Theorem clwwlksnscsh 26940

Description: The set of cyclical shifts of a word representing a closed walk is the set of closed walks represented by cyclical shifts of a word. (Contributed by Alexander van der Vekens, 15-Jun-2018.) (Revised by AV, 30-Apr-2021.)

Assertion

Ref	Expression
clwwlksnscsh	|- ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) -> { y e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) y = ( W cyclShift n ) } = { y e. Word (Vtx ` G ) | E. n e. ( 0 ... N ) y = ( W cyclShift n ) } )

Distinct variable groups:   n, G, y   n, N, y   n, W, y

Proof of Theorem clwwlksnscsh

Dummy variables w x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . . 4 |- ( y = x -> ( y = ( W cyclShift n ) <-> x = ( W cyclShift n ) ) )
2	1	rexbidv 3052	. . 3 |- ( y = x -> ( E. n e. ( 0 ... N ) y = ( W cyclShift n ) <-> E. n e. ( 0 ... N ) x = ( W cyclShift n ) ) )
3	2	cbvrabv 3199	. 2 |- { y e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) y = ( W cyclShift n ) } = { x e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) x = ( W cyclShift n ) }
4		eqid 2622	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
5	4	clwwlksnwrd 26886	. . . . . . 7 |- ( w e. ( N ClWWalksN G ) -> w e. Word (Vtx ` G ) )
6	5	ad2antrl 764	. . . . . 6 |- ( ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) /\ ( w e. ( N ClWWalksN G ) /\ E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) ) -> w e. Word (Vtx ` G ) )
7		simprr 796	. . . . . 6 |- ( ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) /\ ( w e. ( N ClWWalksN G ) /\ E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) ) -> E. n e. ( 0 ... N ) w = ( W cyclShift n ) )
8	6, 7	jca 554	. . . . 5 |- ( ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) /\ ( w e. ( N ClWWalksN G ) /\ E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) ) -> ( w e. Word (Vtx ` G ) /\ E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) )
9		simprr 796	. . . . . . . . . . . . 13 |- ( ( ( w e. Word (Vtx ` G ) /\ n e. ( 0 ... N ) ) /\ ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) ) -> W e. ( N ClWWalksN G ) )
10		simpllr 799	. . . . . . . . . . . . 13 |- ( ( ( ( w e. Word (Vtx ` G ) /\ n e. ( 0 ... N ) ) /\ ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) ) /\ w = ( W cyclShift n ) ) -> n e. ( 0 ... N ) )
11		clwwnisshclwwsn 26930	. . . . . . . . . . . . 13 |- ( ( W e. ( N ClWWalksN G ) /\ n e. ( 0 ... N ) ) -> ( W cyclShift n ) e. ( N ClWWalksN G ) )
12	9, 10, 11	syl2an2r 876	. . . . . . . . . . . 12 |- ( ( ( ( w e. Word (Vtx ` G ) /\ n e. ( 0 ... N ) ) /\ ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) ) /\ w = ( W cyclShift n ) ) -> ( W cyclShift n ) e. ( N ClWWalksN G ) )
13		eleq1 2689	. . . . . . . . . . . . 13 |- ( w = ( W cyclShift n ) -> ( w e. ( N ClWWalksN G ) <-> ( W cyclShift n ) e. ( N ClWWalksN G ) ) )
14	13	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ( w e. Word (Vtx ` G ) /\ n e. ( 0 ... N ) ) /\ ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) ) /\ w = ( W cyclShift n ) ) -> ( w e. ( N ClWWalksN G ) <-> ( W cyclShift n ) e. ( N ClWWalksN G ) ) )
15	12, 14	mpbird 247	. . . . . . . . . . 11 |- ( ( ( ( w e. Word (Vtx ` G ) /\ n e. ( 0 ... N ) ) /\ ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) ) /\ w = ( W cyclShift n ) ) -> w e. ( N ClWWalksN G ) )
16	15	exp31 630	. . . . . . . . . 10 |- ( ( w e. Word (Vtx ` G ) /\ n e. ( 0 ... N ) ) -> ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) -> ( w = ( W cyclShift n ) -> w e. ( N ClWWalksN G ) ) ) )
