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Theorem erclwwlkneq 25025
Description: Two classes are equivalent regarding  .~ if both are words of the same fixed length and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by Alexander van der Vekens, 14-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
erclwwlkneq  |-  ( ( T  e.  X  /\  U  e.  Y )  ->  ( T  .~  U  <->  ( T  e.  W  /\  U  e.  W  /\  E. n  e.  ( 0 ... N ) T  =  ( U cyclShift  n ) ) ) )
Distinct variable groups:    t, E, u    t, N, u    n, V, t, u    t, W, u    T, n, t, u    U, n, t, u
Allowed substitution hints:    .~ ( u, t, n)    E( n)    N( n)    W( n)    X( u, t, n)    Y( u, t, n)

Proof of Theorem erclwwlkneq
StepHypRef Expression
1 eleq1 2526 . . . 4  |-  ( t  =  T  ->  (
t  e.  W  <->  T  e.  W ) )
21adantr 463 . . 3  |-  ( ( t  =  T  /\  u  =  U )  ->  ( t  e.  W  <->  T  e.  W ) )
3 eleq1 2526 . . . 4  |-  ( u  =  U  ->  (
u  e.  W  <->  U  e.  W ) )
43adantl 464 . . 3  |-  ( ( t  =  T  /\  u  =  U )  ->  ( u  e.  W  <->  U  e.  W ) )
5 simpl 455 . . . . 5  |-  ( ( t  =  T  /\  u  =  U )  ->  t  =  T )
6 oveq1 6277 . . . . . 6  |-  ( u  =  U  ->  (
u cyclShift  n )  =  ( U cyclShift  n ) )
76adantl 464 . . . . 5  |-  ( ( t  =  T  /\  u  =  U )  ->  ( u cyclShift  n )  =  ( U cyclShift  n ) )
85, 7eqeq12d 2476 . . . 4  |-  ( ( t  =  T  /\  u  =  U )  ->  ( t  =  ( u cyclShift  n )  <->  T  =  ( U cyclShift  n ) ) )
98rexbidv 2965 . . 3  |-  ( ( t  =  T  /\  u  =  U )  ->  ( E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n )  <->  E. n  e.  ( 0 ... N
) T  =  ( U cyclShift  n ) ) )
102, 4, 93anbi123d 1297 . 2  |-  ( ( t  =  T  /\  u  =  U )  ->  ( ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) )  <->  ( T  e.  W  /\  U  e.  W  /\  E. n  e.  ( 0 ... N
) T  =  ( U cyclShift  n ) ) ) )
11 erclwwlkn.r . 2  |-  .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
1210, 11brabga 4750 1  |-  ( ( T  e.  X  /\  U  e.  Y )  ->  ( T  .~  U  <->  ( T  e.  W  /\  U  e.  W  /\  E. n  e.  ( 0 ... N ) T  =  ( U cyclShift  n ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805   class class class wbr 4439   {copab 4496   ` cfv 5570  (class class class)co 6270   0cc0 9481   ...cfz 11675   cyclShift ccsh 12750   ClWWalksN cclwwlkn 24951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  erclwwlkneqlen  25026  erclwwlknref  25027  erclwwlknsym  25028  erclwwlkntr  25029  eclclwwlkn1  25034
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