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Theorem hashecclwwlkn1 30513
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 is 1 or equals this length. (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
hashecclwwlkn1  |-  ( ( N  e.  Prime  /\  U  e.  ( W /.  .~  ) )  ->  (
( # `  U )  =  1  \/  ( # `
 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 hashecclwwlkn1
Dummy variables  x  y  m 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 30511 . . . 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 2971 . . . . . . . . . 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  |-  ( ( 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 13771 . . . . . . . . . . 11  |-  ( N  e.  Prime  ->  N  e.  NN )
76nnnn0d 10641 . . . . . . . . . 10  |-  ( N  e.  Prime  ->  N  e. 
NN0 )
81eleq2i 2507 . . . . . . . . . . 11  |-  ( x  e.  W  <->  x  e.  ( ( V ClWWalksN  E ) `
 N ) )
98biimpi 194 . . . . . . . . . 10  |-  ( x  e.  W  ->  x  e.  ( ( V ClWWalksN  E ) `
 N ) )
10 Lemma2 30498 . . . . . . . . . 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 ) } )
117, 9, 10syl2an 477 . . . . . . . . 9  |-  ( ( 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 ) } )
125, 11eqtrd 2475 . . . . . . . 8  |-  ( ( 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
) } )
1312eqeq2d 2454 . . . . . . 7  |-  ( ( 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 ) } ) )
14 clwwlknprop 30440 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `
 x )  =  N ) ) )
15 simpll 753 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  /\  N  e.  NN )  ->  x  e. Word  V )
16 elnnne0 10598 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
17 eqeq1 2449 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  =  ( # `  x
)  ->  ( N  =  0  <->  ( # `  x
)  =  0 ) )
1817eqcoms 2446 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  x )  =  N  ->  ( N  =  0  <->  ( # `  x
)  =  0 ) )
1918adantl 466 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  -> 
( N  =  0  <-> 
( # `  x )  =  0 ) )
20 hasheq0 12136 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  e. Word  V  ->  (
( # `  x )  =  0  <->  x  =  (/) ) )
2119, 20sylan9bbr 700 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( N  =  0  <->  x  =  (/) ) )
2221necon3bid 2648 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( N  =/=  0  <->  x  =/=  (/) ) )
2322biimpcd 224 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  =/=  0  ->  (
( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  x  =/=  (/) ) )
2423adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN0  /\  N  =/=  0 )  -> 
( ( x  e. Word  V  /\  ( N  e. 
NN0  /\  ( # `  x
)  =  N ) )  ->  x  =/=  (/) ) )
2516, 24sylbi 195 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  (
( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  x  =/=  (/) ) )
2625impcom 430 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  /\  N  e.  NN )  ->  x  =/=  (/) )
27 eqcom 2445 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  x )  =  N  <->  N  =  ( # `
 x ) )
2827biimpi 194 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  x )  =  N  ->  N  =  ( # `  x
) )
2928adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  ->  N  =  ( # `  x
) )
3029ad2antlr 726 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  /\  N  e.  NN )  ->  N  =  ( # `  x
) )
3115, 26, 303jca 1168 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  /\  N  e.  NN )  ->  (
x  e. Word  V  /\  x  =/=  (/)  /\  N  =  ( # `  x
) ) )
3231ex 434 . . . . . . . . . . . . . . . 16  |-  ( ( x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( N  e.  NN  ->  ( x  e. Word  V  /\  x  =/=  (/)  /\  N  =  (
# `  x )
) ) )
33323adant1 1006 . . . . . . . . . . . . . . 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 )
) ) )
3414, 33syl 16 . . . . . . . . . . . . . 14  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( N  e.  NN  ->  ( x  e. Word  V  /\  x  =/=  (/)  /\  N  =  (
