Theorem erclwwlkneq 30685

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 2852	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑡 ∈ 𝑊 ↔ 𝑇 ∈ 𝑊))
2	1	adantr 486	. . 3 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑡 ∈ 𝑊 ↔ 𝑇 ∈ 𝑊))
3		eleq1 2852	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ 𝑊 ↔ 𝑈 ∈ 𝑊))
4	3	adantl 487	. . 3 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑢 ∈ 𝑊 ↔ 𝑈 ∈ 𝑊))
5		simpl 488	. . . . 5 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → 𝑡 = 𝑇)
6		oveq1 7447	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛))
7	6	adantl 487	. . . . 5 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛))
8	5, 7	eqeq12d 2779	. . . 4 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑡 = (𝑢 cyclShift 𝑛) ↔ 𝑇 = (𝑈 cyclShift 𝑛)))
9	8	rexbidv 3190	. . 3 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)))
10	2, 4, 9	3anbi123d 1465	. 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 5512	1 ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 ↔ (𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   class class class wbr 5106  {copab 5170  (class class class)co 7440  0cc0 11224  ...cfz 13663   cyclShift ccsh 14964   ClWWalksN cclwwlkn 30642

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-iota 6509  df-fv 6563  df-ov 7443

This theorem is used by:  erclwwlkneqlen  30686  erclwwlknref  30687  erclwwlknsym  30688  erclwwlkntr  30689  eclclwwlkn1  30693