17	16	com23 86	. . . . . . . . 9 |- ( ( w e. Word (Vtx ` G ) /\ n e. ( 0 ... N ) ) -> ( w = ( W cyclShift n ) -> ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) -> w e. ( N ClWWalksN G ) ) ) )
18	17	rexlimdva 3031	. . . . . . . 8 |- ( w e. Word (Vtx ` G ) -> ( E. n e. ( 0 ... N ) w = ( W cyclShift n ) -> ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) -> w e. ( N ClWWalksN G ) ) ) )
19	18	imp 445	. . . . . . 7 |- ( ( w e. Word (Vtx ` G ) /\ E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) -> ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) -> w e. ( N ClWWalksN G ) ) )
20	19	impcom 446	. . . . . 6 |- ( ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) /\ ( w e. Word (Vtx ` G ) /\ E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) ) -> w e. ( N ClWWalksN G ) )
21		simprr 796	. . . . . 6 |- ( ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) /\ ( w e. Word (Vtx ` G ) /\ E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) ) -> E. n e. ( 0 ... N ) w = ( W cyclShift n ) )
22	20, 21	jca 554	. . . . 5 |- ( ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) /\ ( w e. Word (Vtx ` G ) /\ E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) ) -> ( w e. ( N ClWWalksN G ) /\ E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) )
23	8, 22	impbida 877	. . . 4 |- ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) -> ( ( w e. ( N ClWWalksN G ) /\ E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) <-> ( w e. Word (Vtx ` G ) /\ E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) ) )
24		eqeq1 2626	. . . . . 6 |- ( x = w -> ( x = ( W cyclShift n ) <-> w = ( W cyclShift n ) ) )
25	24	rexbidv 3052	. . . . 5 |- ( x = w -> ( E. n e. ( 0 ... N ) x = ( W cyclShift n ) <-> E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) )
26	25	elrab 3363	. . . 4 |- ( w e. { x e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) x = ( W cyclShift n ) } <-> ( w e. ( N ClWWalksN G ) /\ E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) )
27		eqeq1 2626	. . . . . 6 |- ( y = w -> ( y = ( W cyclShift n ) <-> w = ( W cyclShift n ) ) )
28	27	rexbidv 3052	. . . . 5 |- ( y = w -> ( E. n e. ( 0 ... N ) y = ( W cyclShift n ) <-> E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) )
29	28	elrab 3363	. . . 4 |- ( w e. { y e. Word (Vtx ` G ) | E. n e. ( 0 ... N ) y = ( W cyclShift n ) } <-> ( w e. Word (Vtx ` G ) /\ E. n e. ( 0 ... N ) w = ( W cyclShift n ) ) )
30	23, 26, 29	3bitr4g 303	. . 3 |- ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) -> ( w e. { x e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) x = ( W cyclShift n ) } <-> w e. { y e. Word (Vtx ` G ) | E. n e. ( 0 ... N ) y = ( W cyclShift n ) } ) )
31	30	eqrdv 2620	. 2 |- ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) -> { x e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) x = ( W cyclShift n ) } = { y e. Word (Vtx ` G ) | E. n e. ( 0 ... N ) y = ( W cyclShift n ) } )
32	3, 31	syl5eq 2668	1 |- ( ( N e. NN0 /\ W e. ( N ClWWalksN G ) ) -> { y e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) y = ( W cyclShift n ) } = { y e. Word (Vtx ` G ) | E. n e. ( 0 ... N ) y = ( W cyclShift n ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   ` cfv 5888  (class class class)co 6650   0cc0 9936   NN0cn0 11292   ...cfz 12326  Word cword 13291   cyclShift ccsh 13534  Vtxcvtx 25874   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-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-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-substr 13303  df-csh 13535  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  hashecclwwlksn1  26954  umgrhashecclwwlk  26955