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Theorem usghashecclwwlk 24511
Description: The size of every equivalence class of the equivalence relation over the set of closed walks (defined as words) with a fixed length which is a prime number equals this length (in an undirected simple graph). (Contributed by Alexander van der Vekens, 17-Jun-2018.)
Hypotheses
Ref Expression
erclwwlkn.w  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
erclwwlkn.r  |-  .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
Assertion
Ref Expression
usghashecclwwlk  |-  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  ( U  e.  ( W /.  .~  )  ->  ( # `  U
)  =  N ) )
Distinct variable groups:    t, E, u    t, N, u    n, V, t, u    t, W, u    n, N    n, W    n, E    U, n, u
Allowed substitution hints:    .~ ( u, t, n)    U( t)

Proof of Theorem usghashecclwwlk
Dummy variables  x  y  m  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 erclwwlkn.w . . . . 5  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
2 erclwwlkn.r . . . . 5  |-  .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
31, 2eclclwwlkn1 24508 . . . 4  |-  ( U  e.  ( W /.  .~  )  ->  ( U  e.  ( W /.  .~  ) 
<->  E. x  e.  W  U  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) } ) )
4 rabeq 3107 . . . . . . . . . 10  |-  ( W  =  ( ( V ClWWalksN  E ) `  N
)  ->  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) }  =  {
y  e.  ( ( V ClWWalksN  E ) `  N
)  |  E. n  e.  ( 0 ... N
) y  =  ( x cyclShift  n ) } )
51, 4mp1i 12 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  N  e.  Prime )  /\  x  e.  W )  ->  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) }  =  { y  e.  ( ( V ClWWalksN  E ) `  N )  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) } )
6 prmnn 14075 . . . . . . . . . . . 12  |-  ( N  e.  Prime  ->  N  e.  NN )
76nnnn0d 10848 . . . . . . . . . . 11  |-  ( N  e.  Prime  ->  N  e. 
NN0 )
87adantl 466 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  N  e. 
NN0 )
91eleq2i 2545 . . . . . . . . . . 11  |-  ( x  e.  W  <->  x  e.  ( ( V ClWWalksN  E ) `
 N ) )
109biimpi 194 . . . . . . . . . 10  |-  ( x  e.  W  ->  x  e.  ( ( V ClWWalksN  E ) `
 N ) )
11 clwwlknscsh 24495 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  x  e.  ( ( V ClWWalksN  E ) `  N
) )  ->  { y  e.  ( ( V ClWWalksN  E ) `  N
)  |  E. n  e.  ( 0 ... N
) y  =  ( x cyclShift  n ) }  =  { y  e. Word  V  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) } )
128, 10, 11syl2an 477 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  N  e.  Prime )  /\  x  e.  W )  ->  { y  e.  ( ( V ClWWalksN  E ) `  N )  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) }  =  {
y  e. Word  V  |  E. n  e.  (
0 ... N ) y  =  ( x cyclShift  n
) } )
135, 12eqtrd 2508 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  N  e.  Prime )  /\  x  e.  W )  ->  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) }  =  { y  e. Word  V  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) } )
1413eqeq2d 2481 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  N  e.  Prime )  /\  x  e.  W )  ->  ( U  =  {
y  e.  W  |  E. n  e.  (
0 ... N ) y  =  ( x cyclShift  n
) }  <->  U  =  { y  e. Word  V  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) } ) )
156adantl 466 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  N  e.  NN )
16 clwwlknprop 24448 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 x )  =  N ) ) )
17 simpll 753 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  /\  N  e.  NN )  ->  x  e. Word  V )
18 elnnne0 10805 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
19 eqeq1 2471 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  =  ( # `  x
)  ->  ( N  =  0  <->  ( # `  x
)  =  0 ) )
2019eqcoms 2479 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  x )  =  N  ->  ( N  =  0  <->  ( # `  x
)  =  0 ) )
2120adantl 466 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  -> 
( N  =  0  <-> 
( # `  x )  =  0 ) )
22 hasheq0 12397 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  e. Word  V  ->  (
( # `  x )  =  0  <->  x  =  (/) ) )
2321, 22sylan9bbr 700 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( N  =  0  <->  x  =  (/) ) )
2423necon3bid 2725 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( N  =/=  0  <->  x  =/=  (/) ) )
2524biimpcd 224 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  =/=  0  ->  (
( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  x  =/=  (/) ) )
2625adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN0  /\  N  =/=  0 )  -> 
( ( x  e. Word  V  /\  ( N  e. 
