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Theorem clwlkfoclwwlk 25572
Description: 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.)
Hypotheses
Ref Expression
clwlkfclwwlk.1  |-  A  =  ( 1st `  c
)
clwlkfclwwlk.2  |-  B  =  ( 2nd `  c
)
clwlkfclwwlk.c  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
clwlkfclwwlk.f  |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `  A
) >. ) )
Assertion
Ref Expression
clwlkfoclwwlk  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  F : C -onto-> ( ( V ClWWalksN  E ) `  N
) )
Distinct variable groups:    E, c    N, c    V, c    C, c    F, c
Allowed substitution hints:    A( c)    B( c)

Proof of Theorem clwlkfoclwwlk
Dummy variables  f  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwlkfclwwlk.1 . . 3  |-  A  =  ( 1st `  c
)
2 clwlkfclwwlk.2 . . 3  |-  B  =  ( 2nd `  c
)
3 clwlkfclwwlk.c . . 3  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
4 clwlkfclwwlk.f . . 3  |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `  A
) >. ) )
51, 2, 3, 4clwlkfclwwlk 25571 . 2  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  F : C
--> ( ( V ClWWalksN  E ) `
 N ) )
6 clwwlknprop 25499 . . . . 5  |-  ( w  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )
76adantl 467 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  w  e.  ( ( V ClWWalksN  E ) `
 N ) )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )
8 simpl 458 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  ( # `  w )  =  N )  ->  N  e.  NN0 )
98anim2i 571 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
10 df-3an 984 . . . . . . . . . . 11  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
119, 10sylibr 215 . . . . . . . . . 10  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) )  ->  ( V  e.  _V  /\  E  e. 
_V  /\  N  e.  NN0 ) )
12113adant2 1024 . . . . . . . . 9  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) )  ->  ( V  e.  _V  /\  E  e. 
_V  /\  N  e.  NN0 ) )
1312adantl 467 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  ( V  e.  _V  /\  E  e.  _V  /\  N  e. 
NN0 ) )
14 isclwwlkn 25496 . . . . . . . 8  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
w  e.  ( ( V ClWWalksN  E ) `  N
)  <->  ( w  e.  ( V ClWWalks  E )  /\  ( # `  w
)  =  N ) ) )
1513, 14syl 17 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  (
w  e.  ( ( V ClWWalksN  E ) `  N
)  <->  ( w  e.  ( V ClWWalks  E )  /\  ( # `  w
)  =  N ) ) )
16 simpl1 1008 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  V USGrph  E )
17 simpr2 1012 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  w  e. Word  V )
18 eleq1 2495 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  N  ->  ( (
# `  w )  e.  Prime 
<->  N  e.  Prime )
)
19 prmnn 14625 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  e.  Prime  ->  ( # `  w
)  e.  NN )
2019nnge1d 10660 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  e.  Prime  ->  1  <_  (
# `  w )
)
2118, 20syl6bir 232 . . . . . . . . . . . . . . 15  |-  ( (
# `  w )  =  N  ->  ( N  e.  Prime  ->  1  <_ 
( # `  w ) ) )
2221adantl 467 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  ( # `  w )  =  N )  -> 
( N  e.  Prime  -> 
1  <_  ( # `  w
) ) )
23223ad2ant3 1028 . . . . . . . . . . . . 13  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) )  ->  ( N  e.  Prime  ->  1  <_  (
# `  w )
) )
2423com12 32 . . . . . . . . . . . 12  |-  ( N  e.  Prime  ->  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) )  ->  1  <_  (
