Theorem eleclclwwlksnlem2 26939

Description: Lemma 2 for eleclclwwlksn 26953. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 30-Apr-2021.)

Hypothesis

Ref	Expression
erclwwlksn1.w	|- W = ( N ClWWalksN G )

Assertion

Ref	Expression
eleclclwwlksnlem2	|- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( E. m e. ( 0 ... N ) Y = ( x cyclShift m ) <-> E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) )

Distinct variable groups:   m, n, G   m, N, n   m, X, n   m, Y, n   k, m, n   x, m, n

Allowed substitution hints:   G( x, k)   N( x, k)   W( x, k, m, n)   X( x, k)   Y( x, k)

Proof of Theorem eleclclwwlksnlem2

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) -> k e. ( 0 ... N ) )
2	1	anim1i 592	. . . 4 |- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( k e. ( 0 ... N ) /\ ( X e. W /\ x e. W ) ) )
3	2	adantr 481	. . 3 |- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. m e. ( 0 ... N ) Y = ( x cyclShift m ) ) -> ( k e. ( 0 ... N ) /\ ( X e. W /\ x e. W ) ) )
4		simpr 477	. . . . 5 |- ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) -> X = ( x cyclShift k ) )
5	4	adantr 481	. . . 4 |- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> X = ( x cyclShift k ) )
6	5	anim1i 592	. . 3 |- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. m e. ( 0 ... N ) Y = ( x cyclShift m ) ) -> ( X = ( x cyclShift k ) /\ E. m e. ( 0 ... N ) Y = ( x cyclShift m ) ) )
7		erclwwlksn1.w	. . . 4 |- W = ( N ClWWalksN G )
8	7	eleclclwwlksnlem1 26938	. . 3 |- ( ( k e. ( 0 ... N ) /\ ( X e. W /\ x e. W ) ) -> ( ( X = ( x cyclShift k ) /\ E. m e. ( 0 ... N ) Y = ( x cyclShift m ) ) -> E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) )
9	3, 6, 8	sylc 65	. 2 |- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. m e. ( 0 ... N ) Y = ( x cyclShift m ) ) -> E. n e. ( 0 ... N ) Y = ( X cyclShift n ) )
10		eqid 2622	. . . . . . . . . . . 12 |- (Vtx ` G ) = (Vtx ` G )
11	10	clwwlknbp 26885	. . . . . . . . . . 11 |- ( x e. ( N ClWWalksN G ) -> ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) )
12	11, 7	eleq2s 2719	. . . . . . . . . 10 |- ( x e. W -> ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) )
13		fznn0sub2 12446	. . . . . . . . . . . 12 |- ( k e. ( 0 ... N ) -> ( N - k ) e. ( 0 ... N ) )
14		oveq1 6657	. . . . . . . . . . . . 13 |- ( ( # ` x ) = N -> ( ( # ` x ) - k ) = ( N - k ) )
15	14	eleq1d 2686	. . . . . . . . . . . 12 |- ( ( # ` x ) = N -> ( ( ( # ` x ) - k ) e. ( 0 ... N ) <-> ( N - k ) e. ( 0 ... N ) ) )
16	13, 15	syl5ibr 236	. . . . . . . . . . 11 |- ( ( # ` x ) = N -> ( k e. ( 0 ... N ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) )
17	16	adantl 482	. . . . . . . . . 10 |- ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) -> ( k e. ( 0 ... N ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) )
18	12, 17	syl 17	. . . . . . . . 9 |- ( x e. W -> ( k e. ( 0 ... N ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) )
19	18	adantl 482	. . . . . . . 8 |- ( ( X e. W /\ x e. W ) -> ( k e. ( 0 ... N ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) )
20	19	com12 32	. . . . . . 7 |- ( k e. ( 0 ... N ) -> ( ( X e. W /\ x e. W ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) )
21	20	adantr 481	. . . . . 6 |- ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) -> ( ( X e. W /\ x e. W ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) ) )
22	21	imp 445	. . . . 5 |- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) )
23	22	adantr 481	. . . 4 |- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) -> ( ( # ` x ) - k ) e. ( 0 ... N ) )
24		simpr 477	. . . . . 6 |- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( X e. W /\ x e. W ) )
25	24	ancomd 467	. . . . 5 |- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( x e. W /\ X e. W ) )
26	25	adantr 481	. . . 4 |- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) -> ( x e. W /\ X e. W ) )
27	23, 26	jca 554	. . 3 |- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) -> ( ( ( # ` x ) - k ) e. ( 0 ... N ) /\ ( x e. W /\ X e. W ) ) )
28		simpll 790	. . . . . . . . . . . . 13 |- ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) /\ k e. ( 0 ... N ) ) -> x e. Word (Vtx ` G ) )
