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Theorem clwlkfoclwwlk 24518
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 24517 . 2  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  F : C
--> ( ( V ClWWalksN  E ) `
 N ) )
6 clwwlknprop 24445 . . . . 5  |-  ( w  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 w )  =  N ) ) )
76adantl 466 . . . 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 457 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  ( # `  w )  =  N )  ->  N  e.  NN0 )
98anim2i 569 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
10 df-3an 975 . . . . . . . . . . 11  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
119, 10sylibr 212 . . . . . . . . . 10  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) )  ->  ( V  e.  _V  /\  E  e. 
_V  /\  N  e.  NN0 ) )
12113adant2 1015 . . . . . . . . 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 466 . . . . . . . 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 24442 . . . . . . . 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 16 . . . . . . 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 999 . . . . . . . . . 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 1003 . . . . . . . . . 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 2539 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  N  ->  ( (
# `  w )  e.  Prime 
<->  N  e.  Prime )
)
19 prmnn 14072 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  e.  Prime  ->  ( # `  w
)  e.  NN )
2019nnge1d 10574 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  e.  Prime  ->  1  <_  (
# `  w )
)
2118, 20syl6bir 229 . . . . . . . . . . . . . . 15  |-  ( (
# `  w )  =  N  ->  ( N  e.  Prime  ->  1  <_ 
( # `  w ) ) )
2221adantl 466 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  ( # `  w )  =  N )  -> 
( N  e.  Prime  -> 
1  <_  ( # `  w
) ) )
23223ad2ant3 1019 . . . . . . . . . . . . 13  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) )  ->  ( N  e.  Prime  ->  1  <_  (
# `  w )
) )
2423com12 31 . . . . . . . . . . . 12  |-  ( N  e.  Prime  ->  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) )  ->  1  <_  (
# `  w )
) )
25243ad2ant3 1019 . . . . . . . . . . 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 429 . . . . . . . . . 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 24463 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  w  e. Word  V  /\  1  <_ 
( # `  w ) )  ->  ( E. f  f ( V ClWalks  E ) ( w concat  <" ( w ` 
0 ) "> ) 
<->  w  e.  ( V ClWWalks  E ) ) )
2816, 17, 26, 27syl3anc 1228 . . . . . . . . 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 concat  <" ( w ` 
0 ) "> ) 
<->  w  e.  ( V ClWWalks  E ) ) )
2928bicomd 201 . . . . . . . 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 concat  <" (
w `  0 ) "> ) ) )
3029anbi1d 704 . . . . . . 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 concat  <" ( w ` 
0 ) "> )  /\  ( # `  w
)  =  N ) ) )
3115, 30bitrd 253 . . . . . 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 concat  <" (
w `  0 ) "> )  /\  ( # `
 w )  =  N ) ) )
32 df-br 4448 . . . . . . . . . . 11  |-  ( f ( V ClWalks  E )
( w concat  <" (
w `  0 ) "> )  <->  <. f ,  ( w concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )
33 simpl 457 . . . . . . . . . . . . . . 15  |-  ( (
<. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )
34 prmnn 14072 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  Prime  ->  N  e.  NN )
3534nnge1d 10574 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  Prime  ->  1  <_  N )
36353ad2ant3 1019 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  1  <_  N )
3736adantr 465 . . . . . . . . . . . . . . . . . . . . 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 4451 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
# `  w )  =  N  ->  ( 1  <_  ( # `  w
)  <->  1  <_  N
) )
3938adantl 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( N  e.  NN0  /\  ( # `  w )  =  N )  -> 
( 1  <_  ( # `
 w )  <->  1  <_  N ) )
40393ad2ant3 1019 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  w  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  w
)  =  N ) )  ->  ( 1  <_  ( # `  w
)  <->  1  <_  N
) )
4140adantl 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  <_  ( # `  w
)  <->  1  <_  N
) )
4237, 41mpbird 232 . . . . . . . . . . . . . . . . . . . 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 532 . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . 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 24431 . . . . . . . . . . . . . . . . . 18  |-  ( <.