# `  w )
) )
25243ad2ant3 1028 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) )  ->  1  <_  (
# `  w )
) )
2625imp 430 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  1  <_  ( # `  w
) )
27 clwlkisclwwlk2 25517 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  w  e. Word  V  /\  1  <_ 
( # `  w ) )  ->  ( E. f  f ( V ClWalks  E ) ( w ++ 
<" ( w ` 
0 ) "> ) 
<->  w  e.  ( V ClWWalks  E ) ) )
2816, 17, 26, 27syl3anc 1264 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  ( E. f  f ( V ClWalks  E ) ( w ++ 
<" ( w ` 
0 ) "> ) 
<->  w  e.  ( V ClWWalks  E ) ) )
2928bicomd 204 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  (
w  e.  ( V ClWWalks  E )  <->  E. f 
f ( V ClWalks  E
) ( w ++  <" ( w `  0
) "> )
) )
3029anbi1d 709 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  (
( w  e.  ( V ClWWalks  E )  /\  ( # `
 w )  =  N )  <->  ( E. f  f ( V ClWalks  E ) ( w ++ 
<" ( w ` 
0 ) "> )  /\  ( # `  w
)  =  N ) ) )
3115, 30bitrd 256 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  (
w  e.  ( ( V ClWWalksN  E ) `  N
)  <->  ( E. f 
f ( V ClWalks  E
) ( w ++  <" ( w `  0
) "> )  /\  ( # `  w
)  =  N ) ) )
32 df-br 4424 . . . . . . . . . . 11  |-  ( f ( V ClWalks  E )
( w ++  <" (
w `  0 ) "> )  <->  <. f ,  ( w ++  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )
33 simpl 458 . . . . . . . . . . . . . . 15  |-  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  ->  <. f ,  ( w ++  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )
34 prmnn 14625 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  Prime  ->  N  e.  NN )
3534nnge1d 10660 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  Prime  ->  1  <_  N )
36353ad2ant3 1028 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  1  <_  N )
3736adantr 466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  1  <_  N )
38 breq2 4427 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
# `  w )  =  N  ->  ( 1  <_  ( # `  w
)  <->  1  <_  N
) )
3938adantl 467 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( N  e.  NN0  /\  ( # `  w )  =  N )  -> 
( 1  <_  ( # `
 w )  <->  1  <_  N ) )
40393ad2ant3 1028 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) )  ->  ( 1  <_  ( # `  w
)  <->  1  <_  N
) )
4140adantl 467 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  (
1  <_  ( # `  w
)  <->  1  <_  N
) )
4237, 41mpbird 235 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  1  <_  ( # `  w
) )
4317, 42jca 534 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  (
w  e. Word  V  /\  1  <_  ( # `  w
) ) )
4443adantr 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N )  ->  ( w  e. Word  V  /\  1  <_  ( # `
 w ) ) )
45 clwlkswlks 25485 . . . . . . . . . . . . . . . . . 18  |-  ( <.
f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V Walks  E
) )
46 wlklenvclwlk 25566 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  ( <. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V Walks 
E )  ->  ( # `
 f )  =  ( # `  w
) ) )
4744, 45, 46syl2im 39 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N )  ->  ( <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
)  ->  ( # `  f
)  =  ( # `  w ) ) )
4847impcom 431 . . . . . . . . . . . . . . . 16  |-  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  ->  ( # `  f
)  =  ( # `  w ) )
49 vex 3083 . . . . . . . . . . . . . . . . . . . 20  |-  f  e. 
