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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremclwwlknfi 24601 If there is only a finite number of vertices, the number of closed walk of fixed length (as words) is also finite. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
 |-  ( ( V  e.  Fin  /\  E  e.  X  /\  N  e.  NN0 )  ->  ( ( V ClWWalksN  E ) `
  N )  e. 
 Fin )
 
Theoremclwlkisclwwlklem2a1 24602* Lemma 1 for clwlkisclwwlklem2a 24608. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
 |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `  P ) )  ->  ( ( ( lastS  `  P )  =  ( P `  0
 )  /\  ( A. i  e.  ( 0..^ ( ( ( ( # `  P )  -  1 )  -  0
 )  -  1 ) ) { ( P `
  i ) ,  ( P `  (
 i  +  1 ) ) }  e.  ran  E 
 /\  { ( P `  ( ( # `  P )  -  2 ) ) ,  ( P `  0 ) }  e.  ran 
 E ) )  ->  A. i  e.  (
 0..^ ( ( # `  P )  -  1
 ) ) { ( P `  i ) ,  ( P `  (
 i  +  1 ) ) }  e.  ran  E ) )
 
Theoremclwlkisclwwlklem2a2 24603* Lemma 3 for clwlkisclwwlklem2a 24608. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
 |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) ) 
 |->  if ( x  < 
 ( ( # `  P )  -  2 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) }
 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  0
 ) } ) ) )   =>    |-  ( ( P  e. Word  V 
 /\  2  <_  ( # `
  P ) ) 
 ->  ( # `  F )  =  ( ( # `
  P )  -  1 ) )
 
Theoremclwlkisclwwlklem2a3 24604* Lemma 3 for clwlkisclwwlklem2a 24608. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
 |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) ) 
 |->  if ( x  < 
 ( ( # `  P )  -  2 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) }
 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  0
 ) } ) ) )   =>    |-  ( ( P  e. Word  V 
 /\  2  <_  ( # `
  P ) ) 
 ->  ( P `  ( # `
  F ) )  =  ( lastS  `  P ) )
 
Theoremclwlkisclwwlklem2fv1 24605* Lemma 4a for clwlkisclwwlklem2a 24608. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
 |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) ) 
 |->  if ( x  < 
 ( ( # `  P )  -  2 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) }
 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  0
 ) } ) ) )   =>    |-  ( ( ( # `  P )  e.  NN0  /\  I  e.  ( 0..^ ( ( # `  P )  -  2 ) ) )  ->  ( F `  I )  =  ( `' E `  { ( P `  I ) ,  ( P `  ( I  +  1 )
 ) } ) )
 
Theoremclwlkisclwwlklem2fv2 24606* Lemma 4b for clwlkisclwwlklem2a 24608. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
 |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) ) 
 |->  if ( x  < 
 ( ( # `  P )  -  2 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) }
 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  0
 ) } ) ) )   =>    |-  ( ( ( # `  P )  e.  NN0  /\  2  <_  ( # `  P ) )  ->  ( F `
  ( ( # `  P )  -  2
 ) )  =  ( `' E `  { ( P `  ( ( # `  P )  -  2
 ) ) ,  ( P `  0 ) }
 ) )
 
Theoremclwlkisclwwlklem2a4 24607* Lemma 4 for clwlkisclwwlklem2a 24608. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
 |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) ) 
 |->  if ( x  < 
 ( ( # `  P )  -  2 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) }
 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  0
 ) } ) ) )   =>    |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `  P ) )  ->  ( ( ( lastS  `  P )  =  ( P `  0
 )  /\  I  e.  ( 0..^ ( ( # `  P )  -  1
 ) ) )  ->  ( { ( P `  I ) ,  ( P `  ( I  +  1 ) ) }  e.  ran  E  ->  ( E `  ( F `  I ) )  =  { ( P `  I ) ,  ( P `  ( I  +  1 ) ) }
 ) ) )
 
Theoremclwlkisclwwlklem2a 24608* Lemma 2 for clwlkisclwwlklem2 24609. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
 |-  F  =  ( x  e.  ( 0..^ ( ( # `  P )  -  1 ) ) 
 |->  if ( x  < 
 ( ( # `  P )  -  2 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  ( x  +  1 ) ) }
 ) ,  ( `' E `  { ( P `  x ) ,  ( P `  0
 ) } ) ) )   =>    |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `  P ) )  ->  ( ( ( lastS  `  P )  =  ( P `  0
 )  /\  ( A. i  e.  ( 0..^ ( ( ( ( # `  P )  -  1 )  -  0
 )  -  1 ) ) { ( P `
  i ) ,  ( P `  (
 i  +  1 ) ) }  e.  ran  E 
 /\  { ( P `  ( ( # `  P )  -  2 ) ) ,  ( P `  0 ) }  e.  ran 
 E ) )  ->  ( ( F  e. Word  dom 
 E  /\  P :
 ( 0 ... ( # `
  F ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
  i ) )  =  { ( P `
  i ) ,  ( P `  (
 i  +  1 ) ) } )  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
 
