Theorem erclwwlksref 26934

Description: .~ is a reflexive relation over the set of closed walks (defined as words). (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
erclwwlksref	|- ( x e. (ClWWalks ` G ) <-> x .~ x )

Distinct variable groups:   n, G, u, w   x, n, u, w

Allowed substitution hints:   .~ ( x, w, u, n)   G( x)

Proof of Theorem erclwwlksref

Step	Hyp	Ref	Expression
1		anidm 676	. . . 4 |- ( ( x e. (ClWWalks ` G ) /\ x e. (ClWWalks ` G ) ) <-> x e. (ClWWalks ` G ) )
2	1	anbi1i 731	. . 3 |- ( ( ( x e. (ClWWalks ` G ) /\ x e. (ClWWalks ` G ) ) /\ E. n e. ( 0 ... ( # ` x ) ) x = ( x cyclShift n ) ) <-> ( x e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` x ) ) x = ( x cyclShift n ) ) )
3		df-3an 1039	. . 3 |- ( ( x e. (ClWWalks ` G ) /\ x e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` x ) ) x = ( x cyclShift n ) ) <-> ( ( x e. (ClWWalks ` G ) /\ x e. (ClWWalks ` G ) ) /\ E. n e. ( 0 ... ( # ` x ) ) x = ( x cyclShift n ) ) )
4		eqid 2622	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
5	4	clwwlkbp 26883	. . . . 5 |- ( x e. (ClWWalks ` G ) -> ( G e. _V /\ x e. Word (Vtx ` G ) /\ x =/= (/) ) )
6		cshw0 13540	. . . . . . 7 |- ( x e. Word (Vtx ` G ) -> ( x cyclShift 0 ) = x )
7		0nn0 11307	. . . . . . . . . 10 |- 0 e. NN0
8	7	a1i 11	. . . . . . . . 9 |- ( x e. Word (Vtx ` G ) -> 0 e. NN0 )
9		lencl 13324	. . . . . . . . 9 |- ( x e. Word (Vtx ` G ) -> ( # ` x ) e. NN0 )
10		hashge0 13176	. . . . . . . . 9 |- ( x e. Word (Vtx ` G ) -> 0 <_ ( # ` x ) )
11		elfz2nn0 12431	. . . . . . . . 9 |- ( 0 e. ( 0 ... ( # ` x ) ) <-> ( 0 e. NN0 /\ ( # ` x ) e. NN0 /\ 0 <_ ( # ` x ) ) )
12	8, 9, 10, 11	syl3anbrc 1246	. . . . . . . 8 |- ( x e. Word (Vtx ` G ) -> 0 e. ( 0 ... ( # ` x ) ) )
13		eqcom 2629	. . . . . . . . 9 |- ( ( x cyclShift 0 ) = x <-> x = ( x cyclShift 0 ) )
14	13	biimpi 206	. . . . . . . 8 |- ( ( x cyclShift 0 ) = x -> x = ( x cyclShift 0 ) )
15		oveq2 6658	. . . . . . . . . 10 |- ( n = 0 -> ( x cyclShift n ) = ( x cyclShift 0 ) )
16	15	eqeq2d 2632	. . . . . . . . 9 |- ( n = 0 -> ( x = ( x cyclShift n ) <-> x = ( x cyclShift 0 ) ) )
17	16	rspcev 3309	. . . . . . . 8 |- ( ( 0 e. ( 0 ... ( # ` x ) ) /\ x = ( x cyclShift 0 ) ) -> E. n e. ( 0 ... ( # ` x ) ) x = ( x cyclShift n ) )
18	12, 14, 17	syl2an 494	. . . . . . 7 |- ( ( x e. Word (Vtx ` G ) /\ ( x cyclShift 0 ) = x ) -> E. n e. ( 0 ... ( # ` x ) ) x = ( x cyclShift n ) )
19	6, 18	mpdan 702	. . . . . 6 |- ( x e. Word (Vtx ` G ) -> E. n e. ( 0 ... ( # ` x ) ) x = ( x cyclShift n ) )
20	19	3ad2ant2 1083	. . . . 5 |- ( ( G e. _V /\ x e. Word (Vtx ` G ) /\ x =/= (/) ) -> E. n e. ( 0 ... ( # ` x ) ) x = ( x cyclShift n ) )
21	5, 20	syl 17	. . . 4 |- ( x e. (ClWWalks ` G ) -> E. n e. ( 0 ... ( # ` x ) ) x = ( x cyclShift n ) )
22	21	pm4.71i 664	. . 3 |- ( x e. (ClWWalks ` G ) <-> ( x e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` x ) ) x = ( x cyclShift n ) ) )
23	2, 3, 22	3bitr4ri 293	. 2 |- ( x e. (ClWWalks ` G ) <-> ( x e. (ClWWalks ` G ) /\ x e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` x ) ) x = ( x cyclShift n ) ) )
24		vex 3203	. . 3 |- x e. _V
25		erclwwlks.r	. . . 4 |- .~ = { <. u , w >. | ( u e. (ClWWalks ` G ) /\ w e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` w ) ) u = ( w cyclShift n ) ) }
26	25	erclwwlkseq 26932	. . 3 |- ( ( x e. _V /\ x e. _V ) -> ( x .~ x <-> ( x e. (ClWWalks ` G ) /\ x e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` x ) ) x = ( x cyclShift n ) ) ) )
27	24, 24, 26	mp2an 708	. 2 |- ( x .~ x <-> ( x e. (ClWWalks ` G ) /\ x e. (ClWWalks ` G ) /\ E. n e. ( 0 ... ( # ` x ) ) x = ( x cyclShift n ) ) )
28	23, 27	bitr4i 267	1 |- ( x e. (ClWWalks ` G ) <-> x .~ x )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   {copab 4712   ` cfv 5888  (class class class)co 6650   0cc0 9936   <_ cle 10075   NN0cn0 11292   ...cfz 12326   #chash 13117  Word cword 13291   cyclShift ccsh 13534  Vtxcvtx 25874  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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-csh 13535  df-clwwlks 26877

This theorem is referenced by:  erclwwlks  26937