_V
50 ovex 6334 . . . . . . . . . . . . . . . . . . . 20  |-  ( w ++ 
<" ( w ` 
0 ) "> )  e.  _V
5149, 50op1st 6816 . . . . . . . . . . . . . . . . . . 19  |-  ( 1st `  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >. )  =  f
5251a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  ( 1st `  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >. )  =  f )
5352fveq2d 5886 . . . . . . . . . . . . . . . . 17  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  ( # `
 ( 1st `  <. f ,  ( w ++  <" ( w `  0
) "> ) >. ) )  =  (
# `  f )
)
5453ad2antrl 732 . . . . . . . . . . . . . . . 16  |-  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  ->  ( # `  ( 1st `  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >. )
)  =  ( # `  f ) )
55 eqcom 2431 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  w )  =  N  <->  N  =  ( # `
 w ) )
5655biimpi 197 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  N  ->  N  =  ( # `  w
) )
5756ad2antll 733 . . . . . . . . . . . . . . . 16  |-  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  ->  N  =  ( # `  w ) )
5848, 54, 573eqtr4d 2473 . . . . . . . . . . . . . . 15  |-  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  ->  ( # `  ( 1st `  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >. )
)  =  N )
591fveq2i 5885 . . . . . . . . . . . . . . . . . 18  |-  ( # `  A )  =  (
# `  ( 1st `  c ) )
6059eqeq1i 2429 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  A )  =  N  <->  ( # `  ( 1st `  c ) )  =  N )
61 fveq2 5882 . . . . . . . . . . . . . . . . . . 19  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  ( 1st `  c )  =  ( 1st `  <. f ,  ( w ++  <" ( w `  0
) "> ) >. ) )
6261fveq2d 5886 . . . . . . . . . . . . . . . . . 18  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  ( # `
 ( 1st `  c
) )  =  (
# `  ( 1st ` 
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >. ) ) )
6362eqeq1d 2424 . . . . . . . . . . . . . . . . 17  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  (
( # `  ( 1st `  c ) )  =  N  <->  ( # `  ( 1st `  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >. )
)  =  N ) )
6460, 63syl5bb 260 . . . . . . . . . . . . . . . 16  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  (
( # `  A )  =  N  <->  ( # `  ( 1st `  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >. )
)  =  N ) )
6564, 3elrab2 3230 . . . . . . . . . . . . . . 15  |-  ( <.
f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  C  <->  ( <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
)  /\  ( # `  ( 1st `  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >. )
)  =  N ) )
6633, 58, 65sylanbrc 668 . . . . . . . . . . . . . 14  |-  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  ->  <. f ,  ( w ++  <" (
w `  0 ) "> ) >.  e.  C
)
6743, 45, 46syl2im 39 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  ( <. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  ( # `
 f )  =  ( # `  w
) ) )
6867ad2antrl 732 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  ->  ( <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
)  ->  ( # `  f
)  =  ( # `  w ) ) )
6968imp 430 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  /\  <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
) )  ->  ( # `
 f )  =  ( # `  w
) )
7069opeq2d 4194 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  /\  <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
) )  ->  <. 0 ,  ( # `  f
) >.  =  <. 0 ,  ( # `  w
) >. )
7170oveq2d 6322 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  /\  <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
) )  ->  (
( w ++  <" (
w `  0 ) "> ) substr  <. 0 ,  ( # `  f
) >. )  =  ( ( w ++  <" (
w `  0 ) "> ) substr  <. 0 ,  ( # `  w
) >. ) )
72 simpr 462 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  /\  <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
) )  ->  <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
) )
73 eqeq2 2437 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  =  ( # `  w
)  ->  ( ( # `
 f )  =  N  <->  ( # `  f
)  =  ( # `  w ) ) )
7473eqcoms 2434 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
# `  w )  =  N  ->  ( (
# `  f )  =  N  <->  ( # `  f
)  =  ( # `  w ) ) )
7574imbi2d 317 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  w )  =  N  ->  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  ( # `
 f )  =  N )  <->  ( <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
)  ->  ( # `  f
)  =  ( # `  w ) ) ) )
7675ad2antll 733 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  ->  ( ( <. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  ( # `
 f )  =  N )  <->  ( <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
)  ->  ( # `  f
)  =  ( # `  w ) ) ) )
7768, 76mpbird 235 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  ->  ( <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
)  ->  ( # `  f
)  =  N ) )
7877imp 430 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  /\  <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
) )  ->  ( # `
 f )  =  N )
7951a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  ( 1st `  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >. )  =  f )
8079fveq2d 5886 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  ( # `
 ( 1st `  <. f ,  ( w ++  <" ( w `  0
) "> ) >. ) )  =  (
# `  f )
)
8162, 80eqtrd 2463 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  ( # `
 ( 1st `  c
) )  =  (
# `  f )
)
8281eqeq1d 2424 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  (
( # `  ( 1st `  c ) )  =  N  <->  ( # `  f
)  =  N ) )
8360, 82syl5bb 260 . . . . . . . . . . . . . . . . . . . . 21  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  (
( # `  A )  =  N  <->  ( # `  f
)  =  N ) )
8483, 3elrab2 3230 . . . . . . . . . . . . . . . . . . . 20  |-  ( <.