Theoremclwlkisclwwlklem2 24609* Lemma for clwlkisclwwlk 24612. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
 |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `  P ) )  ->  ( ( ( lastS  `  P )  =  ( P `  0
 )  /\  ( A. i  e.  ( 0..^ ( ( ( ( # `  P )  -  1 )  -  0
 )  -  1 ) ) { ( P `
  i ) ,  ( P `  (
 i  +  1 ) ) }  e.  ran  E 
 /\  { ( P `  ( ( # `  P )  -  2 ) ) ,  ( P `  0 ) }  e.  ran 
 E ) )  ->  E. f ( ( f  e. Word  dom  E  /\  P : ( 0 ... ( # `  f
 ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f
 ) ) ( E `
  ( f `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }
 )  /\  ( P `  0 )  =  ( P `  ( # `  f ) ) ) ) )
 
Theoremclwlkisclwwlklem1 24610* Lemma for clwlkisclwwlk 24612. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
 |-  ( ( ( V USGrph  E  /\  F  e. Word  dom  E )  /\  ( P : ( 0 ... ( # `  F ) ) --> V  /\  2  <_  ( # `  P ) )  /\  ( A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `  i ) )  =  { ( P `  i ) ,  ( P `  (
 i  +  1 ) ) }  /\  ( P `  0 )  =  ( P `  ( # `
  F ) ) ) )  ->  (
 ( lastS  `  P )  =  ( P `  0
 )  /\  A. i  e.  ( 0..^ ( ( # `  F )  -  1 ) ) {
 ( P `  i
 ) ,  ( P `
  ( i  +  1 ) ) }  e.  ran  E  /\  {
 ( P `  (
 ( # `  F )  -  1 ) ) ,  ( P `  0 ) }  e.  ran 
 E ) )
 
Theoremclwlkisclwwlklem0 24611* Lemma for clwlkisclwwlk 24612. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
 |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `  P ) )  ->  ( E. f ( ( f  e. Word  dom  E  /\  P : ( 0 ... ( # `  f
 ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  f
 ) ) ( E `
  ( f `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }
 )  /\  ( P `  0 )  =  ( P `  ( # `  f ) ) )  <-> 
 ( ( lastS  `  P )  =  ( P `  0 )  /\  ( A. i  e.  ( 0..^ ( ( ( ( # `  P )  -  1 )  -  0
 )  -  1 ) ) { ( P `
  i ) ,  ( P `  (
 i  +  1 ) ) }  e.  ran  E 
 /\  { ( P `  ( ( # `  P )  -  2 ) ) ,  ( P `  0 ) }  e.  ran 
 E ) ) ) )
 
Theoremclwlkisclwwlk 24612* A closed walk as word corresponds to a closed walk in an undirected graph. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
 |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  2  <_  ( # `  P ) )  ->  ( E. f  f ( V ClWalks  E ) P  <->  ( ( lastS  `  P )  =  ( P `  0 )  /\  ( P substr 
 <. 0 ,  ( ( # `  P )  -  1 ) >. )  e.  ( V ClWWalks  E )
 ) ) )
 
Theoremclwlkisclwwlk2 24613* A closed walk corresponds to a closed walk as word in an undirected graph. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
 |-  ( ( V USGrph  E  /\  P  e. Word  V  /\  1  <_  ( # `  P ) )  ->  ( E. f  f ( V ClWalks  E ) ( P concat  <" ( P `  0 ) "> )  <->  P  e.  ( V ClWWalks  E ) ) )
 
Theoremclwwlkisclwwlkn 24614 A closed walk of a fixed length as word is a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.)
 |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  ->  ( P  e.  (
 ( V ClWWalksN  E ) `  N )  ->  P  e.  ( V ClWWalks  E ) ) )
 
Theoremclwwlkssclwwlkn 24615 The closed walks of a fixed length as words are closed walks (in an undirected graph) as words. (Contributed by Alexander van der Vekens, 15-Mar-2018.)
 |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN0 )  ->  ( ( V ClWWalksN  E ) `
  N )  C_  ( V ClWWalks  E ) )
 
Theoremclwwlkel 24616* Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by AV, 28-Sep-2018.) (Proof shortened by AV, 20-Oct-2018.)
 |-  D  =  { w  e.  ( ( V WWalksN  E ) `
  N )  |  ( lastS  `  w )  =  ( w `  0
 ) }   =>    |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  /\  ( P  e. Word  V  /\  ( # `  P )  =  N )  /\  ( A. i  e.  ( 0..^ ( N  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  {
 ( lastS  `  P ) ,  ( P `  0
 ) }  e.  ran  E ) )  ->  ( P concat  <" ( P `
  0 ) "> )  e.  D )
 
Theoremclwwlkf 24617* Lemma 1 for clwwlkbij 24622: F is a function. (Contributed by Alexander van der Vekens, 27-Sep-2018.)
 |-  D  =  { w  e.  ( ( V WWalksN  E ) `
  N )  |  ( lastS  `  w )  =  ( w `  0
 ) }   &    |-  F  =  ( t  e.  D  |->  ( t substr  <. 0 ,  N >. ) )   =>    |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  F : D --> ( ( V ClWWalksN  E ) `  N ) )
 
Theoremclwwlkfv 24618* Lemma 2 for clwwlkbij 24622: the value of function F. (Contributed by Alexander van der Vekens, 28-Sep-2018.)
 |-  D  =  { w  e.  ( ( V WWalksN  E ) `
  N )  |  ( lastS  `  w )  =  ( w `  0
 ) }   &    |-  F  =  ( t  e.  D  |->  ( t substr  <. 0 ,  N >. ) )   =>    |-  ( W  e.  D  ->  ( F `  W )  =  ( W substr  <.
 0 ,  N >. ) )
 
