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Theorem erclwwlkneq 30646
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 465 . . 3  |-  ( ( t  =  T  /\  u  =  U )  ->  ( t  e.  W  <->  T  e.  W ) )
3 eleq1 2526 . . . 4  |-  ( u  =  U  ->  (
u  e.  W  <->  U  e.  W ) )
43adantl 466 . . 3  |-  ( ( t  =  T  /\  u  =  U )  ->  ( u  e.  W  <->  U  e.  W ) )
5 simpl 457 . . . . 5  |-  ( ( t  =  T  /\  u  =  U )  ->  t  =  T )
6 oveq1 6208 . . . . . 6  |-  ( u  =  U  ->  (
u cyclShift  n )  =  ( U cyclShift  n ) )
76adantl 466 . . . . 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 2868 . . 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 1290 . 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 4712 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2800   class class class wbr 4401   {copab 4458   ` cfv 5527  (class class class)co 6201   0cc0 9394   ...cfz 11555   cyclShift ccsh 12544   ClWWalksN cclwwlkn 30563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-iota 5490  df-fv 5535  df-ov 6204
This theorem is referenced by:  erclwwlkneqlen  30647  erclwwlknref  30648  erclwwlknsym  30649  erclwwlkntr  30650  eclclwwlkn1  30655
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