Theorem erclwwlksnsym 26947

Description: .~ is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-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
erclwwlksnsym	|- ( x .~ y -> y .~ x )

Distinct variable groups:   t, W, u   n, N, u, t, x   y, n, t, u, x   n, W

Allowed substitution hints:   .~ ( x, y, u, t, n)   G( x, y, u, t, n)   N( y)   W( x, y)

Proof of Theorem erclwwlksnsym

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. 2 |- x e. _V
2		vex 3203	. 2 |- y e. _V
3		erclwwlksn.w	. . . 4 |- W = ( N ClWWalksN G )
4		erclwwlksn.r	. . . 4 |- .~ = { <. t , u >. | ( t e. W /\ u e. W /\ E. n e. ( 0 ... N ) t = ( u cyclShift n ) ) }
5	3, 4	erclwwlksneqlen 26945	. . 3 |- ( ( x e. _V /\ y e. _V ) -> ( x .~ y -> ( # ` x ) = ( # ` y ) ) )
6	3, 4	erclwwlksneq 26944	. . . 4 |- ( ( x e. _V /\ y e. _V ) -> ( x .~ y <-> ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) ) )
7		simpl2 1065	. . . . . . 7 |- ( ( ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) /\ ( # ` x ) = ( # ` y ) ) -> y e. W )
8		simpl1 1064	. . . . . . 7 |- ( ( ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) /\ ( # ` x ) = ( # ` y ) ) -> x e. W )
9		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 |- (Vtx ` G ) = (Vtx ` G )
10	9	clwwlknbp 26885	. . . . . . . . . . . . . . . . . . 19 |- ( x e. ( N ClWWalksN G ) -> ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) )
11		eqcom 2629	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` x ) = N <-> N = ( # ` x ) )
12	11	biimpi 206	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` x ) = N -> N = ( # ` x ) )
13	10, 12	simpl2im 658	. . . . . . . . . . . . . . . . . 18 |- ( x e. ( N ClWWalksN G ) -> N = ( # ` x ) )
14	13, 3	eleq2s 2719	. . . . . . . . . . . . . . . . 17 |- ( x e. W -> N = ( # ` x ) )
15	14	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( x e. W /\ y e. W ) -> N = ( # ` x ) )
16	15	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) -> N = ( # ` x ) )
17	9	clwwlksnwrd 26886	. . . . . . . . . . . . . . . . . . . . 21 |- ( y e. ( N ClWWalksN G ) -> y e. Word (Vtx ` G ) )
18	17, 3	eleq2s 2719	. . . . . . . . . . . . . . . . . . . 20 |- ( y e. W -> y e. Word (Vtx ` G ) )
19	18	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. W /\ y e. W ) -> y e. Word (Vtx ` G ) )
20	19	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) -> y e. Word (Vtx ` G ) )
21	20	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( N = ( # ` x ) /\ ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) ) -> y e. Word (Vtx ` G ) )
22		simprr 796	. . . . . . . . . . . . . . . . 17 |- ( ( N = ( # ` x ) /\ ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( # ` x ) = ( # ` y ) )
23	21, 22	cshwcshid 13573	. . . . . . . . . . . . . . . 16 |- ( ( N = ( # ` x ) /\ ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( ( n e. ( 0 ... ( # ` y ) ) /\ x = ( y cyclShift n ) ) -> E. m e. ( 0 ... ( # ` x ) ) y = ( x cyclShift m ) ) )
24		oveq2 6658	. . . . . . . . . . . . . . . . . . 19 |- ( N = ( # ` x ) -> ( 0 ... N ) = ( 0 ... ( # ` x ) ) )
25		oveq2 6658	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` x ) = ( # ` y ) -> ( 0 ... ( # ` x ) ) = ( 0 ... ( # ` y ) ) )
26	25	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) -> ( 0 ... ( # ` x ) ) = ( 0 ... ( # ` y ) ) )
27	24, 26	sylan9eq 2676	. . . . . . . . . . . . . . . . . 18 |- ( ( N = ( # ` x ) /\ ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( 0 ... N ) = ( 0 ... ( # ` y ) ) )
28	27	eleq2d 2687	. . . . . . . . . . . . . . . . 17 |- ( ( N = ( # ` x ) /\ ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( n e. ( 0 ... N ) <-> n e. ( 0 ... ( # ` y ) ) ) )
29	28	anbi1d 741	. . . . . . . . . . . . . . . 16 |- ( ( N = ( # ` x ) /\ ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( ( n e. ( 0 ... N ) /\ x = ( y cyclShift n ) ) <-> ( n e. ( 0 ... ( # ` y ) ) /\ x = ( y cyclShift n ) ) ) )