Theoremclwwlkf1 24619* Lemma 3 for clwwlkbij 24622: F is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Proof shortened by AV, 23-Oct-2018.)
 |-  D  =  { w  e.  ( ( V WWalksN  E ) `
  N )  |  ( lastS  `  w )  =  ( w `  0
 ) }   &    |-  F  =  ( t  e.  D  |->  ( t substr  <. 0 ,  N >. ) )   =>    |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  F : D -1-1-> (
 ( V ClWWalksN  E ) `  N ) )
 
Theoremclwwlkfo 24620* Lemma 4 for clwwlkbij 24622: F is an onto function. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
 |-  D  =  { w  e.  ( ( V WWalksN  E ) `
  N )  |  ( lastS  `  w )  =  ( w `  0
 ) }   &    |-  F  =  ( t  e.  D  |->  ( t substr  <. 0 ,  N >. ) )   =>    |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  F : D -onto-> (
 ( V ClWWalksN  E ) `  N ) )
 
Theoremclwwlkf1o 24621* Lemma 5 for clwwlkbij 24622: F is a 1-1 onto function. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
 |-  D  =  { w  e.  ( ( V WWalksN  E ) `
  N )  |  ( lastS  `  w )  =  ( w `  0
 ) }   &    |-  F  =  ( t  e.  D  |->  ( t substr  <. 0 ,  N >. ) )   =>    |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  F : D -1-1-onto-> ( ( V ClWWalksN  E ) `
  N ) )
 
Theoremclwwlkbij 24622* There is a bijection between the set of closed walks of a fixed length represented by walks (as word) and the set of closed walks (as words) of a fixed length. The difference between these two representations is that in the first case the starting vertex is repeated at the end of the word, and in the second case it is not. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
 |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  E. f  f : { w  e.  (
 ( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) } -1-1-onto-> ( ( V ClWWalksN  E ) `
  N ) )
 
Theoremclwwlknwwlkncl 24623* Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
 |-  ( ( N  e.  NN  /\  P  e.  (
 ( V ClWWalksN  E ) `  N ) )  ->  ( P concat  <" ( P `  0 ) "> )  e.  { w  e.  ( ( V WWalksN  E ) `
  N )  |  ( lastS  `  w )  =  ( w `  0
 ) } )
 
Theoremclwwlkvbij 24624* There is a bijection between the set of closed walks of a fixed length starting at a fixed vertex represented by walks (as word) and the set of closed walks (as words) of a fixed length starting at a fixed vertex. The difference between these two representations is that in the first case the starting vertex is repeated at the end of the word, and in the second case it is not. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
 |-  ( ( V  e.  X  /\  E  e.  Y  /\  N  e.  NN )  ->  E. f  f : { w  e.  (
 ( V WWalksN  E ) `  N )  |  ( ( lastS  `  w )  =  ( w `  0
 )  /\  ( w `  0 )  =  S ) } -1-1-onto-> { w  e.  (
 ( V ClWWalksN  E ) `  N )  |  ( w `  0 )  =  S } )
 
Theoremclwwlkext2edg 24625 If a word concatenated with a vertex represents a closed walk in (in a graph), there is an edge between this vertex and the last vertex of the word, and between this vertex and the first vertex of the word. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
 |-  ( ( ( W  e. Word  V  /\  Z  e.  V  /\  N  e.  ( ZZ>=
 `  2 ) ) 
 /\  ( W concat  <" Z "> )  e.  (
 ( V ClWWalksN  E ) `  N ) )  ->  ( { ( lastS  `  W ) ,  Z }  e.  ran  E  /\  { Z ,  ( W `  0 ) }  e.  ran 
 E ) )
 
Theoremwwlkext2clwwlk 24626 If a word represents a walk in (in a graph) and there are edges between the last vertex of the word and another vertex and between this other vertex and the first vertex of the word, then the concatenation of the word representing the walk with this other vertex represents a closed walk. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
 |-  ( ( W  e.  ( ( V WWalksN  E ) `
  N )  /\  Z  e.  V  /\  N  e.  NN0 )  ->  ( ( { ( lastS  `  W ) ,  Z }  e.  ran  E  /\  { Z ,  ( W `
  0 ) }  e.  ran  E )  ->  ( W concat  <" Z "> )  e.  (
 ( V ClWWalksN  E ) `  ( N  +  2
 ) ) ) )
 
Theoremwwlksubclwwlk 24627 Any prefix of a word representing a closed walk represents a word. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
 |-  ( ( M  e.  NN  /\  N  e.  ( ZZ>=
 `  ( M  +  1 ) ) ) 
 ->  ( X  e.  (
 ( V ClWWalksN  E ) `  N )  ->  ( X substr  <. 0 ,  M >. )  e.  ( ( V WWalksN  E ) `  ( M  -  1 ) ) ) )
 
Theoremclwwisshclwwlem1 24628* Lemma 1 for clwwisshclwwlem 24629. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
 |-  ( ( ( L  e.  ( ZZ>= `  2
 )  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  A. i  e.  ( 0..^ ( L  -  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  R  /\  { ( W `  ( L  -  1 ) ) ,  ( W `  0
 ) }  e.  R )  ->  { ( W `
  ( ( A  +  B )  mod  L ) ) ,  ( W `  ( ( ( A  +  1 )  +  B )  mod  L ) ) }  e.  R )
 