f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  C  <->  ( <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
)  /\  ( # `  f
)  =  N ) )
8572, 78, 84sylanbrc 668 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  /\  <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
) )  ->  <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  C )
86 ovex 6334 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w ++  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  f
) >. )  e.  _V
8759opeq2i 4191 . . . . . . . . . . . . . . . . . . . . . 22  |-  <. 0 ,  ( # `  A
) >.  =  <. 0 ,  ( # `  ( 1st `  c ) )
>.
882, 87oveq12i 6318 . . . . . . . . . . . . . . . . . . . . 21  |-  ( B substr  <. 0 ,  ( # `  A ) >. )  =  ( ( 2nd `  c ) substr  <. 0 ,  ( # `  ( 1st `  c ) )
>. )
89 fveq2 5882 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  ( 2nd `  c )  =  ( 2nd `  <. f ,  ( w ++  <" ( w `  0
) "> ) >. ) )
9062opeq2d 4194 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  <. 0 ,  ( # `  ( 1st `  c ) )
>.  =  <. 0 ,  ( # `  ( 1st `  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >. )
) >. )
9189, 90oveq12d 6324 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  (
( 2nd `  c
) substr  <. 0 ,  (
# `  ( 1st `  c ) ) >.
)  =  ( ( 2nd `  <. f ,  ( w ++  <" ( w `  0
) "> ) >. ) substr  <. 0 ,  (
# `  ( 1st ` 
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >. ) ) >.
) )
9249, 50op2nd 6817 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 2nd `  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >. )  =  ( w ++  <" ( w `  0
) "> )
9351fveq2i 5885 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( # `  ( 1st `  <. f ,  ( w ++  <" ( w `  0
) "> ) >. ) )  =  (
# `  f )
9493opeq2i 4191 . . . . . . . . . . . . . . . . . . . . . . 23  |-  <. 0 ,  ( # `  ( 1st `  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >. )
) >.  =  <. 0 ,  ( # `  f
) >.
9592, 94oveq12i 6318 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 2nd `  <. f ,  ( w ++  <" ( w `  0
) "> ) >. ) substr  <. 0 ,  (
# `  ( 1st ` 
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >. ) ) >.
)  =  ( ( w ++  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  f
) >. )
9691, 95syl6eq 2479 . . . . . . . . . . . . . . . . . . . . 21  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  (
( 2nd `  c
) substr  <. 0 ,  (
# `  ( 1st `  c ) ) >.
)  =  ( ( w ++  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  f
) >. ) )
9788, 96syl5eq 2475 . . . . . . . . . . . . . . . . . . . 20  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  ( B substr  <. 0 ,  (
# `  A ) >. )  =  ( ( w ++  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  f
) >. ) )
9897, 4fvmptg 5963 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  C  /\  ( ( w ++  <" ( w `  0
) "> ) substr  <.
