Theorem eleclclwwlknlem2 27842

Description: Lemma 2 for eleclclwwlkn 27857. (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 27841	. . 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 2823	. . . . . . . . . . . 12 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
11	10	clwwlknbp 27815	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
12	11, 7	eleq2s 2933	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
13		fznn0sub2 13017	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...𝑁) → (𝑁 − 𝑘) ∈ (0...𝑁))
14		oveq1 7165	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) = 𝑁 → ((♯‘𝑥) − 𝑘) = (𝑁 − 𝑘))
15	14	eleq1d 2899	. . . . . . . . . . . 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 7166	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
30	29	eleq2d 2900	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (♯‘𝑥) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑥))))
31	30	eqcoms 2831	. . . . . . . . . . . . . . 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 2829	. . . . . . 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 7165	. . . . . . . 8 ⊢ (𝑋 = (𝑥 cyclShift 𝑘) → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)))
44	43	eqcoms 2831	. . . . . . 7 ⊢ ((𝑥 cyclShift 𝑘) = 𝑋 → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)))
45		elfzelz 12911	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(♯‘𝑥)) → 𝑘 ∈ ℤ)
46		2cshwid 14178	. . . . . . . 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 2880	. . . . . 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 2829	. . . 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 27841	. . 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 1537   ∈ wcel 2114  ∃wrex 3141  ‘cfv 6357  (class class class)co 7158  0cc0 10539   − cmin 10872  ℤcz 11984  ...cfz 12895  ♯chash 13693  Word cword 13864   cyclShift ccsh 14152  Vtxcvtx 26783   ClWWalksN cclwwlkn 27804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-sup 8908  df-inf 8909  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-n0 11901  df-z 11985  df-uz 12247  df-rp 12393  df-fz 12896  df-fzo 13037  df-fl 13165  df-mod 13241  df-hash 13694  df-word 13865  df-concat 13925  df-substr 14005  df-pfx 14035  df-csh 14153  df-clwwlk 27762  df-clwwlkn 27805

This theorem is referenced by:  eleclclwwlkn  27857