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Theorem usghashecclwwlk 30434
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 30431 . . . 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 2964 . . . . . . . . . 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 13762 . . . . . . . . . . . 12  |-  ( N  e.  Prime  ->  N  e.  NN )
76nnnn0d 10632 . . . . . . . . . . 11  |-  ( N  e.  Prime  ->  N  e. 
NN0 )
87adantl 463 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  N  e. 
NN0 )
91eleq2i 2505 . . . . . . . . . . 11  |-  ( x  e.  W  <->  x  e.  ( ( V ClWWalksN  E ) `
 N ) )
109biimpi 194 . . . . . . . . . 10  |-  ( x  e.  W  ->  x  e.  ( ( V ClWWalksN  E ) `
 N ) )
11 Lemma2 30418 . . . . . . . . . 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 474 . . . . . . . . 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 2473 . . . . . . . 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 2452 . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  N  e.  NN )
16 clwwlknprop 30360 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 x )  =  N ) ) )
17 simpll 748 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  /\  N  e.  NN )  ->  x  e. Word  V )
18 elnnne0 10589 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
19 eqeq1 2447 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  =  ( # `  x
)  ->  ( N  =  0  <->  ( # `  x
)  =  0 ) )
2019eqcoms 2444 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  x )  =  N  ->  ( N  =  0  <->  ( # `  x
)  =  0 ) )
2120adantl 463 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  -> 
( N  =  0  <-> 
( # `  x )  =  0 ) )
22 hasheq0 12127 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  e. Word  V  ->  (
( # `  x )  =  0  <->  x  =  (/) ) )
2321, 22sylan9bbr 695 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( N  =  0  <->  x  =  (/) ) )
2423necon3bid 2641 . . . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  x )  =  N  <->  N  =  ( # `
 x ) )
3029biimpi 194 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  x )  =  N  ->  N  =  ( # `  x
) )
3130adantl 463 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  ->  N  =  ( # `  x
) )
3231ad2antlr 721 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  /\  N  e.  NN )  ->  N  =  ( # `  x
) )
3317, 28, 323jca 1163 . . . . . . . . . . . . . . . . 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 1001 . . . . . . . . . . . . . . 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 30417 . . . . . . . . . 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 2452 . . . . . . . 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 5688 . . . . . . . . . . . . . . 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 750 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e. Word  V  /\  x  e.  (
( V ClWWalksN  E ) `  ( # `  x ) ) )  /\  ( V USGrph  E  /\  ( # `  x )  e.  Prime ) )  ->  V USGrph  E )
46 prmuz2 13777 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  x )  e.  Prime  ->  ( # `  x
)  e.  ( ZZ>= ` 
2 ) )
4746adantl 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( V USGrph  E  /\  ( # `
 x )  e. 
Prime )  ->  ( # `  x )  e.  (
ZZ>= `  2 ) )
4847adantl 463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e. Word  V  /\  x  e.  (
( V ClWWalksN  E ) `  ( # `  x ) ) )  /\  ( V USGrph  E  /\  ( # `  x )  e.  Prime ) )  ->  ( # `  x
)  e.  ( ZZ>= ` 
2 ) )
49 simplr 749 . . . . . . . . . . . . . . . . 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 30420 . . . . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  =  m  ->  (
x cyclShift  n )  =  ( x cyclShift  m ) )
5352eqeq2d 2452 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  =  m  ->  (
y  =  ( x cyclShift  n )  <->  y  =  ( x cyclShift  m ) ) )
5453cbvrexv 2946 . . . . . . . . . . . . . . . . . . . 20  |-  ( E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n )  <->  E. m  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  m )
)
55 eqeq1 2447 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  =  u  ->  (
y  =  ( x cyclShift  m )  <->  u  =  ( x cyclShift  m ) ) )
56 eqcom 2443 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( u  =  ( x cyclShift  m
)  <->  ( x cyclShift  m
)  =  u )
5755, 56syl6bb 261 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  u  ->  (
y  =  ( x cyclShift  m )  <->  ( x cyclShift  m )  =  u ) )
5857rexbidv 2734 . . . . . . . . . . . . . . . . . . . 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 2969 . . . . . . . . . . . . . . . . . 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 14126 . . . . . . . . . . . . . . . . 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 743 . . . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . . 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 607 . . . . . . . . . . . . 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 1005 . . . . . . . . . . . 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 5688 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  x
)  ->  ( ( V ClWWalksN  E ) `  N
)  =  ( ( V ClWWalksN  E ) `  ( # `
 x ) ) )
6867eleq2d 2508 . . . . . . . . . . . . . . . 16  |-  ( N  =  ( # `  x
)  ->  ( x  e.  ( ( V ClWWalksN  E ) `
 N )  <->  x  e.  ( ( V ClWWalksN  E ) `
 ( # `  x
) ) ) )
69 eleq1 2501 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  x
)  ->  ( N  e.  Prime 
<->  ( # `  x
)  e.  Prime )
)
7069anbi2d 698 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  x
)  ->  ( ( V USGrph  E  /\  N  e. 
Prime )  <->  ( V USGrph  E  /\  ( # `  x
)  e.  Prime )
) )
71 oveq2 6098 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  =  ( # `  x
)  ->  ( 0..^ N )  =  ( 0..^ ( # `  x
) ) )
7271rexeqdv 2922 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  =  ( # `  x
)  ->  ( E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n )  <->  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n )
) )
7372rabbidv 2962 . . . . . . . . . . . . . . . . . . 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 2452 . . . . . . . . . . . . . . . . . 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 2450 . . . . . . . . . . . . . . . . . 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 2444 . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . 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 1006 . . . . . . . . . . . 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 2839 . . . . 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714   {crab 2717   _Vcvv 2970   (/)c0 3634   class class class wbr 4289   {copab 4346   ` cfv 5415  (class class class)co 6090   /.cqs 7096   0cc0 9278   NNcn 10318   2c2 10367   NN0cn0 10575   ZZ>=cuz 10857   ...cfz 11433  ..^cfzo 11544   #chash 12099  Word cword 12217   cyclShift ccsh 12421   Primecprime 13759   USGrph cusg 23199   ClWWalksN cclwwlkn 30339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-disj 4260  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-hash 12100  df-word 12225  df-lsw 12226  df-concat 12227  df-substr 12229  df-reps 12232  df-csh 12422  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-dvds 13532  df-gcd 13687  df-prm 13760  df-phi 13837  df-usgra 23201  df-clwwlk 30341  df-clwwlkn 30342
This theorem is referenced by:  hashclwwlkn  30435
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