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Theorem List for Metamath Proof Explorer - 25501-25600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremclwwlkn2 25501 In an undirected simple graph, a closed walk of length 2 represented as word is a word consisting of 2 symbols representing vertices connected by an edge. (Contributed by Alexander van der Vekens, 19-Sep-2018.)
 |-  ( V USGrph  E  ->  ( W  e.  ( ( V ClWWalksN  E ) `  2
 ) 
 <->  ( ( # `  W )  =  2  /\  W  e. Word  V  /\  {
 ( W `  0
 ) ,  ( W `
  1 ) }  e.  ran  E ) ) )
 
Theoremclwwlknimp 25502* Implications for a set being a closed walk (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.)
 |-  ( W  e.  (
 ( V ClWWalksN  E ) `  N )  ->  ( ( W  e. Word  V  /\  ( # `  W )  =  N )  /\  A. i  e.  ( 0..^ ( N  -  1
 ) ) { ( W `  i ) ,  ( W `  (
 i  +  1 ) ) }  e.  ran  E 
 /\  { ( lastS  `  W ) ,  ( W `  0 ) }  e.  ran 
 E ) )
 
Theoremclwwlksswrd 25503 Closed walks (represented by words) are words. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V ClWWalks  E ) 
 C_ Word  V )
 
Theoremclwwlknfi 25504 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 25505* Lemma 1 for clwlkisclwwlklem2a 25511. (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 25506* Lemma 3 for clwlkisclwwlklem2a 25511. (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 25507* Lemma 3 for clwlkisclwwlklem2a 25511. (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 25508* Lemma 4a for clwlkisclwwlklem2a 25511. (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 25509* Lemma 4b for clwlkisclwwlklem2a 25511. (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 25510* Lemma 4 for clwlkisclwwlklem2a 25511. (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 25511* Lemma 2 for clwlkisclwwlklem2 25512. (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 25512* Lemma for clwlkisclwwlk 25515. (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 25513* Lemma for clwlkisclwwlk 25515. (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 25514* Lemma for clwlkisclwwlk 25515. (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 25515* 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 25516* 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 ++  <" ( P `  0 ) "> )  <->  P  e.  ( V ClWWalks  E ) ) )
 
Theoremclwwlkisclwwlkn 25517 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 25518 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 25519* 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 ++  <" ( P `
  0 ) "> )  e.  D )
 
Theoremclwwlkf 25520* Lemma 1 for clwwlkbij 25525: 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 25521* Lemma 2 for clwwlkbij 25525: 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 25522* Lemma 3 for clwwlkbij 25525: 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 25523* Lemma 4 for clwwlkbij 25525: 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 25524* Lemma 5 for clwwlkbij 25525: 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 25525* 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 25526* 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 ++  <" ( P `  0 ) "> )  e.  { w  e.  ( ( V WWalksN  E ) `
  N )  |  ( lastS  `  w )  =  ( w `  0
 ) } )
 
Theoremclwwlkvbij 25527* 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 25528 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 ++  <" Z "> )  e.  ( ( V ClWWalksN  E ) `
  N ) ) 
 ->  ( { ( lastS  `  W ) ,  Z }  e.  ran  E  /\  { Z ,  ( W `  0 ) }  e.  ran 
 E ) )
 
Theoremwwlkext2clwwlk 25529 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 ++  <" Z "> )  e.  (
 ( V ClWWalksN  E ) `  ( N  +  2
 ) ) ) )
 
Theoremwwlksubclwwlk 25530 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 25531* Lemma 1 for clwwisshclwwlem 25532. (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 25532* Lemma for clwwisshclww 25533. (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 25533 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 25534 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 25535 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 25536  .~ 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 25537* 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 25538* 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 25539*  .~ 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 25540*  .~ 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 25541*  .~ 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 25542*  .~ 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 25543* Lemma 1 for eleclclwwlkn 25559. (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 25544* Lemma 2 for eleclclwwlkn 25559. (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 25545* 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 25546 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 25547* 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 25548  .~ 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 25549* 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 25550* 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 25551*  .~ 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 25552*  .~ 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 25553*  .~ 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 25554*  .~ 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 25555* 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 25556* 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 25557* 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 25558* 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 25559* 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 25560* 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 25561* 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 25562* 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 25563 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 25564 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 25565 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 ++  <" ( W `
  0 ) "> ) >.  e.  ( V Walks  E )  ->  ( # `
  F )  =  ( # `  W ) ) )
 
Theoremclwlkfclwwlk2wrd 25566 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 25567* 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 25568* 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 25569* 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 25570* 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 25571* 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 25572* Lemma 1 for clwlkf1clwwlklem 25575. (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 25573* Lemma 2 for clwlkf1clwwlklem 25575. (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 25574* Lemma 3 for clwlkf1clwwlklem 25575. (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 25575* Lemma for clwlkf1clwwlk 25576. (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 25576* 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 25577* 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 25578* 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 25579* The size of the set of closed walks of length n is divisible by n. This corresponds to statement 9 in [Huneke] p. 2: "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 25580 Extend class notation with walks (of a graph) of length 2 as ordered triple.
 class 2WalksOt
 
Syntaxc2wlkonot 25581 Extend class notation with walks between two vertices (within a graph) of length 2 as ordered triple.
 class 2WalksOnOt
 
Syntaxc2spthot 25582 Extend class notation with paths (of a graph) of length 2 as ordered triple.
 class 2SPathOnOt
 
Syntaxc2pthonot 25583 Extend class notation with simple paths between two vertices (within a graph) of length 2 as ordered triple.
 class 2SPathOnOt
 
Definitiondf-2wlkonot 25584* 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 25585* 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 25586* 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 25587* 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 25588 Lemma for el2wlkonot 25595. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  ( ( f ( V Walks  E ) p 
 /\  ( # `  f
 )  =  2 ) 
 ->  ( p `  1
 )  e.  V )
 
Theoremis2wlkonot 25589* 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 25590* 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 25591* 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 25592* 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 25593* 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 25594* 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 25595* 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 25596* 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 25597* 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 25598* 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 25599* 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 25600 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 ) ) ) )
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