NN0  /\  ( # `  x
)  =  N ) )  ->  x  =/=  (/) ) )
2718, 26sylbi 195 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  (
( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  x  =/=  (/) ) )
2827impcom 430 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  /\  N  e.  NN )  ->  x  =/=  (/) )
29 eqcom 2476 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  x )  =  N  <->  N  =  ( # `
 x ) )
3029biimpi 194 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  x )  =  N  ->  N  =  ( # `  x
) )
3130adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  ->  N  =  ( # `  x
) )
3231ad2antlr 726 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  /\  N  e.  NN )  ->  N  =  ( # `  x
) )
3317, 28, 323jca 1176 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  /\  N  e.  NN )  ->  (
x  e. Word  V  /\  x  =/=  (/)  /\  N  =  ( # `  x
) ) )
3433ex 434 . . . . . . . . . . . . . . . 16  |-  ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( N  e.  NN  ->  ( x  e. Word  V  /\  x  =/=  (/)  /\  N  =  (
# `  x )
) ) )
35343adant1 1014 . . . . . . . . . . . . . . 15  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( N  e.  NN  ->  ( x  e. Word  V  /\  x  =/=  (/)  /\  N  =  (
# `  x )
) ) )
3616, 35syl 16 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( N  e.  NN  ->  ( x  e. Word  V  /\  x  =/=  (/)  /\  N  =  (
# `  x )
) ) )
3736com12 31 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  (
x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( x  e. Word  V  /\  x  =/=  (/)  /\  N  =  (
# `  x )
) ) )
389, 37syl5bi 217 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
x  e.  W  -> 
( x  e. Word  V  /\  x  =/=  (/)  /\  N  =  ( # `  x
) ) ) )
3915, 38syl 16 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  ( x  e.  W  ->  (
x  e. Word  V  /\  x  =/=  (/)  /\  N  =  ( # `  x
) ) ) )
4039imp 429 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  N  e.  Prime )  /\  x  e.  W )  ->  ( x  e. Word  V  /\  x  =/=  (/)  /\  N  =  ( # `  x
) ) )
41 scshwfzeqfzo 12753 . . . . . . . . . 10  |-  ( ( x  e. Word  V  /\  x  =/=  (/)  /\  N  =  ( # `  x
) )  ->  { y  e. Word  V  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) }  =  {
y  e. Word  V  |  E. n  e.  (
0..^ N ) y  =  ( x cyclShift  n
) } )
4240, 41syl 16 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  N  e.  Prime )  /\  x  e.  W )  ->  { y  e. Word  V  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) }  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n
) } )
4342eqeq2d 2481 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  N  e.  Prime )  /\  x  e.  W )  ->  ( U  =  {
y  e. Word  V  |  E. n  e.  (
0 ... N ) y  =  ( x cyclShift  n
) }  <->  U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n
) } ) )
44 fveq2 5864 . . . . . . . . . . . . . . 15  |-  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x ) ) y  =  ( x cyclShift  n
) }  ->  ( # `
 U )  =  ( # `  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } ) )
45 simprl 755 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e. Word  V  /\  x  e.  (
( V ClWWalksN  E ) `  ( # `  x ) ) )  /\  ( V USGrph  E  /\  ( # `  x )  e.  Prime ) )  ->  V USGrph  E )
46 prmuz2 14090 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  x )  e.  Prime  ->  ( # `  x
)  e.  ( ZZ>= ` 
2 ) )
4746adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( V USGrph  E  /\  ( # `
 x )  e. 
