with the vertex degree function.
 class VDeg
 
Definitiondf-umgra 23269* Define the class of all undirected multigraphs. A multigraph is a pair  <. V ,  E >. where  E is a function into subsets of  V of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |- UMGrph  =  { <. v ,  e >.  |  e : dom  e --> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 } }
 
Definitiondf-eupa 23270* Define the set of all Eulerian paths on an undirected multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |- EulPaths  =  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( v UMGrph  e  /\  E. n  e.  NN0  ( f : ( 1 ... n ) -1-1-onto-> dom  e  /\  p : ( 0 ... n ) --> v  /\  A. k  e.  ( 1
 ... n ) ( e `  ( f `
  k ) )  =  { ( p `
  ( k  -  1 ) ) ,  ( p `  k
 ) } ) ) } )
 
Definitiondf-vdgr 23271* Define the vertex degree function for an undirected multigraph. We have to double-count those edges that contain  u "twice" (i.e. self-loops), this being represented as a singleton as the edge's value. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |- VDeg  =  ( v  e.  _V ,  e  e.  _V  |->  ( u  e.  v  |->  ( ( # `  { x  e. 
 dom  e  |  u  e.  ( e `  x ) } )  +  ( # `
  { x  e. 
 dom  e  |  ( e `  x )  =  { u } } ) ) ) )
 
Theoremrelumgra 23272 The class of all undirected multigraphs is a relation. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  Rel UMGrph
 
Theoremisumgra 23273* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremwrdumgra 23274* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V  e.  W  /\  E  e. Word  X )  ->  ( V UMGrph  E  <->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremumgraf2 23275* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremumgraf 23276* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremumgrass 23277 An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  C_  V )
 
Theoremumgran0 23278 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  =/=  (/) )
 
Theoremumgrale 23279 An edge has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( # `
  ( E `  F ) )  <_ 
 2 )
 
Theoremumgrafi 23280 An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  e. 
 Fin )
 
Theoremumgraex 23281* An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  E. x  e.  V  E. y  e.  V  ( E `  F )  =  { x ,  y }
 )
 
Theoremumgrares 23282 A subgraph of a graph (formed by removing some edges from the original graph) is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V UMGrph  E  ->  V UMGrph  ( E  |`  A ) )
 
Theoremumgra0 23283 The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V  e.  W  ->  V UMGrph  (/) )
 
Theoremumgra1 23284 The graph with one edge. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V ) )  ->  V UMGrph  { <. A ,  { B ,  C } >. } )
 
Theoremumgraun 23285 If  <. V ,  E >. and  <. V ,  F >. are graphs, then  <. V ,  E  u.  F >. is a graph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V UMGrph  E )   &    |-  ( ph  ->  V UMGrph  F )   =>    |-  ( ph  ->  V UMGrph  ( E  u.  F ) )
 
Theoremreleupa 23286 The set  ( V EulPaths  E ) of all Eulerian paths on  <. V ,  E >. is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  Rel  ( V EulPaths  E )
 
Theoremiseupa 23287* The property " <. F ,  P >. is an Eulerian path on the graph  <. V ,  E >.". An Eulerian path is defined as bijection  F from the edges to a set  1 ... N a function  P :
( 0 ... N
) --> V into the vertices such that for each 
1  <_  k  <_  N,  F ( k ) is an edge from  P ( k  -  1 ) to  P
( k ). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  ( dom  E  =  A  ->  ( F ( V EulPaths  E ) P  <->  ( V UMGrph  E  /\  E. n  e.  NN0  ( F : ( 1
 ... n ) -1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1
 ... n ) ( E `  ( F `
  k ) )  =  { ( P `
  ( k  -  1 ) ) ,  ( P `  k
 ) } ) ) ) )
 
Theoremeupagra 23288 If an eulerian path exists, then 
<. V ,  E >. is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F ( V EulPaths  E ) P  ->  V UMGrph  E )
 
Theoremeupai 23289* Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  ( ( ( # `  F )  e.  NN0  /\  F : ( 1
 ... ( # `  F ) ) -1-1-onto-> A  /\  P :
 ( 0 ... ( # `
  F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  ( k  -  1
 ) ) ,  ( P `  k ) }
 ) )
 
Theoremeupacl 23290 An Eulerian path has length 
# ( F ), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F ( V EulPaths  E ) P  ->  ( # `  F )  e.  NN0 )
 
Theoremeupaf1o 23291 The  F function in an Eulerian path is a bijection from a one-based sequence to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  F : ( 1 ... ( # `  F ) ) -1-1-onto-> A )
 
Theoremeupafi 23292 Any graph with an Eulerian path is finite. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  (
 ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  A  e.  Fin )
 
Theoremeupapf 23293 The  P function in an Eulerian path is a function from a zero-based finite sequence to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F ( V EulPaths  E ) P  ->  P :
 ( 0 ... ( # `
  F ) ) --> V )
 
Theoremeupaseg 23294 The  N-th edge in an eulerian path is the edge from  P ( N  - 
1 ) to  P ( N ). (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( F ( V EulPaths  E ) P  /\  N  e.  ( 1 ... ( # `  F ) ) )  ->  ( E `  ( F `
  N ) )  =  { ( P `
  ( N  -  1 ) ) ,  ( P `  N ) } )
 
Theoremvdgrfval 23295* The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( V  e.  W  /\  E  Fn  A  /\  A  e.  X )  ->  ( V VDeg  E )  =  ( u  e.  V  |->  ( ( # ` 
 { x  e.  A  |  u  e.  ( E `  x ) }
 )  +  ( # ` 
 { x  e.  A  |  ( E `  x )  =  { u } } ) ) ) )
 
Theoremvdgrval 23296* The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X ) 
 /\  U  e.  V )  ->  ( ( V VDeg 
 E ) `  U )  =  ( ( # `
  { x  e.  A  |  U  e.  ( E `  x ) } )  +  ( # `
  { x  e.  A  |  ( E `
  x )  =  { U } }
 ) ) )
 
Theoremvdgrf 23297 The vertex degree function on finite graphs is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  ->  ( V VDeg  E ) : V --> NN0 )
 
Theoremvdgr0 23298 The degree of a vertex in an empty graph is zero, because there are no edges. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( V  e.  W  /\  U  e.  V ) 
 ->  ( ( V VDeg  (/) ) `  U )  =  0
 )
 
Theoremvdgrun 23299 The degree of a vertex in the union of two graphs on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V UMGrph  E )   &    |-  ( ph  ->  V UMGrph  F )   &    |-  ( ph  ->  U  e.  V )   =>    |-  ( ph  ->  ( ( V VDeg  ( E  u.  F ) ) `
  U )  =  ( ( ( V VDeg 
 E ) `  U )  +  ( ( V VDeg  F ) `  U ) ) )
 
Theoremvdgr1d 23300 The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ph  ->  V  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  U  e.  V )   =>    |-  ( ph  ->  ( ( V VDeg  { <. A ,  { U } >. } ) `  U )  =  2 )
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