30	24	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( N = ( # ` x ) /\ ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( 0 ... N ) = ( 0 ... ( # ` x ) ) )
31	30	rexeqdv 3145	. . . . . . . . . . . . . . . 16 |- ( ( N = ( # ` x ) /\ ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( E. m e. ( 0 ... N ) y = ( x cyclShift m ) <-> E. m e. ( 0 ... ( # ` x ) ) y = ( x cyclShift m ) ) )
32	23, 29, 31	3imtr4d 283	. . . . . . . . . . . . . . 15 |- ( ( N = ( # ` x ) /\ ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) ) -> ( ( n e. ( 0 ... N ) /\ x = ( y cyclShift n ) ) -> E. m e. ( 0 ... N ) y = ( x cyclShift m ) ) )
33	16, 32	mpancom 703	. . . . . . . . . . . . . 14 |- ( ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) -> ( ( n e. ( 0 ... N ) /\ x = ( y cyclShift n ) ) -> E. m e. ( 0 ... N ) y = ( x cyclShift m ) ) )
34	33	expd 452	. . . . . . . . . . . . 13 |- ( ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) -> ( n e. ( 0 ... N ) -> ( x = ( y cyclShift n ) -> E. m e. ( 0 ... N ) y = ( x cyclShift m ) ) ) )
35	34	rexlimdv 3030	. . . . . . . . . . . 12 |- ( ( ( x e. W /\ y e. W ) /\ ( # ` x ) = ( # ` y ) ) -> ( E. n e. ( 0 ... N ) x = ( y cyclShift n ) -> E. m e. ( 0 ... N ) y = ( x cyclShift m ) ) )
36	35	ex 450	. . . . . . . . . . 11 |- ( ( x e. W /\ y e. W ) -> ( ( # ` x ) = ( # ` y ) -> ( E. n e. ( 0 ... N ) x = ( y cyclShift n ) -> E. m e. ( 0 ... N ) y = ( x cyclShift m ) ) ) )
37	36	com23 86	. . . . . . . . . 10 |- ( ( x e. W /\ y e. W ) -> ( E. n e. ( 0 ... N ) x = ( y cyclShift n ) -> ( ( # ` x ) = ( # ` y ) -> E. m e. ( 0 ... N ) y = ( x cyclShift m ) ) ) )
38	37	3impia 1261	. . . . . . . . 9 |- ( ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) -> ( ( # ` x ) = ( # ` y ) -> E. m e. ( 0 ... N ) y = ( x cyclShift m ) ) )
39	38	imp 445	. . . . . . . 8 |- ( ( ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) /\ ( # ` x ) = ( # ` y ) ) -> E. m e. ( 0 ... N ) y = ( x cyclShift m ) )
40		oveq2 6658	. . . . . . . . . 10 |- ( n = m -> ( x cyclShift n ) = ( x cyclShift m ) )
41	40	eqeq2d 2632	. . . . . . . . 9 |- ( n = m -> ( y = ( x cyclShift n ) <-> y = ( x cyclShift m ) ) )
42	41	cbvrexv 3172	. . . . . . . 8 |- ( E. n e. ( 0 ... N ) y = ( x cyclShift n ) <-> E. m e. ( 0 ... N ) y = ( x cyclShift m ) )
43	39, 42	sylibr 224	. . . . . . 7 |- ( ( ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) /\ ( # ` x ) = ( # ` y ) ) -> E. n e. ( 0 ... N ) y = ( x cyclShift n ) )
44	7, 8, 43	3jca 1242	. . . . . 6 |- ( ( ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) /\ ( # ` x ) = ( # ` y ) ) -> ( y e. W /\ x e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) )
45	3, 4	erclwwlksneq 26944	. . . . . . 7 |- ( ( y e. _V /\ x e. _V ) -> ( y .~ x <-> ( y e. W /\ x e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) ) )
46	45	ancoms 469	. . . . . 6 |- ( ( x e. _V /\ y e. _V ) -> ( y .~ x <-> ( y e. W /\ x e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) ) )
47	44, 46	syl5ibr 236	. . . . 5 |- ( ( x e. _V /\ y e. _V ) -> ( ( ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) /\ ( # ` x ) = ( # ` y ) ) -> y .~ x ) )
48	47	expd 452	. . . 4 |- ( ( x e. _V /\ y e. _V ) -> ( ( x e. W /\ y e. W /\ E. n e. ( 0 ... N ) x = ( y cyclShift n ) ) -> ( ( # ` x ) = ( # ` y ) -> y .~ x ) ) )
49	6, 48	sylbid 230	. . 3 |- ( ( x e. _V /\ y e. _V ) -> ( x .~ y -> ( ( # ` x ) = ( # ` y ) -> y .~ x ) ) )
50	5, 49	mpdd 43	. 2 |- ( ( x e. _V /\ y e. _V ) -> ( x .~ y -> y .~ x ) )
51	1, 2, 50	mp2an 708	1 |- ( x .~ y -> y .~ x )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   {copab 4712   ` cfv 5888  (class class class)co 6650   0cc0 9936   ...cfz 12326   #chash 13117  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-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  df-clwwlksn 26878

This theorem is referenced by:  erclwwlksn  26949  eclclwwlksn1  26952