Prime )  ->  ( # `  x )  e.  (
ZZ>= `  2 ) )
4847adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e. Word  V  /\  x  e.  (
( V ClWWalksN  E ) `  ( # `  x ) ) )  /\  ( V USGrph  E  /\  ( # `  x )  e.  Prime ) )  ->  ( # `  x
)  e.  ( ZZ>= ` 
2 ) )
49 simplr 754 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e. Word  V  /\  x  e.  (
( V ClWWalksN  E ) `  ( # `  x ) ) )  /\  ( V USGrph  E  /\  ( # `  x )  e.  Prime ) )  ->  x  e.  ( ( V ClWWalksN  E ) `
 ( # `  x
) ) )
50 usg2cwwkdifex 24497 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  ( # `
 x )  e.  ( ZZ>= `  2 )  /\  x  e.  (
( V ClWWalksN  E ) `  ( # `  x ) ) )  ->  E. i  e.  ( 0..^ ( # `  x ) ) ( x `  i )  =/=  ( x ` 
0 ) )
5145, 48, 49, 50syl3anc 1228 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e. Word  V  /\  x  e.  (
( V ClWWalksN  E ) `  ( # `  x ) ) )  /\  ( V USGrph  E  /\  ( # `  x )  e.  Prime ) )  ->  E. i  e.  ( 0..^ ( # `  x ) ) ( x `  i )  =/=  ( x ` 
0 ) )
52 oveq2 6290 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  =  m  ->  (
x cyclShift  n )  =  ( x cyclShift  m ) )
5352eqeq2d 2481 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  =  m  ->  (
y  =  ( x cyclShift  n )  <->  y  =  ( x cyclShift  m ) ) )
5453cbvrexv 3089 . . . . . . . . . . . . . . . . . . . 20  |-  ( E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n )  <->  E. m  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  m )
)
55 eqeq1 2471 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  =  u  ->  (
y  =  ( x cyclShift  m )  <->  u  =  ( x cyclShift  m ) ) )
56 eqcom 2476 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  ( x cyclShift  m
)  <->  ( x cyclShift  m
)  =  u )
5755, 56syl6bb 261 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  u  ->  (
y  =  ( x cyclShift  m )  <->  ( x cyclShift  m )  =  u ) )
5857rexbidv 2973 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  u  ->  ( E. m  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  m )  <->  E. m  e.  ( 0..^ ( # `  x
) ) ( x cyclShift  m )  =  u ) )
5954, 58syl5bb 257 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  u  ->  ( E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n )  <->  E. m  e.  ( 0..^ ( # `  x
) ) ( x cyclShift  m )  =  u ) )
6059cbvrabv 3112 . . . . . . . . . . . . . . . . . 18  |-  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) }  =  { u  e. Word  V  |  E. m  e.  ( 0..^ ( # `  x ) ) ( x cyclShift  m )  =  u }
6160cshwshashnsame 14442 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e. Word  V  /\  ( # `  x )  e.  Prime )  ->  ( E. i  e.  (
0..^ ( # `  x
) ) ( x `
 i )  =/=  ( x `  0
)  ->  ( # `  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  =  (
# `  x )
) )
6261ad2ant2rl 748 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e. Word  V  /\  x  e.  (
( V ClWWalksN  E ) `  ( # `  x ) ) )  /\  ( V USGrph  E  /\  ( # `  x )  e.  Prime ) )  ->  ( E. i  e.  ( 0..^ ( # `  x
) ) ( x `
 i )  =/=  ( x `  0
)  ->  ( # `  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  =  (
# `  x )
) )
6351, 62mpd 15 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e. Word  V  /\  x  e.  (
( V ClWWalksN  E ) `  ( # `  x ) ) )  /\  ( V USGrph  E  /\  ( # `  x )  e.  Prime ) )  ->  ( # `  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  =  (
# `  x )
)
6444, 63sylan9eqr 2530 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. Word  V  /\  x  e.  ( ( V ClWWalksN  E ) `  ( # `  x
) ) )  /\  ( V USGrph  E  /\  ( # `
 x )  e. 