# `  x )
) ) )
3534com12 31 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  (
x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( x  e. Word  V  /\  x  =/=  (/)  /\  N  =  (
# `  x )
) ) )
368, 35syl5bi 217 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  (
x  e.  W  -> 
( x  e. Word  V  /\  x  =/=  (/)  /\  N  =  ( # `  x
) ) ) )
376, 36syl 16 . . . . . . . . . . 11  |-  ( N  e.  Prime  ->  ( x  e.  W  ->  (
x  e. Word  V  /\  x  =/=  (/)  /\  N  =  ( # `  x
) ) ) )
3837imp 429 . . . . . . . . . 10  |-  ( ( N  e.  Prime  /\  x  e.  W )  ->  (
x  e. Word  V  /\  x  =/=  (/)  /\  N  =  ( # `  x
) ) )
39 scshwfzeqfzo 30497 . . . . . . . . . 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
) } )
4038, 39syl 16 . . . . . . . . 9  |-  ( ( 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
) } )
4140eqeq2d 2454 . . . . . . . 8  |-  ( ( 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
) } ) )
42 oveq2 6104 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( n  =  m  ->  (
x cyclShift  n )  =  ( x cyclShift  m ) )
4342eqeq2d 2454 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  =  m  ->  (
y  =  ( x cyclShift  n )  <->  y  =  ( x cyclShift  m ) ) )
4443cbvrexv 2953 . . . . . . . . . . . . . . . . . . . . 21  |-  ( E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n )  <->  E. m  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  m )
)
45 eqeq1 2449 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  =  u  ->  (
y  =  ( x cyclShift  m )  <->  u  =  ( x cyclShift  m ) ) )
46 eqcom 2445 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( u  =  ( x cyclShift  m
)  <->  ( x cyclShift  m
)  =  u )
4745, 46syl6bb 261 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  =  u  ->  (
y  =  ( x cyclShift  m )  <->  ( x cyclShift  m )  =  u ) )
4847rexbidv 2741 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  u  ->  ( E. m  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  m )  <->  E. m  e.  ( 0..^ ( # `  x
) ) ( x cyclShift  m )  =  u ) )
4944, 48syl5bb 257 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  u  ->  ( E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n )  <->  E. m  e.  ( 0..^ ( # `  x
) ) ( x cyclShift  m )  =  u ) )
5049cbvrabv 2976 . . . . . . . . . . . . . . . . . . 19  |-  { 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 }
5150cshwshash 14136 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e. Word  V  /\  ( # `  x )  e.  Prime )  ->  (
( # `  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  =  (
# `  x )  \/  ( # `  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  =  1 ) )
5251adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e. Word  V  /\  ( # `  x
)  e.  Prime )  /\  U  =  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  ->  (
( # `  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  =  (
# `  x )  \/  ( # `  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  =  1 ) )
5352orcomd 388 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e. Word  V  /\  ( # `  x
)  e.  Prime )  /\  U  =  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  ->  (
( # `  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  =  1  \/  ( # `  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  =  (
# `  x )
) )
54 fveq2 5696 . . . . . . . . . . . . . . . . . . 19  |-  ( 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 ) } ) )
5554eqeq1d 2451 . . . . . . . . . . . . . . . . . 18  |-  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x ) ) y  =  ( x cyclShift  n
) }  ->  (
( # `  U )  =  1  <->  ( # `  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  =  1 ) )
5654eqeq1d 2451 . . . . . . . . . . . . . . . . . 18  |-  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x ) ) y  =  ( x cyclShift  n
) }  ->  (
( # `  U )  =  ( # `  x