f ,  ( w concat  <" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  <. f ,  ( w concat  <" (
w `  0 ) "> ) >.  e.  ( V Walks  E ) )
46 wlklenvclwlk 24512 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  ( <. f ,  ( w concat  <" ( w ` 
0 ) "> ) >.  e.  ( V Walks 
E )  ->  ( # `
 f )  =  ( # `  w
) ) )
4744, 45, 46syl2im 38 . . . . . . . . . . . . . . . . 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E )  ->  ( # `
 f )  =  ( # `  w
) ) )
4847impcom 430 . . . . . . . . . . . . . . . 16  |-  ( (
<. f ,  ( w concat  <" ( 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 3116 . . . . . . . . . . . . . . . . . . . 20  |-  f  e. 
_V
50 ovex 6307 . . . . . . . . . . . . . . . . . . . 20  |-  ( w concat  <" ( w ` 
0 ) "> )  e.  _V
5149, 50op1st 6789 . . . . . . . . . . . . . . . . . . 19  |-  ( 1st `  <. f ,  ( w concat  <" ( 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 concat  <" ( w `
 0 ) "> ) >. )  =  f )
5352fveq2d 5868 . . . . . . . . . . . . . . . . 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 concat  <" (
w `  0 ) "> ) >. )
)  =  ( # `  f ) )
5453ad2antrl 727 . . . . . . . . . . . . . . . 16  |-  ( (
<. f ,  ( w concat  <" ( 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 concat  <" ( w `
 0 ) "> ) >. )
)  =  ( # `  f ) )
55 eqcom 2476 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  w )  =  N  <->  N  =  ( # `
 w ) )
5655biimpi 194 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  =  N  ->  N  =  ( # `  w
) )
5756ad2antll 728 . . . . . . . . . . . . . . . 16  |-  ( (
<. f ,  ( w concat  <" ( 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 2518 . . . . . . . . . . . . . . 15  |-  ( (
<. f ,  ( w concat  <" ( 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 concat  <" ( w `
 0 ) "> ) >. )
)  =  N )
591fveq2i 5867 . . . . . . . . . . . . . . . . . 18  |-  ( # `  A )  =  (
# `  ( 1st `  c ) )
6059eqeq1i 2474 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  A )  =  N  <->  ( # `  ( 1st `  c ) )  =  N )
61 fveq2 5864 . . . . . . . . . . . . . . . . . . 19  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( 1st `  c )  =  ( 1st `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )
)
6261fveq2d 5868 . . . . . . . . . . . . . . . . . 18  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( # `
 ( 1st `  c
) )  =  (
# `  ( 1st ` 
<. f ,  ( w concat  <" ( w ` 
0 ) "> ) >. ) ) )
6362eqeq1d 2469 . . . . . . . . . . . . . . . . 17  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  (
( # `  ( 1st `  c ) )  =  N  <->  ( # `  ( 1st `  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >. )
)  =  N ) )
6460, 63syl5bb 257 . . . . . . . . . . . . . . . 16  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  (
( # `  A )  =  N  <->  ( # `  ( 1st `  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >. )
)  =  N ) )
6564, 3elrab2 3263 . . . . . . . . . . . . . . 15  |-  ( <.
