Theorem eleclclwwlknlem2 30412

Description: Lemma 2 for eleclclwwlkn 30427. (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 487	. . . . 5 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → 𝑘 ∈ (0...𝑁))
2	1	anim1i 626	. . . 4 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑘 ∈ (0...𝑁) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)))
3	2	adantr 485	. . 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 489	. . . . 5 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → 𝑋 = (𝑥 cyclShift 𝑘))
5	4	adantr 485	. . . 4 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → 𝑋 = (𝑥 cyclShift 𝑘))
6	5	anim1i 626	. . 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 30411	. . 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 66	. 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 2763	. . . . . . . . . . . 12 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
11	10	clwwlknbp 30386	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
12	11, 7	eleq2s 2881	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
13		fznn0sub2 13659	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...𝑁) → (𝑁 − 𝑘) ∈ (0...𝑁))
14		oveq1 7417	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) = 𝑁 → ((♯‘𝑥) − 𝑘) = (𝑁 − 𝑘))
15	14	eleq1d 2848	. . . . . . . . . . . 12 ⊢ ((♯‘𝑥) = 𝑁 → (((♯‘𝑥) − 𝑘) ∈ (0...𝑁) ↔ (𝑁 − 𝑘) ∈ (0...𝑁)))
16	13, 15	imbitrrid 249	. . . . . . . . . . 11 ⊢ ((♯‘𝑥) = 𝑁 → (𝑘 ∈ (0...𝑁) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
17	16	adantl 486	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
18	12, 17	syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 → (𝑘 ∈ (0...𝑁) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
19	18	adantl 486	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (𝑘 ∈ (0...𝑁) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
20	19	com12 33	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
21	20	adantr 485	. . . . . 6 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
22	21	imp 411	. . . . 5 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁))
23	22	adantr 485	. . . 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 489	. . . . . 6 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊))
25	24	ancomd 466	. . . . 5 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊))
26	25	adantr 485	. . . 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 520	. . 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 778	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → 𝑥 ∈ Word (Vtx‘𝐺))
29		oveq2 7418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
30	29	eleq2d 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (♯‘𝑥) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑥))))
31	30	eqcoms 2771	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = 𝑁 → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑥))))
32	31	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑥))))
33	32	biimpa 481	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...(♯‘𝑥)))
34	28, 33	jca 520	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))))
35	34	ex 417	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
36	12, 35	syl 18	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → (𝑘 ∈ (0...𝑁) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
37	36	adantl 486	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (𝑘 ∈ (0...𝑁) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
38	37	com12 33	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑁) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
39	38	adantr 485	. . . . . . 7 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
40	39	imp 411	. . . . . 6 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))))
41	4	eqcomd 2769	. . . . . . 7 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → (𝑥 cyclShift 𝑘) = 𝑋)
42	41	adantr 485	. . . . . 6 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑥 cyclShift 𝑘) = 𝑋)
43		oveq1 7417	. . . . . . . 8 ⊢ (𝑋 = (𝑥 cyclShift 𝑘) → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)))
44	43	eqcoms 2771	. . . . . . 7 ⊢ ((𝑥 cyclShift 𝑘) = 𝑋 → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)))
45		elfzelz 13547	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(♯‘𝑥)) → 𝑘 ∈ ℤ)
46		2cshwid 14847	. . . . . . . 8 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ ℤ) → ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
47	45, 46	sylan2 604	. . . . . . 7 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))) → ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
48	44, 47	sylan9eqr 2820	. . . . . 6 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))) ∧ (𝑥 cyclShift 𝑘) = 𝑋) → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
49	40, 42, 48	syl2anc 595	. . . . 5 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
50	49	eqcomd 2769	. . . 4 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → 𝑥 = (𝑋 cyclShift ((♯‘𝑥) − 𝑘)))
51	50	anim1i 626	. . 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 30411	. . 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 66	. 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 812	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 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  ‘cfv 6536  (class class class)co 7410  0cc0 11095   − cmin 11436  ℤcz 12586  ...cfz 13530  ♯chash 14362  Word cword 14546   cyclShift ccsh 14821  Vtxcvtx 29346   ClWWalksN cclwwlkn 30375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-n0 12500  df-z 12587  df-uz 12858  df-rp 13012  df-fz 13531  df-fzo 13679  df-fl 13821  df-mod 13899  df-hash 14363  df-word 14547  df-concat 14604  df-substr 14675  df-pfx 14705  df-csh 14822  df-clwwlk 30333  df-clwwlkn 30376

This theorem is referenced by:  eleclclwwlkn  30427