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Theorem List for Metamath Proof Explorer - 23001-23100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcvmlift 23001* 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 22987 thru cvmliftlem15 23000. (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 23002 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 23003* 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 23004* Lemma for cvmlift2 23018. (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 23005* Lemma for cvmlift2 23018 and cvmlift3 23030. (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 23006* Lemma for cvmlift2 23018. (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 23007* Lemma for cvmlift2 23018. (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 23008* Lemma for cvmlift2 23018. (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 23009* Lemma for cvmlift2 23018. (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 23010* Lemma for cvmlift2 23018. (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 23011* Lemma for cvmlift2 23018. (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 23012* Lemma for cvmlift2 23018. (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 23013* Lemma for cvmlift2 23018. (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 23014* Lemma for cvmlift2 23018. (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 23015* Lemma for cvmlift2 23018. (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 23016* Lemma for cvmlift2 23018. (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 23017* Lemma for cvmlift2 23018. (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 23018* A two-dimensional version of cvmlift 23001. 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 23019* Lemma for cvmliftpht 23020. (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 23020* 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 23021* Lemma for cvmlift3 23030. (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 23022* Lemma for cvmlift2 23018. (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 23023* Lemma for cvmlift2 23018. (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 23024* Lemma for cvmlift2 23018. (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 23025* Lemma for cvmlift2 23018. (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 23026* Lemma for cvmlift3 23030. (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 23027* Lemma for cvmlift3 23030. (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 23028* Lemma for cvmlift2 23018. (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 23029* Lemma for cvmlift2 23018. (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 23030* A general version of cvmlift 23001. 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 )
 )
 
16.3.10  Undirected multigraphs
 
Syntaxcumg 23031 Extend class notation with undirected multigraphs.
 class UMGrph
 
Syntaxceup 23032 Extend class notation with Eulerian paths.
 class EulPaths
 
Syntaxcvdg 23033 Extend class notation with the vertex degree function.
 class VDeg
 
Definitiondf-umgra 23034* 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 23035* 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 23036* 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 23037 The class of all undirected multigraphs is a relation. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  Rel UMGrph
 
Theoremisumgra 23038* 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 23039* 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 23040* 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 23041* 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 23042 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 23043 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 23044 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 23045 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 23046* 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 23047 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 23048 The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V  e.  W  ->  V UMGrph  (/) )
 
Theoremumgra1 23049 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 23050 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 23051 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 23052* 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 23053 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 23054* 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 23055 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 23056 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 23057 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 23058 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 23059 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 23060* 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 23061* 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 23062 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 23063 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 23064 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 23065 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 )
 
Theoremvdgr1b 23066 The vertex degree of a one-edge graph, case 2: an edge from the given vertex to some other vertex contributes one 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  ->  B  e.  V )   &    |-  ( ph  ->  B  =/=  U )   =>    |-  ( ph  ->  (
 ( V VDeg  { <. A ,  { U ,  B } >. } ) `  U )  =  1 )
 
Theoremvdgr1c 23067 The vertex degree of a one-edge graph, case 3: an edge from some other vertex to the given vertex contributes one 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  ->  B  e.  V )   &    |-  ( ph  ->  B  =/=  U )   =>    |-  ( ph  ->  (
 ( V VDeg  { <. A ,  { B ,  U } >. } ) `  U )  =  1 )
 
Theoremvdgr1a 23068 The vertex degree of a one-edge graph, case 1: an edge between two vertices other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ph  ->  V  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  B  =/=  U )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  C  =/=  U )   =>    |-  ( ph  ->  (
 ( V VDeg  { <. A ,  { B ,  C } >. } ) `  U )  =  0 )
 
Theoremeupa0 23069 There is an Eulerian path on the empty graph. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  (
 ( V  e.  W  /\  A  e.  V ) 
 ->  (/) ( V EulPaths  (/) ) { <. 0 ,  A >. } )
 
Theoremeupares 23070 The restriction of an Eulerian path to an initial segment of the path forms an Eulerian path on the subgraph consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  ( ph  ->  G ( V EulPaths  E ) P )   &    |-  ( ph  ->  N  e.  ( 0 ... ( # `
  G ) ) )   &    |-  F  =  ( E  |`  ( G " ( 1 ... N ) ) )   &    |-  H  =  ( G  |`  ( 1
 ... N ) )   &    |-  Q  =  ( P  |`  ( 0 ... N ) )   =>    |-  ( ph  ->  H ( V EulPaths  F ) Q )
 
