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Theorem clwlkfoclwwlk 30523
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 30522 . 2  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  F : C
--> ( ( V ClWWalksN  E ) `
 N ) )
6 clwwlknprop 30440 . . . . 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 967 . . . . . . . . . . 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 1007 . . . . . . . . 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 30437 . . . . . . . 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 991 . . . . . . . . . 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 995 . . . . . . . . . 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 2503 . . . . . . . . . . . . . . . 16  |-  ( (
# `  w )  =  N  ->  ( (
# `  w )  e.  Prime 
<->  N  e.  Prime )
)
19 prmnn 13771 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  w )  e.  Prime  ->  ( # `  w
)  e.  NN )
2019nnge1d 10369 . . . . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . 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 1011 . . . . . . . . . . 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 30457 . . . . . . . . . 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 1218 . . . . . . . . 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 4298 . . . . . . . . . . 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 13771 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  Prime  ->  N  e.  NN )
3534nnge1d 10369 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  Prime  ->  1  <_  N )
36353ad2ant3 1011 . . . . . . . . . . . . . . . . . . . . . 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 4301 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
# `  w )  =  N  ->  ( 1  <_  ( # `  w
)  <->  1  <_  N
) )
3938adantl 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( N  e.  NN0  /\  ( # `  w )  =  N )  -> 
( 1  <_  ( # `
 w )  <->  1  <_  N ) )
40393ad2ant3 1011 . . . . . . . . . . . . . . . . . . . . . 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 30428 . . . . . . . . . . . . . . . . . 18  |-  ( <.
f ,  ( w concat  <" ( w ` 
0 ) "> ) >.  e.  ( V ClWalks  E )  ->  <. f ,  ( w concat  <" (
w `  0 ) "> ) >.  e.  ( V Walks  E ) )
46 wlkp1lenfislenp 30517 . . . . . . . . . . . . . . . . . 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 2980 . . . . . . . . . . . . . . . . . . . 20  |-  f  e. 
_V
50 ovex 6121 . . . . . . . . . . . . . . . . . . . 20  |-  ( w concat  <" ( w ` 
0 ) "> )  e.  _V
5149, 50op1st 6590 . . . . . . . . . . . . . . . . . . 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 5700 . . . . . . . . . . . . . . . . 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 2445 . . . . . . . . . . . . . . . . . 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 2485 . . . . . . . . . . . . . . 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 5699 . . . . . . . . . . . . . . . . . 18  |-  ( # `  A )  =  (
# `  ( 1st `  c ) )
6059eqeq1i 2450 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  A )  =  N  <->  ( # `  ( 1st `  c ) )  =  N )
61 fveq2 5696 . . . . . . . . . . . . . . . . . . 19  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( 1st `  c )  =  ( 1st `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )
)
6261fveq2d 5700 . . . . . . . . . . . . . . . . . 18  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( # `
 ( 1st `  c
) )  =  (
# `  ( 1st ` 
<. f ,  ( w concat  <" ( w ` 
0 ) "> ) >. ) ) )
6362eqeq1d 2451 . . . . . . . . . . . . . . . . 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 3124 . . . . . . . . . . . . . . 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 4071 . . . . . . . . . . . . . . . . . . 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 6112 . . . . . . . . . . . . . . . . . 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 2452 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  =  ( # `  w
)  ->  ( ( # `
 f )  =  N  <->  ( # `  f
)  =  ( # `  w ) ) )
7473eqcoms 2446 . . . . . . . . . . . . . . . . . . . . . . . 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 5700 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( # `
 ( 1st `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )
)  =  ( # `  f ) )
8162, 80eqtrd 2475 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( # `
 ( 1st `  c
) )  =  (
# `  f )
)
8281eqeq1d 2451 . . . . . . . . . . . . . . . . . . . . . 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 3124 . . . . . . . . . . . . . . . . . . . 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 6121 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w concat  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  f
) >. )  e.  _V
8759opeq2i 4068 . . . . . . . . . . . . . . . . . . . . . 22  |-  <. 0 ,  ( # `  A
) >.  =  <. 0 ,  ( # `  ( 1st `  c ) )
>.
