Theorem erclwwlksneq 26944

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 AV, 30-Apr-2021.)

Hypotheses

Ref	Expression
erclwwlksn.w	|- W = ( N ClWWalksN G )
erclwwlksn.r	|- .~ = { <. t , u >. | ( t e. W /\ u e. W /\ E. n e. ( 0 ... N ) t = ( u cyclShift n ) ) }

Assertion

Ref	Expression
erclwwlksneq	|- ( ( 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, W, u   t, N, u   T, n, t, u   U, n, t, u

Allowed substitution hints:   .~ ( u, t, n)   G( u, t, n)   N( n)   W( n)   X( u, t, n)   Y( u, t, n)

Proof of Theorem erclwwlksneq

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . 4 |- ( t = T -> ( t e. W <-> T e. W ) )
2	1	adantr 481	. . 3 |- ( ( t = T /\ u = U ) -> ( t e. W <-> T e. W ) )
3		eleq1 2689	. . . 4 |- ( u = U -> ( u e. W <-> U e. W ) )
4	3	adantl 482	. . 3 |- ( ( t = T /\ u = U ) -> ( u e. W <-> U e. W ) )
5		simpl 473	. . . . 5 |- ( ( t = T /\ u = U ) -> t = T )
6		oveq1 6657	. . . . . 6 |- ( u = U -> ( u cyclShift n ) = ( U cyclShift n ) )
7	6	adantl 482	. . . . 5 |- ( ( t = T /\ u = U ) -> ( u cyclShift n ) = ( U cyclShift n ) )
8	5, 7	eqeq12d 2637	. . . 4 |- ( ( t = T /\ u = U ) -> ( t = ( u cyclShift n ) <-> T = ( U cyclShift n ) ) )
9	8	rexbidv 3052	. . 3 |- ( ( t = T /\ u = U ) -> ( E. n e. ( 0 ... N ) t = ( u cyclShift n ) <-> E. n e. ( 0 ... N ) T = ( U cyclShift n ) ) )
10	2, 4, 9	3anbi123d 1399	. 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		erclwwlksn.r	. 2 |- .~ = { <. t , u >. | ( t e. W /\ u e. W /\ E. n e. ( 0 ... N ) t = ( u cyclShift n ) ) }
12	10, 11	brabga 4989	1 |- ( ( T e. X /\ U e. Y ) -> ( T .~ U <-> ( T e. W /\ U e. W /\ E. n e. ( 0 ... N ) T = ( U cyclShift n ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   {copab 4712  (class class class)co 6650   0cc0 9936   ...cfz 12326   cyclShift ccsh 13534   ClWWalksN cclwwlksn 26876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  erclwwlksneqlen  26945  erclwwlksnref  26946  erclwwlksnsym  26947  erclwwlksntr  26948  eclclwwlksn1  26952