Theoremeupap1 23071 Append one path segment to an Eulerian path (enlarging the graph to add the new edge). (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  -.  B  e.  A )   &    |-  ( ph  ->  G ( V EulPaths  E ) P )   &    |-  ( ph  ->  N  =  ( # `  G ) )   &    |-  F  =  ( E  u.  { <. B ,  { ( P `
  N ) ,  C } >. } )   &    |-  H  =  ( G  u.  { <. ( N  +  1 ) ,  B >. } )   &    |-  Q  =  ( P  u.  { <. ( N  +  1 ) ,  C >. } )   =>    |-  ( ph  ->  H ( V EulPaths  F ) Q )
 
Theoremeupath2lem1 23072 Lemma for eupath2 23075. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( U  e.  V  ->  ( U  e.  if ( A  =  B ,  (/)
 ,  { A ,  B } )  <->  ( A  =/=  B 
 /\  ( U  =  A  \/  U  =  B ) ) ) )
 
Theoremeupath2lem2 23073 Lemma for eupath2 23075. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  B  e.  _V   =>    |-  ( ( B  =/=  C 
 /\  B  =  U )  ->  ( -.  U  e.  if ( A  =  B ,  (/) ,  { A ,  B }
 ) 
 <->  U  e.  if ( A  =  C ,  (/)
 ,  { A ,  C } ) ) )
 
Theoremeupath2lem3 23074* Lemma for eupath2 23075. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F ( V EulPaths  E ) P )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( N  +  1 ) 
 <_  ( # `  F ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( ( V VDeg  ( E  |`  ( F "
 ( 1 ... N ) ) ) ) `
  x ) }  =  if ( ( P `
  0 )  =  ( P `  N ) ,  (/) ,  {
 ( P `  0
 ) ,  ( P `
  N ) }
 ) )   =>    |-  ( ph  ->  ( -.  2  ||  ( ( V VDeg  ( E  |`  ( F
 " ( 1 ... ( N  +  1 ) ) ) ) ) `  U )  <->  U  e.  if (
 ( P `  0
 )  =  ( P `
  ( N  +  1 ) ) ,  (/) ,  { ( P `
  0 ) ,  ( P `  ( N  +  1 )
 ) } ) ) )
 
Theoremeupath2 23075* The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F ( V EulPaths  E ) P )   =>    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( ( V VDeg  E ) `  x ) }  =  if ( ( P `
  0 )  =  ( P `  ( # `
  F ) ) ,  (/) ,  { ( P `  0 ) ,  ( P `  ( # `
  F ) ) } ) )
 
Theoremeupath 23076* A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  (
 ( V EulPaths  E )  =/=  (/)  ->  ( # `  { x  e.  V  |  -.  2  ||  ( ( V VDeg  E ) `  x ) }
 )  e.  { 0 ,  2 } )
 
Theoremvdeg0i 23077 The base case for the induction for calculating the degree of a vertex. The degree of  U in the empty graph is  0. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  V  e.  _V   &    |-  U  e.  V   =>    |-  (
 ( V VDeg  (/) ) `  U )  =  0
 
Theoremumgrabi 23078* Show that an unordered pair is a valid edge in a graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  X  e.  V   &    |-  Y  e.  V   =>    |-  ( ph  ->  { X ,  Y }  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremvdegp1ai 23079* The induction step for a vertex degree calculation. If the degree of  U in the edge set  E is  P, then adding  { X ,  Y } to the edge set, where  X  =/=  U  =/= 
Y, yields degree  P as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  (  T.  ->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )   &    |-  U  e.  V   &    |-  ( ( V VDeg 
 E ) `  U )  =  P   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  Y  e.  V   &    |-  Y  =/=  U   &    |-  F  =  ( E concat  <" { X ,  Y } "> )   =>    |-  ( ( V VDeg  F ) `  U )  =  P
 
Theoremvdegp1bi 23080* The induction step for a vertex degree calculation. If the degree of  U in the edge set  E is  P, then adding  { U ,  X } to the edge set, where 
X  =/=  U, yields degree  P  + 
1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  (  T.  ->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )   &    |-  U  e.  V   &    |-  ( ( V VDeg 
 E ) `  U )  =  P   &    |-  Q  =  ( P  +  1 )   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  F  =  ( E concat  <" { U ,  X } "> )   =>    |-  ( ( V VDeg  F ) `  U )  =  Q
 
