Theorem eleclclwwlknlem2 27585

Description: Lemma 2 for eleclclwwlkn 27600. (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 475	. . . . 5 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → 𝑘 ∈ (0...𝑁))
2	1	anim1i 605	. . . 4 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑘 ∈ (0...𝑁) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)))
3	2	adantr 473	. . 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 477	. . . . 5 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → 𝑋 = (𝑥 cyclShift 𝑘))
5	4	adantr 473	. . . 4 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → 𝑋 = (𝑥 cyclShift 𝑘))
6	5	anim1i 605	. . 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 27584	. . 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 2778	. . . . . . . . . . . 12 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
11	10	clwwlknbp 27550	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
12	11, 7	eleq2s 2884	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
13		fznn0sub2 12830	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...𝑁) → (𝑁 − 𝑘) ∈ (0...𝑁))
14		oveq1 6983	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) = 𝑁 → ((♯‘𝑥) − 𝑘) = (𝑁 − 𝑘))
15	14	eleq1d 2850	. . . . . . . . . . . 12 ⊢ ((♯‘𝑥) = 𝑁 → (((♯‘𝑥) − 𝑘) ∈ (0...𝑁) ↔ (𝑁 − 𝑘) ∈ (0...𝑁)))
16	13, 15	syl5ibr 238	. . . . . . . . . . 11 ⊢ ((♯‘𝑥) = 𝑁 → (𝑘 ∈ (0...𝑁) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
17	16	adantl 474	. . . . . . . . . 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 474	. . . . . . . 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 473	. . . . . 6 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
22	21	imp 398	. . . . 5 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁))
23	22	adantr 473	. . . 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 477	. . . . . 6 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊))
25	24	ancomd 454	. . . . 5 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊))
26	25	adantr 473	. . . 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 504	. . 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 754	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → 𝑥 ∈ Word (Vtx‘𝐺))
29		oveq2 6984	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
30	29	eleq2d 2851	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 = (♯‘𝑥) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑥))))
31	30	eqcoms 2786	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = 𝑁 → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑥))))
32	31	adantl 474	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑥))))
33	32	biimpa 469	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...(♯‘𝑥)))
34	28, 33	jca 504	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))))
35	34	ex 405	. . . . . . . . . . 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 474	. . . . . . . . 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 473	. . . . . . 7 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
40	39	imp 398	. . . . . 6 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))))
41	4	eqcomd 2784	. . . . . . 7 ⊢ ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → (𝑥 cyclShift 𝑘) = 𝑋)
42	41	adantr 473	. . . . . 6 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑥 cyclShift 𝑘) = 𝑋)
43		oveq1 6983	. . . . . . . 8 ⊢ (𝑋 = (𝑥 cyclShift 𝑘) → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)))
44	43	eqcoms 2786	. . . . . . 7 ⊢ ((𝑥 cyclShift 𝑘) = 𝑋 → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)))
45		elfzelz 12724	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(♯‘𝑥)) → 𝑘 ∈ ℤ)
46		2cshwid 14038	. . . . . . . 8 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ ℤ) → ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
47	45, 46	sylan2 583	. . . . . . 7 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))) → ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
48	44, 47	sylan9eqr 2836	. . . . . 6 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))) ∧ (𝑥 cyclShift 𝑘) = 𝑋) → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
49	40, 42, 48	syl2anc 576	. . . . 5 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
50	49	eqcomd 2784	. . . 4 ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → 𝑥 = (𝑋 cyclShift ((♯‘𝑥) − 𝑘)))
51	50	anim1i 605	. . 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 27584	. . 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 788	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 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∃wrex 3089  ‘cfv 6188  (class class class)co 6976  0cc0 10335   − cmin 10670  ℤcz 11793  ...cfz 12708  ♯chash 13505  Word cword 13672   cyclShift ccsh 14007  Vtxcvtx 26484   ClWWalksN cclwwlkn 27539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412  ax-pre-sup 10413

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-map 8208  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-sup 8701  df-inf 8702  df-card 9162  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-div 11099  df-nn 11440  df-2 11503  df-n0 11708  df-z 11794  df-uz 12059  df-rp 12205  df-fz 12709  df-fzo 12850  df-fl 12977  df-mod 13053  df-hash 13506  df-word 13673  df-concat 13734  df-substr 13804  df-pfx 13853  df-csh 14009  df-clwwlk 27488  df-clwwlkn 27540

This theorem is referenced by:  eleclclwwlkn  27600