0 ,  ( # `  f ) >. )  e.  _V )  ->  ( F `  <. f ,  ( w ++  <" (
w `  0 ) "> ) >. )  =  ( ( w ++ 
<" ( w ` 
0 ) "> ) substr  <. 0 ,  (
# `  f ) >. ) )
9985, 86, 98sylancl 666 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  /\  <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
) )  ->  ( F `  <. f ,  ( w ++  <" (
w `  0 ) "> ) >. )  =  ( ( w ++ 
<" ( w ` 
0 ) "> ) substr  <. 0 ,  (
# `  f ) >. ) )
10043ad2antrl 732 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  ->  ( w  e. Word  V  /\  1  <_ 
( # `  w ) ) )
101100adantr 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  /\  <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
) )  ->  (
w  e. Word  V  /\  1  <_  ( # `  w
) ) )
102 simpl 458 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  w  e. Word  V )
103 wrdsymb1 12709 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  (
w `  0 )  e.  V )
104103s1cld 12747 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  <" (
w `  0 ) ">  e. Word  V )
105 eqidd 2423 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  ( # `
 w )  =  ( # `  w
) )
106 swrdccatid 12856 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  e. Word  V  /\  <" ( w ` 
0 ) ">  e. Word  V  /\  ( # `  w )  =  (
# `  w )
)  ->  ( (
w ++  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  w
) >. )  =  w )
107102, 104, 105, 106syl3anc 1264 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  (
( w ++  <" (
w `  0 ) "> ) substr  <. 0 ,  ( # `  w
) >. )  =  w )
108107eqcomd 2430 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  w  =  ( ( w ++ 
<" ( w ` 
0 ) "> ) substr  <. 0 ,  (
# `  w ) >. ) )
109101, 108syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  /\  <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
) )  ->  w  =  ( ( w ++ 
<" ( w ` 
0 ) "> ) substr  <. 0 ,  (
# `  w ) >. ) )
11071, 99, 1093eqtr4rd 2474 . . . . . . . . . . . . . . . . 17  |-  ( ( ( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  /\  <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
) )  ->  w  =  ( F `  <. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >. ) )
111110ex 435 . . . . . . . . . . . . . . . 16  |-  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  ->  ( <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
)  ->  w  =  ( F `  <. f ,  ( w ++  <" ( w `  0
) "> ) >. ) ) )
112111adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( ( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  /\  c  = 
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >. )  ->  ( <. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  w  =  ( F `  <. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >. ) ) )
113 fveq2 5882 . . . . . . . . . . . . . . . . . 18  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  ( F `  c )  =  ( F `  <. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >. ) )
114113eqeq2d 2436 . . . . . . . . . . . . . . . . 17  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  (
w  =  ( F `
 c )  <->  w  =  ( F `  <. f ,  ( w ++  <" ( w `  0
) "> ) >. ) ) )
115114imbi2d 317 . . . . . . . . . . . . . . . 16  |-  ( c  =  <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  ->  (
( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  ->  w  =  ( F `  c ) )  <->  ( <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
)  ->  w  =  ( F `  <. f ,  ( w ++  <" ( w `  0
) "> ) >. ) ) ) )
116115adantl 467 . . . . . . . . . . . . . . 15  |-  ( ( ( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  /\  c  = 
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >. )  ->  (
( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  ->  w  =  ( F `  c ) )  <->  ( <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
)  ->  w  =  ( F `  <. f ,  ( w ++  <" ( w `  0
) "> ) >. ) ) ) )
117112, 116mpbird 235 . . . . . . . . . . . . . 14  |-  ( ( ( <. f ,  ( w ++  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  /\  c  = 
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >. )  ->  ( <. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  w  =  ( F `  c ) ) )
11866, 117rspcimedv 3184 . . . . . . . . . . . . 13  |-  ( (
<. f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  /\  (
( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N ) )  ->  ( <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
)  ->  E. c  e.  C  w  =  ( F `  c ) ) )
119118ex 435 . . . . . . . . . . . 12  |-  ( <.
f ,  ( w ++ 
<" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  (
( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N )  ->  ( <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
)  ->  E. c  e.  C  w  =  ( F `  c ) ) ) )
120119pm2.43b 52 . . . . . . . . . . 11  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N )  ->  ( <. f ,  ( w ++  <" ( w `  0
) "> ) >.  e.  ( V ClWalks  E
)  ->  E. c  e.  C  w  =  ( F `  c ) ) )
12132, 120syl5bi 220 . . . . . . . . . 10  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N )  ->  ( f ( V ClWalks  E ) ( w ++ 
<" ( w ` 
0 ) "> )  ->  E. c  e.  C  w  =  ( F `  c ) ) )
122121exlimdv 1772 . . . . . . . . 9  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  ( ( V  e. 