Theoremclwwisshclwwlem 24629* Lemma for clwwisshclww 24630. (Contributed by AV, 24-Mar-2018.) (Revised by AV, 10-Jun-2018.) (Proof shortened by AV, 2-Nov-2018.)
 |-  ( ( W  e. Word  V 
 /\  N  e.  (
 1..^ ( # `  W ) ) )  ->  ( ( A. i  e.  ( 0..^ ( ( # `  W )  -  1 ) ) {
 ( W `  i
 ) ,  ( W `
  ( i  +  1 ) ) }  e.  ran  E  /\  {
 ( lastS  `  W ) ,  ( W `  0
 ) }  e.  ran  E )  ->  A. j  e.  ( 0..^ ( ( # `  ( W cyclShift  N ) )  -  1 ) ) { ( ( W cyclShift  N ) `  j
 ) ,  ( ( W cyclShift  N ) `  (
 j  +  1 ) ) }  e.  ran  E ) )
 
Theoremclwwisshclww 24630 Cyclically shifting a closed walk as word results in a closed walk as word (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Mar-2018.) (Revised by Alexander van der Vekens, 10-Jun-2018.)
 |-  ( ( W  e.  ( V ClWWalks  E )  /\  N  e.  ( 0..^ ( # `  W ) ) )  ->  ( W cyclShift  N )  e.  ( V ClWWalks  E ) )
 
Theoremclwwisshclwwn 24631 Cyclically shifting a closed walk as word results in a closed walk as word (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jun-2018.)
 |-  ( ( W  e.  ( V ClWWalks  E )  /\  N  e.  ( 0 ... ( # `  W ) ) )  ->  ( W cyclShift  N )  e.  ( V ClWWalks  E )
 )
 
Theoremclwwnisshclwwn 24632 Cyclically shifting a closed walk as word of fixed length results in a closed walk as word of the same length (in an undirected graph). (Contributed by Alexander van der Vekens, 10-Jun-2018.)
 |-  ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N ) )  ->  ( M  e.  ( 0 ...
 N )  ->  ( W cyclShift  M )  e.  (
 ( V ClWWalksN  E ) `  N ) ) )
 
Theoremerclwwlkrel 24633  .~ is a relation. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
 |- 
 .~  =  { <. u ,  w >.  |  ( u  e.  ( V ClWWalks  E )  /\  w  e.  ( V ClWWalks  E )  /\  E. n  e.  (
 0 ... ( # `  w ) ) u  =  ( w cyclShift  n )
 ) }   =>    |- 
 Rel  .~
 
Theoremerclwwlkeq 24634* Two classes are equivalent regarding  .~ if both are words and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.)
 |- 
 .~  =  { <. u ,  w >.  |  ( u  e.  ( V ClWWalks  E )  /\  w  e.  ( V ClWWalks  E )  /\  E. n  e.  (
 0 ... ( # `  w ) ) u  =  ( w cyclShift  n )
 ) }   =>    |-  ( ( U  e.  X  /\  W  e.  Y )  ->  ( U  .~  W 
 <->  ( U  e.  ( V ClWWalks  E )  /\  W  e.  ( V ClWWalks  E )  /\  E. n  e.  (
 0 ... ( # `  W ) ) U  =  ( W cyclShift  n ) ) ) )
 
Theoremerclwwlkeqlen 24635* If two classes are equivalent regarding  .~, then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.)
 |- 
 .~  =  { <. u ,  w >.  |  ( u  e.  ( V ClWWalks  E )  /\  w  e.  ( V ClWWalks  E )  /\  E. n  e.  (
 0 ... ( # `  w ) ) u  =  ( w cyclShift  n )
 ) }   =>    |-  ( ( U  e.  X  /\  W  e.  Y )  ->  ( U  .~  W  ->  ( # `  U )  =  ( # `  W ) ) )
 
Theoremerclwwlkref 24636*  .~ is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.)
 |- 
 .~  =  { <. u ,  w >.  |  ( u  e.  ( V ClWWalks  E )  /\  w  e.  ( V ClWWalks  E )  /\  E. n  e.  (
 0 ... ( # `  w ) ) u  =  ( w cyclShift  n )
 ) }   =>    |-  ( x  e.  ( V ClWWalks  E )  <->  x  .~  x )
 
Theoremerclwwlksym 24637*  .~ is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.)
 |- 
 .~  =  { <. u ,  w >.  |  ( u  e.  ( V ClWWalks  E )  /\  w  e.  ( V ClWWalks  E )  /\  E. n  e.  (
 0 ... ( # `  w ) ) u  =  ( w cyclShift  n )
 ) }   =>    |-  ( x  .~  y  ->  y  .~  x )
 
Theoremerclwwlktr 24638*  .~ is a transitive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.)
 |- 
 .~  =  { <. u ,  w >.  |  ( u  e.  ( V ClWWalks  E )  /\  w  e.  ( V ClWWalks  E )  /\  E. n  e.  (
 0 ... ( # `  w ) ) u  =  ( w cyclShift  n )
 ) }   =>    |-  ( ( x  .~  y  /\  y  .~  z
 )  ->  x  .~  z )
 