Prime ) )  /\  U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  ->  ( # `
 U )  =  ( # `  x
) )
6564exp41 610 . . . . . . . . . . . . 13  |-  ( x  e. Word  V  ->  (
x  e.  ( ( V ClWWalksN  E ) `  ( # `
 x ) )  ->  ( ( V USGrph  E  /\  ( # `  x
)  e.  Prime )  ->  ( U  =  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) }  ->  ( # `  U
)  =  ( # `  x ) ) ) ) )
66653ad2ant2 1018 . . . . . . . . . . . 12  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( x  e.  ( ( V ClWWalksN  E ) `
 ( # `  x
) )  ->  (
( V USGrph  E  /\  ( # `  x )  e.  Prime )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x ) ) y  =  ( x cyclShift  n
) }  ->  ( # `
 U )  =  ( # `  x
) ) ) ) )
67 fveq2 5864 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  x
)  ->  ( ( V ClWWalksN  E ) `  N
)  =  ( ( V ClWWalksN  E ) `  ( # `
 x ) ) )
6867eleq2d 2537 . . . . . . . . . . . . . . . 16  |-  ( N  =  ( # `  x
)  ->  ( x  e.  ( ( V ClWWalksN  E ) `
 N )  <->  x  e.  ( ( V ClWWalksN  E ) `
 ( # `  x
) ) ) )
69 eleq1 2539 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  x
)  ->  ( N  e.  Prime 
<->  ( # `  x
)  e.  Prime )
)
7069anbi2d 703 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  x
)  ->  ( ( V USGrph  E  /\  N  e. 
Prime )  <->  ( V USGrph  E  /\  ( # `  x
)  e.  Prime )
) )
71 oveq2 6290 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  =  ( # `  x
)  ->  ( 0..^ N )  =  ( 0..^ ( # `  x
) ) )
7271rexeqdv 3065 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  =  ( # `  x
)  ->  ( E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n )  <->  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n )
) )
7372rabbidv 3105 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  ( # `  x
)  ->  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n ) }  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )
7473eqeq2d 2481 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  x
)  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n
) }  <->  U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } ) )
75 eqeq2 2482 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  x
)  ->  ( ( # `
 U )  =  N  <->  ( # `  U
)  =  ( # `  x ) ) )
7674, 75imbi12d 320 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  x
)  ->  ( ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n ) }  ->  (
# `  U )  =  N )  <->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) }  ->  ( # `  U
)  =  ( # `  x ) ) ) )
7770, 76imbi12d 320 . . . . . . . . . . . . . . . 16  |-  ( N  =  ( # `  x
)  ->  ( (
( V USGrph  E  /\  N  e.  Prime )  -> 
( U  =  {
y  e. Word  V  |  E. n  e.  (
0..^ N ) y  =  ( x cyclShift  n
) }  ->  ( # `
 U )  =  N ) )  <->  ( ( V USGrph  E  /\  ( # `  x )  e.  Prime )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) }  ->  ( # `  U
)  =  ( # `  x ) ) ) ) )
7868, 77imbi12d 320 . . . . . . . . . . . . . . 15  |-  ( N  =  ( # `  x
)  ->  ( (
x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n ) }  ->  (
# `  U )  =  N ) ) )  <-> 
( x  e.  ( ( V ClWWalksN  E ) `  ( # `  x
) )  ->  (
( V USGrph  E  /\  ( # `  x )  e.  Prime )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x ) ) y  =  ( x cyclShift  n
) }  ->  ( # `
 U )  =  ( # `  x
) ) ) ) ) )
7978eqcoms 2479 . . . . . . . . . . . . . 14  |-  ( (
# `  x )  =  N  ->  ( ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n ) }  ->  (
# `  U )  =  N ) ) )  <-> 
( x  e.  ( ( V ClWWalksN  E ) `  ( # `  x
) )  ->  (
( V USGrph  E  /\  ( # `  x )  e.  Prime )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x ) ) y  =  ( x cyclShift  n
) }  ->  ( # `
 U )  =  ( # `  x
) ) ) ) ) )
8079adantl 466 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  -> 
( ( x  e.  ( ( V ClWWalksN  E ) `
 N )  -> 
( ( V USGrph  E  /\  N  e.  Prime )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n
) }  ->  ( # `
 U )  =  N ) ) )  <-> 
( x  e.  ( ( V ClWWalksN  E ) `  ( # `  x