f ,  ( w concat  <" ( w ` 
0 ) "> ) >.  e.  C  <->  ( <. f ,  ( w concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E )  /\  ( # `
 ( 1st `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )
)  =  N ) )
6633, 58, 65sylanbrc 664 . . . . . . . . . . . . . 14  |-  ( (
<. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  C
)
6743, 45, 46syl2im 38 . . . . . . . . . . . . . . . . . . . . . 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 concat  <" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  ( # `
 f )  =  ( # `  w
) ) )
6867ad2antrl 727 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
<. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E )  ->  ( # `
 f )  =  ( # `  w
) ) )
6968imp 429 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( <. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )  -> 
( # `  f )  =  ( # `  w
) )
7069opeq2d 4220 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( <. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )  ->  <. 0 ,  ( # `  f ) >.  =  <. 0 ,  ( # `  w
) >. )
7170oveq2d 6298 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( <. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )  -> 
( ( w concat  <" (
w `  0 ) "> ) substr  <. 0 ,  ( # `  f
) >. )  =  ( ( w concat  <" (
w `  0 ) "> ) substr  <. 0 ,  ( # `  w
) >. ) )
72 simpr 461 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( <. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )  ->  <. f ,  ( w concat  <" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E ) )
73 eqeq2 2482 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  =  ( # `  w
)  ->  ( ( # `
 f )  =  N  <->  ( # `  f
)  =  ( # `  w ) ) )
7473eqcoms 2479 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
# `  w )  =  N  ->  ( (
# `  f )  =  N  <->  ( # `  f
)  =  ( # `  w ) ) )
7574imbi2d 316 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  w )  =  N  ->  ( (
<. f ,  ( w concat  <" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  ( # `
 f )  =  N )  <->  ( <. f ,  ( w concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E )  ->  ( # `
 f )  =  ( # `  w
) ) ) )
7675ad2antll 728 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
<. f ,  ( w concat  <" ( 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 concat  <" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  ( # `
 f )  =  N )  <->  ( <. f ,  ( w concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E )  ->  ( # `
 f )  =  ( # `  w
) ) ) )
7768, 76mpbird 232 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
<. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E )  ->  ( # `
 f )  =  N ) )
7877imp 429 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( <. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )  -> 
( # `  f )  =  N )
7951a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( 1st `  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >. )  =  f )
8079fveq2d 5868 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( # `
 ( 1st `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )
)  =  ( # `  f ) )
8162, 80eqtrd 2508 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( # `
 ( 1st `  c
) )  =  (
# `  f )
)
8281eqeq1d 2469 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  (
( # `  ( 1st `  c ) )  =  N  <->  ( # `  f
)  =  N ) )
8360, 82syl5bb 257 . . . . . . . . . . . . . . . . . . . . 21  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  (
( # `  A )  =  N  <->  ( # `  f
)  =  N ) )
8483, 3elrab2 3263 . . . . . . . . . . . . . . . . . . . 20  |-  ( <.
f ,  ( w concat  <" ( w ` 
0 ) "> ) >.  e.  C  <->  ( <. f ,  ( w concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E )  /\  ( # `
 f )  =  N ) )
8572, 78, 84sylanbrc 664 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( <. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )  ->  <. f ,  ( w concat  <" ( w ` 
0 ) "> ) >.  e.  C )
86 ovex 6307 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w concat  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  f
) >. )  e.  _V
8759opeq2i 4217 . . . . . . . . . . . . . . . . . . . . . 22  |-  <. 0 ,  ( # `  A
) >.  =  <. 0 ,  ( # `  ( 1st `  c ) )
>.
882, 87oveq12i 6294 . . . . . . . . . . . . . . . . . . . . 21  |-  ( B substr  <. 0 ,  ( # `  A ) >. )  =  ( ( 2nd `  c ) substr  <. 0 ,  ( # `  ( 1st `  c ) )
>. )
89 fveq2 5864 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( 2nd `  c )  =  ( 2nd `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )
)
9062opeq2d 4220 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  <. 0 ,  ( # `  ( 1st `  c ) )
>.  =  <. 0 ,  ( # `  ( 1st `  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >. )
) >. )
9189, 90oveq12d 6300 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  (
( 2nd `  c
) substr  <. 0 ,  (
# `  ( 1st `  c ) ) >.
)  =  ( ( 2nd `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. ) substr  <.