Theoremerclwwlk 24639*  .~ is an equivalence relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.)
 |- 
 .~  =  { <. u ,  w >.  |  ( u  e.  ( V ClWWalks  E )  /\  w  e.  ( V ClWWalks  E )  /\  E. n  e.  (
 0 ... ( # `  w ) ) u  =  ( w cyclShift  n )
 ) }   =>    |- 
 .~  Er  ( V ClWWalks  E )
 
Theoremeleclclwwlknlem1 24640* Lemma 1 for eleclclwwlkn 24656. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   =>    |-  ( ( K  e.  ( 0 ... N )  /\  ( X  e.  W  /\  Y  e.  W ) )  ->  ( ( X  =  ( Y cyclShift  K )  /\  E. m  e.  ( 0 ... N ) Z  =  ( Y cyclShift  m ) )  ->  E. n  e.  (
 0 ... N ) Z  =  ( X cyclShift  n ) ) )
 
Theoremeleclclwwlknlem2 24641* Lemma 2 for eleclclwwlkn 24656. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   =>    |-  ( ( ( k  e.  ( 0 ...
 N )  /\  X  =  ( x cyclShift  k )
 )  /\  ( X  e.  W  /\  x  e.  W ) )  ->  ( E. m  e.  (
 0 ... N ) Y  =  ( x cyclShift  m )  <->  E. n  e.  (
 0 ... N ) Y  =  ( X cyclShift  n ) ) )
 
Theoremclwwlknscsh 24642* The set of cyclical shifts of a word representing a closed walk is the set of closed walks represented by cyclical shifts of a word. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
 |-  ( ( N  e.  NN0  /\  W  e.  ( ( V ClWWalksN  E ) `  N ) )  ->  { y  e.  ( ( V ClWWalksN  E ) `
  N )  | 
 E. n  e.  (
 0 ... N ) y  =  ( W cyclShift  n ) }  =  { y  e. Word  V  |  E. n  e.  ( 0 ... N ) y  =  ( W cyclShift  n ) } )
 
Theoremusg2cwwk2dif 24643 If a word represents a closed walk of length at least 2 in a undirected simple graph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
 |-  ( ( V USGrph  E  /\  N  e.  ( ZZ>= `  2 )  /\  W  e.  ( ( V ClWWalksN  E ) `
  N ) ) 
 ->  ( W `  1
 )  =/=  ( W `  0 ) )
 
Theoremusg2cwwkdifex 24644* If a word represents a closed walk of length at least 2 in a undirected simple graph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
 |-  ( ( V USGrph  E  /\  N  e.  ( ZZ>= `  2 )  /\  W  e.  ( ( V ClWWalksN  E ) `
  N ) ) 
 ->  E. i  e.  (
 0..^ N ) ( W `  i )  =/=  ( W `  0 ) )
 
Theoremerclwwlknrel 24645  .~ is a relation. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |- 
 Rel  .~
 
Theoremerclwwlkneq 24646* Two classes are equivalent regarding  .~ if both are words of the same fixed length and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |-  ( ( T  e.  X  /\  U  e.  Y )  ->  ( T  .~  U 
 <->  ( T  e.  W  /\  U  e.  W  /\  E. n  e.  ( 0
 ... N ) T  =  ( U cyclShift  n ) ) ) )
 
Theoremerclwwlkneqlen 24647* If two classes are equivalent regarding  .~, then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |-  ( ( T  e.  X  /\  U  e.  Y )  ->  ( T  .~  U  ->  ( # `  T )  =  ( # `  U ) ) )
 
Theoremerclwwlknref 24648*  .~ is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 26-Mar-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |-  ( x  e.  W  <->  x 
 .~  x )
 
Theoremerclwwlknsym 24649*  .~ is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |-  ( x  .~  y  ->  y  .~  x )
 
Theoremerclwwlkntr 24650*  .~ is a transitive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |-  ( ( x  .~  y  /\  y  .~  z
 )  ->  x  .~  z )
 
Theoremerclwwlkn 24651*  .~ is an equivalence relation over the set of closed walks (defined as words) with a fixed length. (Contributed by Alexander van der Vekens, 10-Apr-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |- 
 .~  Er  W
 
Theoremqerclwwlknfi 24652* The quotient set of the set of closed walks (defined as words) with a fixed length according to the equivalence relation  .~ is finite. (Contributed by Alexander van der Vekens, 10-Apr-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |-  ( ( V  e.  Fin  /\  E  e.  X  /\  N  e.  NN0 )  ->  ( W /.  .~  )  e.  Fin )
 
Theoremhashclwwlkn0 24653* The number of closed walks (defined as words) with a fixed length is the sum of the sizes of all equivalence classes according to  .~. (Contributed by Alexander van der Vekens, 10-Apr-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |-  ( ( V  e.  Fin  /\  E  e.  X  /\  N  e.  NN0 )  ->  ( # `  W )  =  sum_ x  e.  ( W /.  .~  ) ( # `  x ) )
 
Theoremeclclwwlkn0 24654* An equivalence class according to 
.~. (Contributed by Alexander van der Vekens, 12-Apr-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |-  ( B  e.  X  ->  ( B  e.  ( W /.  .~  )  <->  E. x  e.  W  B  =  { y  |  x  .~  y }
 ) )
 
Theoremeclclwwlkn1 24655* An equivalence class according to 
.~. (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by Alexander van der Vekens, 15-Jun-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |-  ( B  e.  X  ->  ( B  e.  ( W /.  .~  )  <->  E. x  e.  W  B  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) } )
 )
 