)  <->  ( # `  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  =  (
# `  x )
) )
5755, 56orbi12d 709 . . . . . . . . . . . . . . . . 17  |-  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x ) ) y  =  ( x cyclShift  n
) }  ->  (
( ( # `  U
)  =  1  \/  ( # `  U
)  =  ( # `  x ) )  <->  ( ( # `
 { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x ) ) y  =  ( x cyclShift  n
) } )  =  1  \/  ( # `  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  =  (
# `  x )
) ) )
5857adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e. Word  V  /\  ( # `  x
)  e.  Prime )  /\  U  =  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  ->  (
( ( # `  U
)  =  1  \/  ( # `  U
)  =  ( # `  x ) )  <->  ( ( # `
 { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x ) ) y  =  ( x cyclShift  n
) } )  =  1  \/  ( # `  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  =  (
# `  x )
) ) )
5953, 58mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e. Word  V  /\  ( # `  x
)  e.  Prime )  /\  U  =  {
y  e. Word  V  |  E. n  e.  (
0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) } )  ->  (
( # `  U )  =  1  \/  ( # `
 U )  =  ( # `  x
) ) )
6059ex 434 . . . . . . . . . . . . . 14  |-  ( ( x  e. Word  V  /\  ( # `  x )  e.  Prime )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x ) ) y  =  ( x cyclShift  n
) }  ->  (
( # `  U )  =  1  \/  ( # `
 U )  =  ( # `  x
) ) ) )
6160ex 434 . . . . . . . . . . . . 13  |-  ( x  e. Word  V  ->  (
( # `  x )  e.  Prime  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x ) ) y  =  ( x cyclShift  n
) }  ->  (
( # `  U )  =  1  \/  ( # `
 U )  =  ( # `  x
) ) ) ) )
62613ad2ant2 1010 . . . . . . . . . . . 12  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( ( # `
 x )  e. 
Prime  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U )  =  1  \/  ( # `  U
)  =  ( # `  x ) ) ) ) )
63 eleq1 2503 . . . . . . . . . . . . . . . 16  |-  ( N  =  ( # `  x
)  ->  ( N  e.  Prime 
<->  ( # `  x
)  e.  Prime )
)
64 oveq2 6104 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  =  ( # `  x
)  ->  ( 0..^ N )  =  ( 0..^ ( # `  x
) ) )
6564rexeqdv 2929 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  ( # `  x
)  ->  ( E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n )  <->  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n )
) )
6665rabbidv 2969 . . . . . . . . . . . . . . . . . 18  |-  ( 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 ) } )
6766eqeq2d 2454 . . . . . . . . . . . . . . . . 17  |-  ( 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 ) } ) )
68 eqeq2 2452 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  x
)  ->  ( ( # `
 U )  =  N  <->  ( # `  U
)  =  ( # `  x ) ) )
6968orbi2d 701 . . . . . . . . . . . . . . . . 17  |-  ( N  =  ( # `  x
)  ->  ( (
( # `  U )  =  1  \/  ( # `
 U )  =  N )  <->  ( ( # `
 U )  =  1  \/  ( # `  U )  =  (
# `  x )
) ) )
7067, 69imbi12d 320 . . . . . . . . . . . . . . . 16  |-  ( N  =  ( # `  x
)  ->  ( ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U
)  =  1  \/  ( # `  U
)  =  N ) )  <->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U )  =  1  \/  ( # `  U
)  =  ( # `  x ) ) ) ) )
7163, 70imbi12d 320 . . . . . . . . . . . . . . 15  |-  ( N  =  ( # `  x
)  ->  ( ( N  e.  Prime  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U
)  =  1  \/  ( # `  U
)  =  N ) ) )  <->  ( ( # `
 x )  e. 
Prime  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U )  =  1  \/  ( # `  U
)  =  ( # `  x ) ) ) ) ) )
7271eqcoms 2446 . . . . . . . . . . . . . 14  |-  ( (
# `  x )  =  N  ->  ( ( N  e.  Prime  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U )  =  1  \/  ( # `  U
)  =  N ) ) )  <->  ( ( # `
 x )  e. 
Prime  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U )  =  1  \/  ( # `  U
)  =  ( # `  x ) ) ) ) ) )
7372adantl 466 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  -> 
( ( N  e. 
