Theorem erclwwlkseq 26932

Description: Two classes are equivalent regarding .~ if both are words and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.)

Hypothesis

Ref	Expression
erclwwlks.r	|- .~ = { <. u , w >. | ( u e. (ClWWalks ` G ) /\ w e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` w ) ) u = ( w cyclShift n ) ) }

Assertion

Ref	Expression
erclwwlkseq	|- ( ( U e. X /\ W e. Y ) -> ( U .~ W <-> ( U e. (ClWWalks ` G ) /\ W e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` W ) ) U = ( W cyclShift n ) ) ) )

Distinct variable groups:   n, G, u, w   U, n, u, w   n, W, u, w

Allowed substitution hints:   .~ ( w, u, n)   X( w, u, n)   Y( w, u, n)

Proof of Theorem erclwwlkseq

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . 4 |- ( u = U -> ( u e. (ClWWalks ` G ) <-> U e. (ClWWalks ` G ) ) )
2	1	adantr 481	. . 3 |- ( ( u = U /\ w = W ) -> ( u e. (ClWWalks ` G ) <-> U e. (ClWWalks ` G ) ) )
3		eleq1 2689	. . . 4 |- ( w = W -> ( w e. (ClWWalks ` G ) <-> W e. (ClWWalks ` G ) ) )
4	3	adantl 482	. . 3 |- ( ( u = U /\ w = W ) -> ( w e. (ClWWalks ` G ) <-> W e. (ClWWalks ` G ) ) )
5		fveq2 6191	. . . . . 6 |- ( w = W -> ( # ` w ) = ( # ` W ) )
6	5	oveq2d 6666	. . . . 5 |- ( w = W -> ( 0 ... ( # ` w ) ) = ( 0 ... ( # ` W ) ) )
7	6	adantl 482	. . . 4 |- ( ( u = U /\ w = W ) -> ( 0 ... ( # ` w ) ) = ( 0 ... ( # ` W ) ) )
8		simpl 473	. . . . 5 |- ( ( u = U /\ w = W ) -> u = U )
9		oveq1 6657	. . . . . 6 |- ( w = W -> ( w cyclShift n ) = ( W cyclShift n ) )
10	9	adantl 482	. . . . 5 |- ( ( u = U /\ w = W ) -> ( w cyclShift n ) = ( W cyclShift n ) )
11	8, 10	eqeq12d 2637	. . . 4 |- ( ( u = U /\ w = W ) -> ( u = ( w cyclShift n ) <-> U = ( W cyclShift n ) ) )
12	7, 11	rexeqbidv 3153	. . 3 |- ( ( u = U /\ w = W ) -> ( E. n e. ( 0 ... ( # ` w ) ) u = ( w cyclShift n ) <-> E. n e. ( 0 ... ( # ` W ) ) U = ( W cyclShift n ) ) )
13	2, 4, 12	3anbi123d 1399	. 2 |- ( ( u = U /\ w = W ) -> ( ( u e. (ClWWalks ` G ) /\ w e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` w ) ) u = ( w cyclShift n ) ) <-> ( U e. (ClWWalks ` G ) /\ W e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` W ) ) U = ( W cyclShift n ) ) ) )
14		erclwwlks.r	. 2 |- .~ = { <. u , w >. | ( u e. (ClWWalks ` G ) /\ w e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` w ) ) u = ( w cyclShift n ) ) }
15	13, 14	brabga 4989	1 |- ( ( U e. X /\ W e. Y ) -> ( U .~ W <-> ( U e. (ClWWalks ` G ) /\ W e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` W ) ) U = ( W 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   ` cfv 5888  (class class class)co 6650   0cc0 9936   ...cfz 12326   #chash 13117   cyclShift ccsh 13534  ClWWalkscclwwlks 26875

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:  erclwwlkseqlen  26933  erclwwlksref  26934  erclwwlkssym  26935  erclwwlkstr  26936