0 ,  ( # `  ( 1st `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )
) >. ) )
9249, 50op2nd 6790 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 2nd `  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >. )  =  ( w concat  <" (
w `  0 ) "> )
9351fveq2i 5867 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( # `  ( 1st `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )
)  =  ( # `  f )
9493opeq2i 4217 . . . . . . . . . . . . . . . . . . . . . . 23  |-  <. 0 ,  ( # `  ( 1st `  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >. )
) >.  =  <. 0 ,  ( # `  f
) >.
9592, 94oveq12i 6294 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 2nd `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. ) substr  <.
0 ,  ( # `  ( 1st `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )
) >. )  =  ( ( w concat  <" (
w `  0 ) "> ) substr  <. 0 ,  ( # `  f
) >. )
9691, 95syl6eq 2524 . . . . . . . . . . . . . . . . . . . . 21  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  (
( 2nd `  c
) substr  <. 0 ,  (
# `  ( 1st `  c ) ) >.
)  =  ( ( w concat  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  f
) >. ) )
9788, 96syl5eq 2520 . . . . . . . . . . . . . . . . . . . 20  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( B substr  <. 0 ,  (
# `  A ) >. )  =  ( ( w concat  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  f
) >. ) )
9897, 4fvmptg 5946 . . . . . . . . . . . . . . . . . . 19  |-  ( (
<. f ,  ( w concat  <" ( w ` 
0 ) "> ) >.  e.  C  /\  ( ( w concat  <" (
w `  0 ) "> ) substr  <. 0 ,  ( # `  f
) >. )  e.  _V )  ->  ( F `  <. f ,  ( w concat  <" ( w ` 
0 ) "> ) >. )  =  ( ( w concat  <" (
w `  0 ) "> ) substr  <. 0 ,  ( # `  f
) >. ) )
9985, 86, 98sylancl 662 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( <. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )  -> 
( F `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )  =  ( ( w concat  <" ( w ` 
0 ) "> ) substr  <. 0 ,  (
# `  f ) >. ) )
10043ad2antrl 727 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
<. f ,  ( w concat  <" ( 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 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( <. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )  -> 
( w  e. Word  V  /\  1  <_  ( # `  w ) ) )
102 simpl 457 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  w  e. Word  V )
103 wrdsymb1 12537 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  (
w `  0 )  e.  V )
104103s1cld 12572 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  <" (
w `  0 ) ">  e. Word  V )
105 eqidd 2468 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  ( # `
 w )  =  ( # `  w
) )
106 swrdccatid 12679 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  e. Word  V  /\  <" ( w ` 
0 ) ">  e. Word  V  /\  ( # `  w )  =  (
# `  w )
)  ->  ( (
w concat  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  w
) >. )  =  w )
107102, 104, 105, 106syl3anc 1228 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  (
( w concat  <" (
w `  0 ) "> ) substr  <. 0 ,  ( # `  w
) >. )  =  w )
108107eqcomd 2475 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  w  =  ( ( w concat  <" ( w ` 
0 ) "> ) substr  <. 0 ,  (
# `  w ) >. ) )
109101, 108syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( <. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )  ->  w  =  ( (
w concat  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  w
) >. ) )
11071, 99, 1093eqtr4rd 2519 . . . . . . . . . . . . . . . . 17  |-  ( ( ( <. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E ) )  ->  w  =  ( F `  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >. )
)
111110ex 434 . . . . . . . . . . . . . . . 16  |-  ( (
<. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E )  ->  w  =  ( F `  <. f ,  ( w concat  <" ( w ` 
0 ) "> ) >. ) ) )
112111adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( <. f ,  ( w concat  <" ( 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 concat  <" ( w ` 
