Theorem erclwwlkneq 30099

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 2832	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑡 ∈ 𝑊 ↔ 𝑇 ∈ 𝑊))
2	1	adantr 480	. . 3 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑡 ∈ 𝑊 ↔ 𝑇 ∈ 𝑊))
3		eleq1 2832	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ 𝑊 ↔ 𝑈 ∈ 𝑊))
4	3	adantl 481	. . 3 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑢 ∈ 𝑊 ↔ 𝑈 ∈ 𝑊))
5		simpl 482	. . . . 5 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → 𝑡 = 𝑇)
6		oveq1 7455	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛))
7	6	adantl 481	. . . . 5 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑢 cyclShift 𝑛) = (𝑈 cyclShift 𝑛))
8	5, 7	eqeq12d 2756	. . . 4 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (𝑡 = (𝑢 cyclShift 𝑛) ↔ 𝑇 = (𝑈 cyclShift 𝑛)))
9	8	rexbidv 3185	. . 3 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → (∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)))
10	2, 4, 9	3anbi123d 1436	. 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 5553	1 ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 ↔ (𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   class class class wbr 5166  {copab 5228  (class class class)co 7448  0cc0 11184  ...cfz 13567   cyclShift ccsh 14836   ClWWalksN cclwwlkn 30056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-iota 6525  df-fv 6581  df-ov 7451

This theorem is referenced by:  erclwwlkneqlen  30100  erclwwlknref  30101  erclwwlknsym  30102  erclwwlkntr  30103  eclclwwlkn1  30107