882, 87oveq12i 6108 . . . . . . . . . . . . . . . . . . . . 21  |-  ( B substr  <. 0 ,  ( # `  A ) >. )  =  ( ( 2nd `  c ) substr  <. 0 ,  ( # `  ( 1st `  c ) )
>. )
89 fveq2 5696 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( 2nd `  c )  =  ( 2nd `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )
)
9062opeq2d 4071 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  <. 0 ,  ( # `  ( 1st `  c ) )
>.  =  <. 0 ,  ( # `  ( 1st `  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >. )
) >. )
9189, 90oveq12d 6114 . . . . . . . . . . . . . . . . . . . . . 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 6591 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 2nd `  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >. )  =  ( w concat  <" (
w `  0 ) "> )
9351fveq2i 5699 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( # `  ( 1st `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )
)  =  ( # `  f )
9493opeq2i 4068 . . . . . . . . . . . . . . . . . . . . . . 23  |-  <. 0 ,  ( # `  ( 1st `  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >. )
) >.  =  <. 0 ,  ( # `  f
) >.
9592, 94oveq12i 6108 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 2nd `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. ) substr  <.
0 ,  ( # `  ( 1st `  <. f ,  ( w concat  <" (
w `  0 ) "> ) >. )
) >. )  =  ( ( w concat  <" (
w `  0 ) "> ) substr  <. 0 ,  ( # `  f
) >. )
9691, 95syl6eq 2491 . . . . . . . . . . . . . . . . . . . . 21  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  (
( 2nd `  c
) substr  <. 0 ,  (
# `  ( 1st `  c ) ) >.
)  =  ( ( w concat  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  f
) >. ) )
9788, 96syl5eq 2487 . . . . . . . . . . . . . . . . . . . 20  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( B substr  <. 0 ,  (
# `  A ) >. )  =  ( ( w concat  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  f
) >. ) )
9897, 4fvmptg 5777 . . . . . . . . . . . . . . . . . . 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 12267 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  (
w `  0 )  e.  V )
104103s1cld 12299 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  <" (
w `  0 ) ">  e. Word  V )
105 eqidd 2444 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  ( # `
 w )  =  ( # `  w
) )
106 swrdccatid 12393 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( w  e. Word  V  /\  <" ( w ` 
0 ) ">  e. Word  V  /\  ( # `  w )  =  (
# `  w )
)  ->  ( (
w concat  <" ( w `
 0 ) "> ) substr  <. 0 ,  ( # `  w
) >. )  =  w )
107102, 104, 105, 106syl3anc 1218 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w  e. Word  V  /\  1  <_  ( # `  w
) )  ->  (
( w concat  <" (
w `  0 ) "> ) substr  <. 0 ,  ( # `  w
) >. )  =  w )
108107eqcomd 2448 . . . . . . . . . . . . . . . . . . 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 2486 . . . . . . . . . . . . . . . . 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 5696 . . . . . . . . . . . . . . . . . 18  |-  ( c  =  <. f ,  ( w concat  <" ( w `
 0 ) "> ) >.  ->  ( F `  c )  =  ( F `  <. f ,  ( w concat  <" ( w ` 
0 ) "> ) >. ) )
114113eqeq2d 2454 . . . . . . . . . . . . . . . . 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 3080 . . . . . . . . . . . . 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 1690 . . . . . . . . 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 2804 . 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 5863 . 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 965    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2720   E.wrex 2721   {crab 2724   _Vcvv 2977   <.cop 3888   class class class wbr 4297    e. cmpt 4355   -->wf 5419   -onto->wfo 5421   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581   Fincfn 7315   0cc0 9287   1c1 9288    <_ cle 9424   NN0cn0 10584   #chash 12108  Word cword 12226   concat cconcat 12228   <"cs1 12229   substr csubstr 12230   Primecprime 13768   USGrph cusg 23269   Walks cwalk 23410   ClWalks cclwlk 30417   ClWWalks cclwwlk 30418   ClWWalksN cclwwlkn 30419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-hash 12109  df-word 12234  df-lsw 12235  df-concat 12236  df-s1 12237  df-substr 12238  df-dvds 13541  df-prm 13769  df-usgra 23271  df-wlk 23420  df-clwlk 30420  df-clwwlk 30421  df-clwwlkn 30422
This theorem is referenced by:  clwlkf1oclwwlk  30529
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