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Theorem hashecclwwlkn1 24960
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 24958 . . . 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 3103 . . . . . . . . . 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 14231 . . . . . . . . . . 11  |-  ( N  e.  Prime  ->  N  e.  NN )
76nnnn0d 10873 . . . . . . . . . 10  |-  ( N  e.  Prime  ->  N  e. 
NN0 )
81eleq2i 2535 . . . . . . . . . . 11  |-  ( x  e.  W  <->  x  e.  ( ( V ClWWalksN  E ) `
 N ) )
98biimpi 194 . . . . . . . . . 10  |-  ( x  e.  W  ->  x  e.  ( ( V ClWWalksN  E ) `
 N ) )
10 clwwlknscsh 24945 . . . . . . . . . 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 2498 . . . . . . . 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 2471 . . . . . . 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 24898 . . . . . . . . . . . . . . 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 10830 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
17 eqeq1 2461 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  =  ( # `  x
)  ->  ( N  =  0  <->  ( # `  x
)  =  0 ) )
1817eqcoms 2469 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  x )  =  N  ->  ( N  =  0  <->  ( # `  x
)  =  0 ) )
1918adantl 466 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  -> 
( N  =  0  <-> 
( # `  x )  =  0 ) )
20 hasheq0 12435 . . . . . . . . . . . . . . . . . . . . . . . 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 2715 . . . . . . . . . . . . . . . . . . . . . 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 2466 . . . . . . . . . . . . . . . . . . . . 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 1176 . . . . . . . . . . . . . . . . 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 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 )
) ) )
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 12805 . . . . . . . . . 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 2471 . . . . . . . 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 6304 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( n  =  m  ->  (
x cyclShift  n )  =  ( x cyclShift  m ) )
4342eqeq2d 2471 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  =  m  ->  (
y  =  ( x cyclShift  n )  <->  y  =  ( x cyclShift  m ) ) )
4443cbvrexv 3085 . . . . . . . . . . . . . . . . . . . . 21  |-  ( E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n )  <->  E. m  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  m )
)
45 eqeq1 2461 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  =  u  ->  (
y  =  ( x cyclShift  m )  <->  u  =  ( x cyclShift  m ) ) )
46 eqcom 2466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( u  =  ( x cyclShift  m
)  <->  ( x cyclShift  m
)  =  u )
4745, 46syl6bb 261 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  =  u  ->  (
y  =  ( x cyclShift  m )  <->  ( x cyclShift  m )  =  u ) )
4847rexbidv 2968 . . . . . . . . . . . . . . . . . . . . 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 3108 . . . . . . . . . . . . . . . . . . 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 14600 . . . . . . . . . . . . . . . . . 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 5872 . . . . . . . . . . . . . . . . . . 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 2459 . . . . . . . . . . . . . . . . . 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 2459 . . . . . . . . . . . . . . . . . 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 1018 . . . . . . . . . . . 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 2529 . . . . . . . . . . . . . . . 16  |-  ( N  =  ( # `  x
)  ->  ( N  e.  Prime 
<->  ( # `  x
)  e.  Prime )
)
64 oveq2 6304 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  =  ( # `  x
)  ->  ( 0..^ N )  =  ( 0..^ ( # `  x
) ) )
6564rexeqdv 3061 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  ( # `  x
)  ->  ( E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n )  <->  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n )
) )
6665rabbidv 3101 . . . . . . . . . . . . . . . . . 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 2471 . . . . . . . . . . . . . . . . 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 2472 . . . . . . . . . . . . . . . . . 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 2469 . . . . . . . . . . . . . 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 1019 . . . . . . . . . . . 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 2949 . . . . 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   {crab 2811   _Vcvv 3109   (/)c0 3793   {copab 4514   ` cfv 5594  (class class class)co 6296   /.cqs 7328   0cc0 9509   1c1 9510   NNcn 10556   NN0cn0 10816   ...cfz 11697  ..^cfzo 11820   #chash 12407  Word cword 12537   cyclShift ccsh 12770   Primecprime 14228   ClWWalksN cclwwlkn 24875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-ec 7331  df-qs 7335  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-hash 12408  df-word 12545  df-lsw 12546  df-concat 12547  df-substr 12549  df-reps 12552  df-csh 12771  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-sum 13520  df-dvds 13998  df-gcd 14156  df-prm 14229  df-phi 14307  df-clwwlk 24877  df-clwwlkn 24878
This theorem is referenced by: (None)
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