Theoremeleclclwwlkn 24656* A member of an equivalence class according to  .~. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by Alexander van der Vekens, 15-Jun-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |-  ( ( B  e.  ( W /.  .~  )  /\  X  e.  B ) 
 ->  ( Y  e.  B  <->  ( Y  e.  W  /\  E. n  e.  ( 0
 ... N ) Y  =  ( X cyclShift  n ) ) ) )
 
Theoremhashecclwwlkn1 24657* The size of every equivalence class of the equivalence relation over the set of closed walks (defined as words) with a fixed length which is a prime number is 1 or equals this length. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |-  ( ( N  e.  Prime  /\  U  e.  ( W /.  .~  ) ) 
 ->  ( ( # `  U )  =  1  \/  ( # `  U )  =  N ) )
 
Theoremusghashecclwwlk 24658* The size of every equivalence class of the equivalence relation over the set of closed walks (defined as words) with a fixed length which is a prime number equals this length (in an undirected simple graph). (Contributed by Alexander van der Vekens, 17-Jun-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |-  ( ( V USGrph  E  /\  N  e.  Prime )  ->  ( U  e.  ( W /.  .~  )  ->  ( # `  U )  =  N ) )
 
Theoremhashclwwlkn 24659* The size of the set of closed walks (defined as words) with a fixed length which is a prime number is the product of the number of equivalence classes for 
.~ over the set of closed walks and the fixed length. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
 |-  W  =  ( ( V ClWWalksN  E ) `  N )   &    |- 
 .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0
 ... N ) t  =  ( u cyclShift  n ) ) }   =>    |-  ( ( V  e.  Fin  /\  V USGrph  E  /\  N  e.  Prime )  ->  ( # `  W )  =  ( ( # `
  ( W /.  .~  ) )  x.  N ) )
 
Theoremclwwlkndivn 24660 The size of the set of closed walks (defined as words) of length n is divisible by n. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
 |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  ->  N  ||  ( # `  (
 ( V ClWWalksN  E ) `  N ) ) )
 
Theoremwlklenvp1 24661 The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018.)
 |-  ( F ( V Walks  E ) P  ->  ( # `  P )  =  ( ( # `  F )  +  1 )
 )
 
Theoremwlklenvclwlk 24662 The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.)
 |-  ( ( W  e. Word  V 
 /\  1  <_  ( # `
  W ) ) 
 ->  ( <. F ,  ( W concat  <" ( W `
  0 ) "> ) >.  e.  ( V Walks  E )  ->  ( # `
  F )  =  ( # `  W ) ) )
 
Theoremclwlkfclwwlk2wrd 24663 The second component of a closed walk is a word over the "vertices". (Contributed by Alexander van der Vekens, 25-Jun-2018.)
 |-  A  =  ( 1st `  c )   &    |-  B  =  ( 2nd `  c )   &    |-  C  =  { c  e.  ( V ClWalks  E )  |  ( # `  A )  =  N }   &    |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `
  A ) >. ) )   =>    |-  ( c  e.  C  ->  B  e. Word  V )
 
Theoremclwlkfclwwlk1hashn 24664* The size of the first component of a closed walk. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
 |-  A  =  ( 1st `  c )   &    |-  B  =  ( 2nd `  c )   &    |-  C  =  { c  e.  ( V ClWalks  E )  |  ( # `  A )  =  N }   &    |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `
  A ) >. ) )   =>    |-  ( W  e.  C  ->  ( # `  ( 1st `  W ) )  =  N )
 
Theoremclwlkfclwwlk1hash 24665* The size of the first component of a closed walk is an integer in the range between 0 and the size of the second component. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
 |-  A  =  ( 1st `  c )   &    |-  B  =  ( 2nd `  c )   &    |-  C  =  { c  e.  ( V ClWalks  E )  |  ( # `  A )  =  N }   &    |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `
  A ) >. ) )   =>    |-  ( c  e.  C  ->  ( # `  A )  e.  ( 0 ... ( # `  B ) ) )
 
Theoremclwlkfclwwlk2sswd 24666* The size of a subword of the second component of a closed walk with length of the size of the second component. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
 |-  A  =  ( 1st `  c )   &    |-  B  =  ( 2nd `  c )   &    |-  C  =  { c  e.  ( V ClWalks  E )  |  ( # `  A )  =  N }   &    |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `
  A ) >. ) )   =>    |-  ( c  e.  C  ->  ( # `  A )  =  ( # `  ( B substr 
 <. 0 ,  ( # `  A ) >. ) ) )
 
Theoremclwlkfclwwlk 24667* There is a function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 25-Jun-2018.)
 |-  A  =  ( 1st `  c )   &    |-  B  =  ( 2nd `  c )   &    |-  C  =  { c  e.  ( V ClWalks  E )  |  ( # `  A )  =  N }   &    |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `
  A ) >. ) )   =>    |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  ->  F : C --> ( ( V ClWWalksN  E ) `  N ) )
 
Theoremclwlkfoclwwlk 24668* There is an onto function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 30-Jun-2018.)
 |-  A  =  ( 1st `  c )   &    |-  B  =  ( 2nd `  c )   &    |-  C  =  { c  e.  ( V ClWalks  E )  |  ( # `  A )  =  N }   &    |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `
  A ) >. ) )   =>    |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  ->  F : C -onto-> ( ( V ClWWalksN  E ) `  N ) )
 