_V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )  /\  ( # `  w
)  =  N )  ->  ( E. f 
f ( V ClWalks  E
) ( w ++  <" ( w `  0
) "> )  ->  E. c  e.  C  w  =  ( F `  c ) ) )
123122ex 435 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  (
( # `  w )  =  N  ->  ( E. f  f ( V ClWalks  E ) ( w ++ 
<" ( w ` 
0 ) "> )  ->  E. c  e.  C  w  =  ( F `  c ) ) ) )
124123com23 81 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  ( E. f  f ( V ClWalks  E ) ( w ++ 
<" ( w ` 
0 ) "> )  ->  ( ( # `  w )  =  N  ->  E. c  e.  C  w  =  ( F `  c ) ) ) )
125124impd 432 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  (
( E. f  f ( V ClWalks  E )
( w ++  <" (
w `  0 ) "> )  /\  ( # `
 w )  =  N )  ->  E. c  e.  C  w  =  ( F `  c ) ) )
12631, 125sylbid 218 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  (
( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) ) )  ->  (
w  e.  ( ( V ClWWalksN  E ) `  N
)  ->  E. c  e.  C  w  =  ( F `  c ) ) )
127126impancom 441 . . . 4  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  w  e.  ( ( V ClWWalksN  E ) `
 N ) )  ->  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) )  ->  E. c  e.  C  w  =  ( F `  c ) ) )
1287, 127mpd 15 . . 3  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  Prime )  /\  w  e.  ( ( V ClWWalksN  E ) `
 N ) )  ->  E. c  e.  C  w  =  ( F `  c ) )
129128ralrimiva 2836 . 2  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  A. w  e.  ( ( V ClWWalksN  E ) `
 N ) E. c  e.  C  w  =  ( F `  c ) )
130 dffo3 6053 . 2  |-  ( F : C -onto-> ( ( V ClWWalksN  E ) `  N
)  <->  ( F : C
--> ( ( V ClWWalksN  E ) `
 N )  /\  A. w  e.  ( ( V ClWWalksN  E ) `  N
) E. c  e.  C  w  =  ( F `  c ) ) )
1315, 129, 130sylanbrc 668 1  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  F : C -onto-> ( ( V ClWWalksN  E ) `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872   A.wral 2771   E.wrex 2772   {crab 2775   _Vcvv 3080   <.cop 4004   class class class wbr 4423    |-> cmpt 4482   -->wf 5597   -onto->wfo 5599   ` cfv 5601  (class class class)co 6306   1stc1st 6806   2ndc2nd 6807   Fincfn 7581   0cc0 9547   1c1 9548    <_ cle 9684   NN0cn0 10877   #chash 12522  Word cword 12661   ++ cconcat 12663   <"cs1 12664   substr csubstr 12665   Primecprime 14622   USGrph cusg 25056   Walks cwalk 25225   ClWalks cclwlk 25474   ClWWalks cclwwlk 25475   ClWWalksN cclwwlkn 25476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-2o 7195  df-oadd 7198  df-er 7375  df-map 7486  df-pm 7487  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-card 8382  df-cda 8606  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-nn 10618  df-2 10676  df-n0 10878  df-z 10946  df-uz 11168  df-rp 11311  df-fz 11793  df-fzo 11924  df-hash 12523  df-word 12669  df-lsw 12670  df-concat 12671  df-s1 12672  df-substr 12673  df-dvds 14306  df-prm 14623  df-usgra 25059  df-wlk 25235  df-clwlk 25477  df-clwwlk 25478  df-clwwlkn 25479
This theorem is referenced by:  clwlkf1oclwwlk  25578
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