Theorem eleclclwwlknlem2 27832

Description: Lemma 2 for eleclclwwlkn 27847. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 30-Apr-2021.)

Hypothesis

Ref	Expression
erclwwlkn1.w	⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)

Assertion

Ref	Expression
eleclclwwlknlem2	⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚) ↔ ∃𝑛 ∈ (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 eleclclwwlknlem2

Step	Hyp	Ref	Expression
1		simpl 485	. . . . 5 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → 𝑘 ∈ (0...𝑁))
2	1	anim1i 616	. . . 4 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑘 ∈ (0...𝑁) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)))
3	2	adantr 483	. . 3 ⊢ ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)) → (𝑘 ∈ (0...𝑁) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)))
4		simpr 487	. . . . 5 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → 𝑋 = (𝑥 cyclShift 𝑘))
5	4	adantr 483	. . . 4 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → 𝑋 = (𝑥 cyclShift 𝑘))
6	5	anim1i 616	. . 3 ⊢ ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)) → (𝑋 = (𝑥 cyclShift 𝑘) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)))
7		erclwwlkn1.w	. . . 4 ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)
8	7	eleclclwwlknlem1 27831	. . 3 ⊢ ((𝑘 ∈ (0...𝑁) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → ((𝑋 = (𝑥 cyclShift 𝑘) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
9	3, 6, 8	sylc 65	. 2 ⊢ ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))
10		eqid 2819	. . . . . . . . . . . 12 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
11	10	clwwlknbp 27805	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
12	11, 7	eleq2s 2929	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
13		fznn0sub2 13006	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...𝑁) → (𝑁 − 𝑘) ∈ (0...𝑁))
14		oveq1 7155	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) = 𝑁 → ((♯‘𝑥) − 𝑘) = (𝑁 − 𝑘))
15	14	eleq1d 2895	. . . . . . . . . . . 12 ⊢ ((♯‘𝑥) = 𝑁 → (((♯‘𝑥) − 𝑘) ∈ (0...𝑁) ↔ (𝑁 − 𝑘) ∈ (0...𝑁)))
16	13, 15	syl5ibr 248	. . . . . . . . . . 11 ⊢ ((♯‘𝑥) = 𝑁 → (𝑘 ∈ (0...𝑁) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
17	16	adantl 484	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
18	12, 17	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 → (𝑘 ∈ (0...𝑁) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
19	18	adantl 484	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (𝑘 ∈ (0...𝑁) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
20	19	com12 32	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
21	20	adantr 483	. . . . . 6 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
22	21	imp 409	. . . . 5 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁))
23	22	adantr 483	. . . 4 ⊢ ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁))
24		simpr 487	. . . . . 6 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊))
25	24	ancomd 464	. . . . 5 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊))
26	25	adantr 483	. . . 4 ⊢ ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → (𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊))
27	23, 26	jca 514	. . 3 ⊢ ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → (((♯‘𝑥) − 𝑘) ∈ (0...𝑁) ∧ (𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊)))
28		simpll 765	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → 𝑥 ∈ Word (Vtx‘𝐺))
29		oveq2 7156	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
30	29	eleq2d 2896	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (♯‘𝑥) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑥))))
31	30	eqcoms 2827	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = 𝑁 → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑥))))
32	31	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑥))))
33	32	biimpa 479	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...(♯‘𝑥)))
34	28, 33	jca 514	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))))
35	34	ex 415	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
36	12, 35	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → (𝑘 ∈ (0...𝑁) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
37	36	adantl 484	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (𝑘 ∈ (0...𝑁) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
38	37	com12 32	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑁) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
39	38	adantr 483	. . . . . . 7 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
40	39	imp 409	. . . . . 6 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))))
41	4	eqcomd 2825	. . . . . . 7 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → (𝑥 cyclShift 𝑘) = 𝑋)
42	41	adantr 483	. . . . . 6 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑥 cyclShift 𝑘) = 𝑋)
43		oveq1 7155	. . . . . . . 8 ⊢ (𝑋 = (𝑥 cyclShift 𝑘) → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)))
44	43	eqcoms 2827	. . . . . . 7 ⊢ ((𝑥 cyclShift 𝑘) = 𝑋 → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)))
45		elfzelz 12900	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(♯‘𝑥)) → 𝑘 ∈ ℤ)
46		2cshwid 14168	. . . . . . . 8 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ ℤ) → ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
47	45, 46	sylan2 594	. . . . . . 7 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))) → ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
48	44, 47	sylan9eqr 2876	. . . . . 6 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))) ∧ (𝑥 cyclShift 𝑘) = 𝑋) → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
49	40, 42, 48	syl2anc 586	. . . . 5 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
50	49	eqcomd 2825	. . . 4 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → 𝑥 = (𝑋 cyclShift ((♯‘𝑥) − 𝑘)))
51	50	anim1i 616	. . 3 ⊢ ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → (𝑥 = (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
52	7	eleclclwwlknlem1 27831	. . 3 ⊢ ((((♯‘𝑥) − 𝑘) ∈ (0...𝑁) ∧ (𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊)) → ((𝑥 = (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)))
53	27, 51, 52	sylc 65	. 2 ⊢ ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚))
54	9, 53	impbida 799	1 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ∃wrex 3137  ‘cfv 6348  (class class class)co 7148  0cc0 10529   − cmin 10862  ℤcz 11973  ...cfz 12884  ♯chash 13682  Word cword 13853   cyclShift ccsh 14142  Vtxcvtx 26773   ClWWalksN cclwwlkn 27794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12885  df-fzo 13026  df-fl 13154  df-mod 13230  df-hash 13683  df-word 13854  df-concat 13915  df-substr 13995  df-pfx 14025  df-csh 14143  df-clwwlk 27752  df-clwwlkn 27795

This theorem is referenced by:  eleclclwwlkn  27847