Theoremclwlkf1clwwlklem1 24669* Lemma 1 for clwlkf1clwwlklem 24672. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
 |-  A  =  ( 1st `  c )   &    |-  B  =  ( 2nd `  c )   &    |-  C  =  { c  e.  ( V ClWalks  E )  |  ( # `  A )  =  N }   &    |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `
  A ) >. ) )   =>    |-  ( W  e.  C  ->  N  <_  ( # `  ( 2nd `  W ) ) )
 
Theoremclwlkf1clwwlklem2 24670* Lemma 2 for clwlkf1clwwlklem 24672. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
 |-  A  =  ( 1st `  c )   &    |-  B  =  ( 2nd `  c )   &    |-  C  =  { c  e.  ( V ClWalks  E )  |  ( # `  A )  =  N }   &    |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `
  A ) >. ) )   =>    |-  ( W  e.  C  ->  ( ( 2nd `  W ) `  0 )  =  ( ( 2nd `  W ) `  N ) )
 
Theoremclwlkf1clwwlklem3 24671* Lemma 3 for clwlkf1clwwlklem 24672. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
 |-  A  =  ( 1st `  c )   &    |-  B  =  ( 2nd `  c )   &    |-  C  =  { c  e.  ( V ClWalks  E )  |  ( # `  A )  =  N }   &    |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `
  A ) >. ) )   =>    |-  ( W  e.  C  ->  ( 2nd `  W )  e. Word  V )
 
Theoremclwlkf1clwwlklem 24672* Lemma for clwlkf1clwwlk 24673. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
 |-  A  =  ( 1st `  c )   &    |-  B  =  ( 2nd `  c )   &    |-  C  =  { c  e.  ( V ClWalks  E )  |  ( # `  A )  =  N }   &    |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `
  A ) >. ) )   =>    |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C ) 
 ->  ( ( ( 2nd `  U ) substr  <. 0 ,  ( # `  ( 1st `  U ) )
 >. )  =  (
 ( 2nd `  W ) substr  <.
 0 ,  ( # `  ( 1st `  W ) ) >. )  ->  A. y  e.  (
 0 ... N ) ( ( 2nd `  U ) `  y )  =  ( ( 2nd `  W ) `  y ) ) )
 
Theoremclwlkf1clwwlk 24673* There is a one-to-one function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 5-Jul-2018.)
 |-  A  =  ( 1st `  c )   &    |-  B  =  ( 2nd `  c )   &    |-  C  =  { c  e.  ( V ClWalks  E )  |  ( # `  A )  =  N }   &    |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `
  A ) >. ) )   =>    |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  ->  F : C -1-1-> ( ( V ClWWalksN  E ) `  N ) )
 
Theoremclwlkf1oclwwlk 24674* There is a one-to-one onto function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 5-Jul-2018.)
 |-  A  =  ( 1st `  c )   &    |-  B  =  ( 2nd `  c )   &    |-  C  =  { c  e.  ( V ClWalks  E )  |  ( # `  A )  =  N }   &    |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `
  A ) >. ) )   =>    |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  ->  F : C -1-1-onto-> ( ( V ClWWalksN  E ) `
  N ) )
 
Theoremclwlksizeeq 24675* The size of the set of closed walks (defined as words) of length n corresponds to the size of the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 6-Jul-2018.)
 |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  ->  ( # `  ( ( V ClWWalksN  E ) `  N ) )  =  ( # `
  { c  e.  ( V ClWalks  E )  |  ( # `  ( 1st `  c ) )  =  N } )
 )
 
Theoremclwlkndivn 24676* The size of the set of closed walks of length n is divisible by n. This corresponds to Huneke's observation "It follows that, if p is a prime number, then the number of closed walks of length p is divisible by p". (Contributed by Alexander van der Vekens, 6-Jul-2018.)
 |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  ->  N  ||  ( # `  { c  e.  ( V ClWalks  E )  |  ( # `  ( 1st `  c ) )  =  N } )
 )
 
16.1.5.7  Walks/paths of length 2 as ordered triples
 
Syntaxc2wlkot 24677 Extend class notation with walks (of a graph) of length 2 as ordered triple.
 class 2WalksOt
 
Syntaxc2wlkonot 24678 Extend class notation with walks between two vertices (within a graph) of length 2 as ordered triple.
 class 2WalksOnOt
 
Syntaxc2spthot 24679 Extend class notation with paths (of a graph) of length 2 as ordered triple.
 class 2SPathOnOt
 
Syntaxc2pthonot 24680 Extend class notation with simple paths between two vertices (within a graph) of length 2 as ordered triple.
 class 2SPathOnOt
 
Definitiondf-2wlkonot 24681* Define the collection of walks of length 2 with particular endpoints as ordered triple (in a graph). (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |- 2WalksOnOt  =  ( v  e.  _V ,  e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  { t  e.  (
 ( v  X.  v
 )  X.  v )  |  E. f E. p ( f ( a ( v WalkOn  e ) b ) p  /\  ( # `  f )  =  2  /\  (
 ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
 ) )  =  ( p `  1 ) 
 /\  ( 2nd `  t
 )  =  b ) ) } ) )
 
Definitiondf-2wlksot 24682* Define the collection of all walks of length 2 as ordered triple (in a graph). (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |- 2WalksOt  =  ( v  e.  _V ,  e  e.  _V  |->  { t  e.  ( ( v  X.  v )  X.  v )  | 
 E. a  e.  v  E. b  e.  v  t  e.  ( a
 ( v 2WalksOnOt  e ) b ) } )
 
