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Theorem hashecclwwlkn1 24657
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 24655 . . . 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 3112 . . . . . . . . . 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 14096 . . . . . . . . . . 11  |-  ( N  e.  Prime  ->  N  e.  NN )
76nnnn0d 10864 . . . . . . . . . 10  |-  ( N  e.  Prime  ->  N  e. 
NN0 )
81eleq2i 2545 . . . . . . . . . . 11  |-  ( x  e.  W  <->  x  e.  ( ( V ClWWalksN  E ) `
 N ) )
98biimpi 194 . . . . . . . . . 10  |-  ( x  e.  W  ->  x  e.  ( ( V ClWWalksN  E ) `
 N ) )
10 clwwlknscsh 24642 . . . . . . . . . 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 2508 . . . . . . . 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 2481 . . . . . . 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 24595 . . . . . . . . . . . . . . 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 10821 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
17 eqeq1 2471 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( N  =  ( # `  x
)  ->  ( N  =  0  <->  ( # `  x
)  =  0 ) )
1817eqcoms 2479 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  x )  =  N  ->  ( N  =  0  <->  ( # `  x
)  =  0 ) )
1918adantl 466 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  NN0  /\  ( # `  x )  =  N )  -> 
( N  =  0  <-> 
( # `  x )  =  0 ) )
20 hasheq0 12413 . . . . . . . . . . . . . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . . . . . . . . . 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 12774 . . . . . . . . . 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 2481 . . . . . . . 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 6303 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( n  =  m  ->  (
x cyclShift  n )  =  ( x cyclShift  m ) )
4342eqeq2d 2481 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  =  m  ->  (
y  =  ( x cyclShift  n )  <->  y  =  ( x cyclShift  m ) ) )
4443cbvrexv 3094 . . . . . . . . . . . . . . . . . . . . 21  |-  ( E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n )  <->  E. m  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  m )
)
45 eqeq1 2471 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  =  u  ->  (
y  =  ( x cyclShift  m )  <->  u  =  ( x cyclShift  m ) ) )
46 eqcom 2476 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( u  =  ( x cyclShift  m
)  <->  ( x cyclShift  m
)  =  u )
4745, 46syl6bb 261 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  =  u  ->  (
y  =  ( x cyclShift  m )  <->  ( x cyclShift  m )  =  u ) )
4847rexbidv 2978 . . . . . . . . . . . . . . . . . . . . 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 3117 . . . . . . . . . . . . . . . . . . 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 14464 . . . . . . . . . . . . . . . . . 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 2469 . . . . . . . . . . . . . . . . . 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 2469 . . . . . . . . . . . . . . . . . 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 2539 . . . . . . . . . . . . . . . 16  |-  ( N  =  ( # `  x
)  ->  ( N  e.  Prime 
<->  ( # `  x
)  e.  Prime )
)
64 oveq2 6303 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  =  ( # `  x
)  ->  ( 0..^ N )  =  ( 0..^ ( # `  x
) ) )
6564rexeqdv 3070 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  ( # `  x
)  ->  ( E. n  e.  ( 0..^ N ) y  =  ( x cyclShift  n )  <->  E. n  e.  ( 0..^ ( # `  x
) ) y  =  ( x cyclShift  n )
) )
6665rabbidv 3110 . . . . . . . . . . . . . . . . . 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 2481 . . . . . . . . . . . . . . . . 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 2482 . . . . . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . . . 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 2959 . . . . 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 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   {crab 2821   _Vcvv 3118   (/)c0 3790   {copab 4510   ` cfv 5594  (class class class)co 6295   /.cqs 7322   0cc0 9504   1c1 9505   NNcn 10548   NN0cn0 10807   ...cfz 11684  ..^cfzo 11804   #chash 12385  Word cword 12515   cyclShift ccsh 12739   Primecprime 14093   ClWWalksN cclwwlkn 24572
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-ec 7325  df-qs 7329  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-hash 12386  df-word 12523  df-lsw 12524  df-concat 12525  df-substr 12527  df-reps 12530  df-csh 12740  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-dvds 13865  df-gcd 14021  df-prm 14094  df-phi 14172  df-clwwlk 24574  df-clwwlkn 24575
This theorem is referenced by: (None)
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