Theoremvdegp1ci 23081* The induction step for a vertex degree calculation. If the degree of  U in the edge set  E is  P, then adding  { X ,  U } to the edge set, where  X  =/=  U, yields degree  P  + 
1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  (  T.  ->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )   &    |-  U  e.  V   &    |-  ( ( V VDeg 
 E ) `  U )  =  P   &    |-  Q  =  ( P  +  1 )   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  F  =  ( E concat  <" { X ,  U } "> )   =>    |-  ( ( V VDeg  F ) `  U )  =  Q
 
Theoremkonigsberg 23082 The Konigsberg Bridge problem. If  <. V ,  E >. is the graph on four vertices  0 ,  1 ,  2 ,  3, with edges  { 0 ,  1 } ,  { 0 ,  2 } ,  { 0 ,  3 } ,  {
1 ,  2 } ,  { 1 ,  2 } ,  {
2 ,  3 } ,  { 2 ,  3 }, then vertices  0 ,  1 ,  3 each have degree three, and  2 has degree five, so there are four vertices of odd degree and thus by eupath 23076 the graph cannot have an Eulerian path. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   =>    |-  ( V EulPaths  E )  =  (/)
 
16.3.11  Normal numbers
 
Theoremsnmlff 23083* The function  F from snmlval 23085 is a mapping from positive integers to real numbers in the range 
[ 0 ,  1 ]. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  F  =  ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
 )  /  n )
 )   =>    |-  F : NN --> ( 0 [,] 1 )
 
Theoremsnmlfval 23084* The function  F from snmlval 23085 maps  N to the relative density of  B in the first  N digits of the digit string of  A in base  R. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  F  =  ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
 )  /  n )
 )   =>    |-  ( N  e.  NN  ->  ( F `  N )  =  ( ( # `
  { k  e.  ( 1 ... N )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
 )  /  N )
 )
 
Theoremsnmlval 23085* The property " A is simply normal in base  R". A number is simply normal if each digit  0  <_  b  <  R occurs in the base-  R digit string of  A with frequency  1  /  R (which is consistent with the expectation in an infinite random string of numbers selected from  0 ... R  -  1). (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  S  =  ( r  e.  ( ZZ>=
 `  2 )  |->  { x  e.  RR  |  A. b  e.  (
 0 ... ( r  -  1 ) ) ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( x  x.  ( r ^ k
 ) )  mod  r
 ) )  =  b } )  /  n ) )  ~~>  ( 1  /  r ) } )   =>    |-  ( A  e.  ( S `  R )  <->  ( R  e.  ( ZZ>= `  2 )  /\  A  e.  RR  /\  A. b  e.  ( 0
 ... ( R  -  1 ) ) ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  b }
 )  /  n )
 )  ~~>  ( 1  /  R ) ) )
 
Theoremsnmlflim 23086* If  A is simply normal, then the function  F of relative density of  B in the digit string converges to  1  /  R, i.e. the set of occurences of  B in the digit string has natural density  1  /  R. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  S  =  ( r  e.  ( ZZ>=
 `  2 )  |->  { x  e.  RR  |  A. b  e.  (
 0 ... ( r  -  1 ) ) ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( x  x.  ( r ^ k
 ) )  mod  r
 ) )  =  b } )  /  n ) )  ~~>  ( 1  /  r ) } )   &    |-  F  =  ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
 )  /  n )
 )   =>    |-  ( ( A  e.  ( S `  R ) 
 /\  B  e.  (
 0 ... ( R  -  1 ) ) ) 
 ->  F  ~~>  ( 1  /  R ) )
 
16.3.12  Godel-sets of formulas
 
Syntaxcgoe 23087 The Godel-set of membership.
 class  e.g
 
Syntaxcgna 23088 The Godel-set for the Sheffer stroke.
 class  | g
 
Syntaxcgol 23089 The Godel-set of universal quantification. (Note that this is not a wff.)
 class  A.g N U
 
Syntaxcsat 23090 The satisfaction function.
 class  Sat
 
Syntaxcfmla 23091 The formula set predicate.
 class  Fmla
 
Syntaxcsate 23092 The  e.-satisfaction function.
 class  Sat E
 
Syntaxcprv 23093 The "proves" relation.
 class  |=
 
Definitiondf-goel 23094 Define the Godel-set of membership. Here the arguments  x  =  <. N ,  P >. correspond to vN and vP , so  ( (/)  e.g 
1o ) actually means v0  e. v1 , not  0  e.  1. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  e.g  =  ( x  e.  ( om  X.  om )  |->  <. (/)
 ,  x >. )
 