Definitiondf-2spthonot 24683* Define the collection of simple paths of length 2 with particular endpoints as ordered triple (in a graph) . (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |- 2SPathOnOt  =  ( v  e.  _V ,  e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  { t  e.  (
 ( v  X.  v
 )  X.  v )  |  E. f E. p ( f ( a ( v SPathOn  e ) b ) p  /\  ( # `  f )  =  2  /\  (
 ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
 ) )  =  ( p `  1 ) 
 /\  ( 2nd `  t
 )  =  b ) ) } ) )
 
Definitiondf-2spthsot 24684* Define the collection of all simple paths of length 2 as ordered triple. (in a graph) (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |- 2SPathOnOt  =  ( v  e.  _V ,  e  e.  _V  |->  { t  e.  ( ( v  X.  v )  X.  v )  | 
 E. a  e.  v  E. b  e.  v  t  e.  ( a
 ( v 2SPathOnOt  e ) b ) } )
 
Theoremel2wlkonotlem 24685 Lemma for el2wlkonot 24692. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  ( ( f ( V Walks  E ) p 
 /\  ( # `  f
 )  =  2 ) 
 ->  ( p `  1
 )  e.  V )
 
Theoremis2wlkonot 24686* The set of walks of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V 2WalksOnOt  E )  =  ( a  e.  V ,  b  e.  V  |->  { t  e.  (
 ( V  X.  V )  X.  V )  | 
 E. f E. p ( f ( a ( V WalkOn  E )
 b ) p  /\  ( # `  f )  =  2  /\  (
 ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
 ) )  =  ( p `  1 ) 
 /\  ( 2nd `  t
 )  =  b ) ) } ) )
 
Theoremis2spthonot 24687* The set of simple paths of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V 2SPathOnOt  E )  =  ( a  e.  V ,  b  e.  V  |->  { t  e.  (
 ( V  X.  V )  X.  V )  | 
 E. f E. p ( f ( a ( V SPathOn  E )
 b ) p  /\  ( # `  f )  =  2  /\  (
 ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
 ) )  =  ( p `  1 ) 
 /\  ( 2nd `  t
 )  =  b ) ) } ) )
 
Theorem2wlkonot 24688* The set of walks of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V )
 )  ->  ( A ( V 2WalksOnOt  E ) B )  =  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V WalkOn  E ) B ) p  /\  ( # `  f )  =  2  /\  (
 ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
 ) )  =  ( p `  1 ) 
 /\  ( 2nd `  t
 )  =  B ) ) } )
 
Theorem2spthonot 24689* The set of simple paths of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V )
 )  ->  ( A ( V 2SPathOnOt  E ) B )  =  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  (
 ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
 ) )  =  ( p `  1 ) 
 /\  ( 2nd `  t
 )  =  B ) ) } )
 
Theorem2wlksot 24690* The set of walks of length 2 (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V 2WalksOt  E )  =  { t  e.  ( ( V  X.  V )  X.  V )  |  E. a  e.  V  E. b  e.  V  t  e.  (
 a ( V 2WalksOnOt  E ) b ) } )
 
Theorem2spthsot 24691* The set of simple paths of length 2 (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 28-Feb-2018.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V 2SPathOnOt  E )  =  { t  e.  ( ( V  X.  V )  X.  V )  |  E. a  e.  V  E. b  e.  V  t  e.  (
 a ( V 2SPathOnOt  E ) b ) } )
 
Theoremel2wlkonot 24692* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
 )  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V Walks  E ) p 
 /\  ( # `  f
 )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2
 ) ) ) ) ) )
 
Theoremel2spthonot 24693* A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
 )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1
 )  /\  C  =  ( p `  2 ) ) ) ) ) )
 
Theoremel2spthonot0 24694* A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 9-Mar-2018.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
 )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) ) )
 
Theoremel2wlkonotot0 24695* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
 )  ->  ( <. A ,  B ,  C >.  e.  ( R ( V 2WalksOnOt  E ) S )  <-> 
 ( A  =  R  /\  C  =  S  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 
 /\  ( A  =  ( p `  0 ) 
 /\  B  =  ( p `  1 ) 
 /\  C  =  ( p `  2 ) ) ) ) ) )
 
Theoremel2wlkonotot 24696* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
 )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 
 /\  ( A  =  ( p `  0 ) 
 /\  B  =  ( p `  1 ) 
 /\  C  =  ( p `  2 ) ) ) ) )
 
Theoremel2wlkonotot1 24697 A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 8-Mar-2018.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
 )  ->  ( <. A ,  B ,  C >.  e.  ( R ( V 2WalksOnOt  E ) S )  <-> 
 ( A  =  R  /\  C  =  S  /\  <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C ) ) ) )
 
Theorem2wlkonot3v 24698 If an ordered triple represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
 |-  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  (
 ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V ) 
 /\  T  e.  (
 ( V  X.  V )  X.  V ) ) )
 
Theorem2spthonot3v 24699 If an ordered triple represents a simple path of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |-  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  (
 ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V ) 
 /\  T  e.  (
 ( V  X.  V )  X.  V ) ) )
 
Theorem2wlkonotv 24700 If an ordered tripple represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
 |-  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  ->  (
 ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 ) )
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