Theorem erclwwlkneq 28431

Description: Two classes are equivalent regarding ∼ if both are words of the same fixed length and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 30-Apr-2021.)

Hypotheses

Ref	Expression
erclwwlkn.w	⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)
erclwwlkn.r	⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}

Assertion

Ref	Expression
erclwwlkneq	⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 ↔ (𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))

Distinct variable groups:   𝑡,𝑊,𝑢   𝑡,𝑁,𝑢   𝑇,𝑛,𝑡,𝑢   𝑈,𝑛,𝑡,𝑢

Allowed substitution hints:   ∼ (𝑢,𝑡,𝑛)   𝐺(𝑢,𝑡,𝑛)   𝑁(𝑛)   𝑊(𝑛)   𝑋(𝑢,𝑡,𝑛)   𝑌(𝑢,𝑡,𝑛)

Proof of Theorem erclwwlkneq

Step	Hyp	Ref	Expression
1		eleq1 2826	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑡 ∈ 𝑊 ↔ 𝑇 ∈ 𝑊))
2	1	adantr 481	. . 3 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑡 ∈ 𝑊 ↔ 𝑇 ∈ 𝑊))
3		eleq1 2826	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ 𝑊 ↔ 𝑈 ∈ 𝑊))
4	3	adantl 482	. . 3 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑢 ∈ 𝑊 ↔ 𝑈 ∈ 𝑊))
5		simpl 483	. . . . 5 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → 𝑡 = 𝑇)
6		oveq1 7282	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛))
7	6	adantl 482	. . . . 5 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛))
8	5, 7	eqeq12d 2754	. . . 4 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑡 = (𝑢 cyclShift 𝑛) ↔ 𝑇 = (𝑈 cyclShift 𝑛)))
9	8	rexbidv 3226	. . 3 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)))
10	2, 4, 9	3anbi123d 1435	. 2 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → ((𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛)) ↔ (𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))
11		erclwwlkn.r	. 2 ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
12	10, 11	brabga 5447	1 ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 ↔ (𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   class class class wbr 5074  {copab 5136  (class class class)co 7275  0cc0 10871  ...cfz 13239   cyclShift ccsh 14501   ClWWalksN cclwwlkn 28388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by:  erclwwlkneqlen  28432  erclwwlknref  28433  erclwwlknsym  28434  erclwwlkntr  28435  eclclwwlkn1  28439