Definitiondf-gona 23095 Define the Godel-set for the Sheffer stroke NAND. Here the arguments  x  =  <. U ,  V >. are also Godel-sets corresponding to smaller formulae. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  | g  =  ( x  e.  ( _V  X.  _V )  |->  <. 1o ,  x >. )
 
Definitiondf-goal 23096 Define the Godel-set of universal quantification. Here  N  e.  om corresponds to vN , and  U represents another formula, and this expression is  [ A. x ph ]  =  A.g N U where 
x is the  N-th variable,  U  =  [ ph ] is the code for  ph. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  A.g N U  =  <. 2o ,  <. N ,  U >. >.
 
Definitiondf-sat 23097* Define the satisfaction predicate. This recursive construction builds up a function over wff codes and simultaneously defines the set of assignments to all variables from  M that makes the coded wff true in the model  M, where  e. is interpreted as the binary relation  E on  M. The interpretation of the statement  S  e.  ( ( ( M  Sat  E ) `  n ) `  U ) is that for the model  <. M ,  E >.,  S : om --> M is an valuation of the variables (v0  =  ( S `  (/) ), v1  =  ( S `  1o ), etc.) and  U is a code for a wff using  =  ,  e.  ,  \/  ,  -.  ,  A. that is true under the assignment  S. The function is defined by finite recursion;  ( ( M  Sat  E ) `  n ) only operates on wffs of depth at most  n  e.  om, and  ( ( M  Sat  E ) `  om )  =  U_ n  e.  om ( ( M  Sat  E ) `  n ) operates on all wffs. The coding scheme for the wffs is defined so that
  • vi  e. vj is coded as  <. (/) ,  <. i ,  j >. >.,
  •  ( ph  -/\  ps ) is coded as  <. 1o ,  <. ph ,  ps >. >., and
  •  A. vi  ph is coded as  <. 2o ,  <. i ,  ph >. >..

(Contributed by Mario Carneiro, 14-Jul-2013.)

 |-  Sat  =  ( m  e.  _V ,  e  e.  _V  |->  ( rec ( ( f  e.  _V  |->  ( f  u.  { <. x ,  y >.  |  E. u  e.  f  ( E. v  e.  f  ( x  =  ( ( 1st `  u )  | g  ( 1st `  v
 ) )  /\  y  =  ( ( m  ^m  om )  \  ( ( 2nd `  u )  i^i  ( 2nd `  v
 ) ) ) )  \/  E. i  e. 
 om  ( x  = 
 A.g i ( 1st `  u )  /\  y  =  { a  e.  ( m  ^m  om )  | 
 A. z  e.  m  ( { <. i ,  z >. }  u.  ( a  |`  ( om  \  {
 i } ) ) )  e.  ( 2nd `  u ) } )
 ) } ) ) ,  { <. x ,  y >.  |  E. i  e.  om  E. j  e. 
 om  ( x  =  ( i  e.g  j
 )  /\  y  =  { a  e.  ( m  ^m  om )  |  ( a `  i
 ) e ( a `
  j ) }
 ) } )  |`  suc  om ) )
 
Definitiondf-sate 23098* A simplified version of the satisfaction predicate, using the standard membership relation and eliminating the extra variable  n. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  Sat E  =  ( m  e. 
 _V ,  u  e. 
 _V  |->  ( ( ( m  Sat  (  _E 
 i^i  ( m  X.  m ) ) ) `
  om ) `  u ) )
 
Definitiondf-fmla 23099 Define the predicate which defines the set of valid Godel formulas. The parameter  n defines the maximum height of the formulas: the set  ( Fmla `  (/) ) is all formulas of the form  x  =  y or  x  e.  y (which in our coding scheme is the set  ( { (/) ,  1o }  X.  ( om  X.  om ) ); see df-sat 23097 for the full coding scheme), and each extra level adds to the complexity of the formulas in  ( Fmla `  n
).  ( Fmla `  om )  =  U_ n  e. 
om ( Fmla `  n
) is the set of all valid formulas. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  Fmla  =  ( n  e.  suc  om 
 |->  dom  ( ( (/)  Sat  (/) ) `  n ) )
 
Syntaxcgon 23100 The Godel-set of negation. (Note that this is not a wff.)
 class  -.g U
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