Prime  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n
) }  ->  (
( # `  U )  =  1  \/  ( # `
 U )  =  N ) ) )  <-> 
( ( # `  x
)  e.  Prime  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U )  =  1  \/  ( # `  U
)  =  ( # `  x ) ) ) ) ) )
74733ad2ant3 1011 . . . . . . . . . . . 12  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( ( N  e.  Prime  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U
)  =  1  \/  ( # `  U
)  =  N ) ) )  <->  ( ( # `
 x )  e. 
Prime  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U )  =  1  \/  ( # `  U
)  =  ( # `  x ) ) ) ) ) )
7562, 74mpbird 232 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  x  e. Word  V  /\  ( N  e.  NN0  /\  ( # `  x
)  =  N ) )  ->  ( N  e.  Prime  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n
) }  ->  (
( # `  U )  =  1  \/  ( # `
 U )  =  N ) ) ) )
7614, 75syl 16 . . . . . . . . . 10  |-  ( x  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( N  e.  Prime  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n
) }  ->  (
( # `  U )  =  1  \/  ( # `
 U )  =  N ) ) ) )
778, 76sylbi 195 . . . . . . . . 9  |-  ( x  e.  W  ->  ( N  e.  Prime  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U
)  =  1  \/  ( # `  U
)  =  N ) ) ) )
7877impcom 430 . . . . . . . 8  |-  ( ( N  e.  Prime  /\  x  e.  W )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U
)  =  1  \/  ( # `  U
)  =  N ) ) )
7941, 78sylbid 215 . . . . . . 7  |-  ( ( N  e.  Prime  /\  x  e.  W )  ->  ( U  =  { y  e. Word  V  |  E. n  e.  ( 0 ... N
) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U
)  =  1  \/  ( # `  U
)  =  N ) ) )
8013, 79sylbid 215 . . . . . 6  |-  ( ( N  e.  Prime  /\  x  e.  W )  ->  ( U  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n
) }  ->  (
( # `  U )  =  1  \/  ( # `
 U )  =  N ) ) )
8180rexlimdva 2846 . . . . 5  |-  ( N  e.  Prime  ->  ( E. x  e.  W  U  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) }  ->  ( ( # `  U
)  =  1  \/  ( # `  U
)  =  N ) ) )
8281com12 31 . . . 4  |-  ( E. x  e.  W  U  =  { y  e.  W  |  E. n  e.  ( 0 ... N ) y  =  ( x cyclShift  n ) }  ->  ( N  e.  Prime  ->  ( ( # `  U
)  =  1  \/  ( # `  U
)  =  N ) ) )
833, 82syl6bi 228 . . 3  |-  ( U  e.  ( W /.  .~  )  ->  ( U  e.  ( W /.  .~  )  ->  ( N  e. 
Prime  ->  ( ( # `  U )  =  1  \/  ( # `  U
)  =  N ) ) ) )
8483pm2.43i 47 . 2  |-  ( U  e.  ( W /.  .~  )  ->  ( N  e.  Prime  ->  ( ( # `
 U )  =  1  \/  ( # `  U )  =  N ) ) )
8584impcom 430 1  |-  ( ( N  e.  Prime  /\  U  e.  ( W /.  .~  ) )  ->  (
( # `  U )  =  1  \/  ( # `
 U )  =  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   E.wrex 2721   {crab 2724   _Vcvv 2977   (/)c0 3642   {copab 4354   ` cfv 5423  (class class class)co 6096   /.cqs 7105   0cc0 9287   1c1 9288   NNcn 10327   NN0cn0 10584   ...cfz 11442  ..^cfzo 11553   #chash 12108  Word cword 12226   cyclShift ccsh 12430   Primecprime 13768   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-inf2 7852  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  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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-disj 4268  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-se 4685  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-isom 5432  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-ec 7108  df-qs 7112  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  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-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-hash 12109  df-word 12234  df-lsw 12235  df-concat 12236  df-substr 12238  df-reps 12241  df-csh 12431  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-dvds 13541  df-gcd 13696  df-prm 13769  df-phi 13846  df-clwwlk 30421  df-clwwlkn 30422
This theorem is referenced by: (None)
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