29		oveq2 6658	. . . . . . . . . . . . . . . . 17 |- ( N = ( # ` x ) -> ( 0 ... N ) = ( 0 ... ( # ` x ) ) )
30	29	eleq2d 2687	. . . . . . . . . . . . . . . 16 |- ( N = ( # ` x ) -> ( k e. ( 0 ... N ) <-> k e. ( 0 ... ( # ` x ) ) ) )
31	30	eqcoms 2630	. . . . . . . . . . . . . . 15 |- ( ( # ` x ) = N -> ( k e. ( 0 ... N ) <-> k e. ( 0 ... ( # ` x ) ) ) )
32	31	adantl 482	. . . . . . . . . . . . . 14 |- ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) -> ( k e. ( 0 ... N ) <-> k e. ( 0 ... ( # ` x ) ) ) )
33	32	biimpa 501	. . . . . . . . . . . . 13 |- ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) /\ k e. ( 0 ... N ) ) -> k e. ( 0 ... ( # ` x ) ) )
34	28, 33	jca 554	. . . . . . . . . . . 12 |- ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) /\ k e. ( 0 ... N ) ) -> ( x e. Word (Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) )
35	34	ex 450	. . . . . . . . . . 11 |- ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) -> ( k e. ( 0 ... N ) -> ( x e. Word (Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) ) )
36	12, 35	syl 17	. . . . . . . . . 10 |- ( x e. W -> ( k e. ( 0 ... N ) -> ( x e. Word (Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) ) )
37	36	adantl 482	. . . . . . . . 9 |- ( ( X e. W /\ x e. W ) -> ( k e. ( 0 ... N ) -> ( x e. Word (Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) ) )
38	37	com12 32	. . . . . . . 8 |- ( k e. ( 0 ... N ) -> ( ( X e. W /\ x e. W ) -> ( x e. Word (Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) ) )
39	38	adantr 481	. . . . . . 7 |- ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) -> ( ( X e. W /\ x e. W ) -> ( x e. Word (Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) ) )
40	39	imp 445	. . . . . 6 |- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( x e. Word (Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) )
41	4	eqcomd 2628	. . . . . . 7 |- ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) -> ( x cyclShift k ) = X )
42	41	adantr 481	. . . . . 6 |- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( x cyclShift k ) = X )
43		oveq1 6657	. . . . . . . 8 |- ( X = ( x cyclShift k ) -> ( X cyclShift ( ( # ` x ) - k ) ) = ( ( x cyclShift k ) cyclShift ( ( # ` x ) - k ) ) )
44	43	eqcoms 2630	. . . . . . 7 |- ( ( x cyclShift k ) = X -> ( X cyclShift ( ( # ` x ) - k ) ) = ( ( x cyclShift k ) cyclShift ( ( # ` x ) - k ) ) )
45		elfzelz 12342	. . . . . . . 8 |- ( k e. ( 0 ... ( # ` x ) ) -> k e. ZZ )
46		2cshwid 13560	. . . . . . . 8 |- ( ( x e. Word (Vtx ` G ) /\ k e. ZZ ) -> ( ( x cyclShift k ) cyclShift ( ( # ` x ) - k ) ) = x )
47	45, 46	sylan2 491	. . . . . . 7 |- ( ( x e. Word (Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) -> ( ( x cyclShift k ) cyclShift ( ( # ` x ) - k ) ) = x )
48	44, 47	sylan9eqr 2678	. . . . . 6 |- ( ( ( x e. Word (Vtx ` G ) /\ k e. ( 0 ... ( # ` x ) ) ) /\ ( x cyclShift k ) = X ) -> ( X cyclShift ( ( # ` x ) - k ) ) = x )
49	40, 42, 48	syl2anc 693	. . . . 5 |- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( X cyclShift ( ( # ` x ) - k ) ) = x )
50	49	eqcomd 2628	. . . 4 |- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> x = ( X cyclShift ( ( # ` x ) - k ) ) )
51	50	anim1i 592	. . 3 |- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) -> ( x = ( X cyclShift ( ( # ` x ) - k ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) )
52	7	eleclclwwlksnlem1 26938	. . 3 |- ( ( ( ( # ` x ) - k ) e. ( 0 ... N ) /\ ( x e. W /\ X e. W ) ) -> ( ( x = ( X cyclShift ( ( # ` x ) - k ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) -> E. m e. ( 0 ... N ) Y = ( x cyclShift m ) ) )
53	27, 51, 52	sylc 65	. 2 |- ( ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) /\ E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) -> E. m e. ( 0 ... N ) Y = ( x cyclShift m ) )
54	9, 53	impbida 877	1 |- ( ( ( k e. ( 0 ... N ) /\ X = ( x cyclShift k ) ) /\ ( X e. W /\ x e. W ) ) -> ( E. m e. ( 0 ... N ) Y = ( x cyclShift m ) <-> E. n e. ( 0 ... N ) Y = ( X cyclShift n ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   ` cfv 5888  (class class class)co 6650   0cc0 9936   - cmin 10266   ZZcz 11377   ...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:  eleclclwwlksn  26953