) )  ->  (
( V USGrph  E  /\  ( # `  x )  e.  Prime )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x ) ) y  =  ( x cyclShift  n
) }  ->  ( # `
 U )  =  ( # `  x
) ) ) ) ) )
81803ad2ant3 1019 . . . . . . . . . . . 12  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( (
x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n ) }  ->  (
# `  U )  =  N ) ) )  <-> 
( x  e.  ( ( V ClWWalksN  E ) `  ( # `  x
) )  ->  (
( V USGrph  E  /\  ( # `  x )  e.  Prime )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x ) ) y  =  ( x cyclShift  n
) }  ->  ( # `
 U )  =  ( # `  x
) ) ) ) ) )
8266, 81mpbird 232 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( x  e.  ( ( V ClWWalksN  E ) `
 N )  -> 
( ( V USGrph  E  /\  N  e.  Prime )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n
) }  ->  ( # `
 U )  =  N ) ) ) )
8316, 82mpcom 36 . . . . . . . . . 10  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n ) }  ->  (
# `  U )  =  N ) ) )
849, 83sylbi 195 . . . . . . . . 9  |-  ( x  e.  W  ->  (
( V USGrph  E  /\  N  e.  Prime )  -> 
( U  =  {
y  e. Word  V  |  E. n  e.  (
0..^ N ) y  =  ( x cyclShift  n
) }  ->  ( # `
 U )  =  N ) ) )
8584impcom 430 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  N  e.  Prime )  /\  x  e.  W )  ->  ( U  =  {
y  e. Word  V  |  E. n  e.  (
0..^ N ) y  =  ( x cyclShift  n
) }  ->  ( # `
 U )  =  N ) )
8643, 85sylbid 215 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  N  e.  Prime )  /\  x  e.  W )  ->  ( U  =  {
y  e. Word  V  |  E. n  e.  (
0 ... N ) y  =  ( x cyclShift  n
) }  ->  ( # `
 U )  =  N ) )
8714, 86sylbid 215 . . . . . 6  |-  ( ( ( V USGrph  E  /\  N  e.  Prime )  /\  x  e.  W )  ->  ( U  =  {
y  e.  W  |  E. n  e.  (
0 ... N ) y  =  ( x cyclShift  n
) }  ->  ( # `
 U )  =  N ) )
8887rexlimdva 2955 . . . . 5  |-  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  ( E. x  e.  W  U  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) }  ->  (
# `  U )  =  N ) )
8988com12 31 . . . 4  |-  ( E. x  e.  W  U  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) }  ->  ( ( V USGrph  E  /\  N  e.  Prime )  -> 
( # `  U )  =  N ) )
903, 89syl6bi 228 . . 3  |-  ( U  e.  ( W /.  .~  )  ->  ( U  e.  ( W /.  .~  )  ->  ( ( V USGrph  E  /\  N  e.  Prime )  ->  ( # `  U
)  =  N ) ) )
9190pm2.43i 47 . 2  |-  ( U  e.  ( W /.  .~  )  ->  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  ( # `  U )  =  N ) )
9291com12 31 1  |-  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  ( U  e.  ( W /.  .~  )  ->  ( # `  U
)  =  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   {crab 2818   _Vcvv 3113   (/)c0 3785   class class class wbr 4447   {copab 4504   ` cfv 5586  (class class class)co 6282   /.cqs 7307   0cc0 9488   NNcn 10532   2c2 10581   NN0cn0 10791   ZZ>=cuz 11078   ...cfz 11668  ..^cfzo 11788   #chash 12369  Word cword 12496   cyclShift ccsh 12718   Primecprime 14072   USGrph cusg 24006   ClWWalksN cclwwlkn 24425
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-inf2 8054  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  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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-disj 4418  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-se 4839  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-isom 5595  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-ec 7310  df-qs 7314  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  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-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-hash 12370  df-word 12504  df-lsw 12505  df-concat 12506  df-substr 12508  df-reps 12511  df-csh 12719  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-dvds 13844  df-gcd 14000  df-prm 14073  df-phi 14151  df-usgra 24009  df-clwwlk 24427  df-clwwlkn 24428
This theorem is referenced by:  hashclwwlkn  24512
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