0 ) "> ) >. )  ->  ( <. f ,  ( w concat  <" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  w  =  ( F `  <. f ,  ( w concat  <" ( w ` 
0 ) "> ) >. ) ) )
113 fveq2 5864 . . . . . . . . . . . . . . . . . 18  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( F `  c )  =  ( F `  <. f ,  ( w concat  <" ( w ` 
0 ) "> ) >. ) )
114113eqeq2d 2481 . . . . . . . . . . . . . . . . 17  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  (
w  =  ( F `
 c )  <->  w  =  ( F `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )
) )
115114imbi2d 316 . . . . . . . . . . . . . . . 16  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  (
( <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  ->  w  =  ( F `  c ) )  <->  ( <. f ,  ( w concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E )  ->  w  =  ( F `  <. f ,  ( w concat  <" ( w ` 
0 ) "> ) >. ) ) ) )
116115adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( <. f ,  ( w concat  <" ( 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 concat  <" ( w ` 
0 ) "> ) >. )  ->  (
( <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  e.  ( V ClWalks  E )  ->  w  =  ( F `  c ) )  <->  ( <. f ,  ( w concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E )  ->  w  =  ( F `  <. f ,  ( w concat  <" ( w ` 
0 ) "> ) >. ) ) ) )
117112, 116mpbird 232 . . . . . . . . . . . . . 14  |-  ( ( ( <. f ,  ( w concat  <" ( 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 concat  <" ( w ` 
0 ) "> ) >. )  ->  ( <. f ,  ( w concat  <" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  w  =  ( F `  c ) ) )
11866, 117rspcimedv 3216 . . . . . . . . . . . . 13  |-  ( (
<. f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E )  ->  E. c  e.  C  w  =  ( F `  c ) ) )
119118ex 434 . . . . . . . . . . . 12  |-  ( <.
f ,  ( w concat  <" ( 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E )  ->  E. c  e.  C  w  =  ( F `  c ) ) ) )
120119pm2.43b 50 . . . . . . . . . . 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 concat  <" (
w `  0 ) "> ) >.  e.  ( V ClWalks  E )  ->  E. c  e.  C  w  =  ( F `  c ) ) )
12132, 120syl5bi 217 . . . . . . . . . 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 concat  <" ( w ` 
0 ) "> )  ->  E. c  e.  C  w  =  ( F `  c ) ) )
122121exlimdv 1700 . . . . . . . . 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 concat  <" (
w `  0 ) "> )  ->  E. c  e.  C  w  =  ( F `  c ) ) )
123122ex 434 . . . . . . . 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 concat  <" ( w ` 
0 ) "> )  ->  E. c  e.  C  w  =  ( F `  c ) ) ) )
124123com23 78 . . . . . . 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 concat  <" ( w ` 
0 ) "> )  ->  ( ( # `  w )  =  N  ->  E. c  e.  C  w  =  ( F `  c ) ) ) )
125124impd 431 . . . . . 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 concat  <" (
w `  0 ) "> )  /\  ( # `
 w )  =  N )  ->  E. c  e.  C  w  =  ( F `  c ) ) )
12631, 125sylbid 215 . . . . 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 440 . . . 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 2878 . 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 6034 . 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 664 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 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113   <.cop 4033   class class class wbr 4447    |-> cmpt 4505   -->wf 5582   -onto->wfo 5584   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   Fincfn 7513   0cc0 9488   1c1 9489    <_ cle 9625   NN0cn0 10791   #chash 12367  Word cword 12494   concat cconcat 12496   <"cs1 12497   substr csubstr 12498   Primecprime 14069   USGrph cusg 24003   Walks cwalk 24171   ClWalks cclwlk 24420   ClWWalks cclwwlk 24421   ClWWalksN cclwwlkn 24422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12368  df-word 12502  df-lsw 12503  df-concat 12504  df-s1 12505  df-substr 12506  df-dvds 13841  df-prm 14070  df-usgra 24006  df-wlk 24181  df-clwlk 24423  df-clwwlk 24424  df-clwwlkn 24425
This theorem is referenced by:  clwlkf1oclwwlk  24524
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