Theorem erclwwlkseq 41239

Description: Two classes are equivalent regarding ∼ if both are words and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.)

Hypothesis

Ref	Expression
erclwwlks.r	⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalkS‘𝐺) ∧ 𝑤 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}

Assertion

Ref	Expression
erclwwlkseq	⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 ↔ (𝑈 ∈ (ClWWalkS‘𝐺) ∧ 𝑊 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))))

Distinct variable groups:   𝑛,𝐺,𝑢,𝑤   𝑈,𝑛,𝑢,𝑤   𝑛,𝑊,𝑢,𝑤

Allowed substitution hints:   ∼ (𝑤,𝑢,𝑛)   𝑋(𝑤,𝑢,𝑛)   𝑌(𝑤,𝑢,𝑛)

Proof of Theorem erclwwlkseq

Step	Hyp	Ref	Expression
1		eleq1 2676	. . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 ∈ (ClWWalkS‘𝐺) ↔ 𝑈 ∈ (ClWWalkS‘𝐺)))
2	1	adantr 480	. . 3 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (𝑢 ∈ (ClWWalkS‘𝐺) ↔ 𝑈 ∈ (ClWWalkS‘𝐺)))
3		eleq1 2676	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑤 ∈ (ClWWalkS‘𝐺) ↔ 𝑊 ∈ (ClWWalkS‘𝐺)))
4	3	adantl 481	. . 3 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (𝑤 ∈ (ClWWalkS‘𝐺) ↔ 𝑊 ∈ (ClWWalkS‘𝐺)))
5		fveq2 6103	. . . . . 6 ⊢ (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
6	5	oveq2d 6565	. . . . 5 ⊢ (𝑤 = 𝑊 → (0...(#‘𝑤)) = (0...(#‘𝑊)))
7	6	adantl 481	. . . 4 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (0...(#‘𝑤)) = (0...(#‘𝑊)))
8		simpl 472	. . . . 5 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → 𝑢 = 𝑈)
9		oveq1 6556	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑤 cyclShift 𝑛) = (𝑊 cyclShift 𝑛))
10	9	adantl 481	. . . . 5 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (𝑤 cyclShift 𝑛) = (𝑊 cyclShift 𝑛))
11	8, 10	eqeq12d 2625	. . . 4 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (𝑢 = (𝑤 cyclShift 𝑛) ↔ 𝑈 = (𝑊 cyclShift 𝑛)))
12	7, 11	rexeqbidv 3130	. . 3 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)))
13	2, 4, 12	3anbi123d 1391	. 2 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → ((𝑢 ∈ (ClWWalkS‘𝐺) ∧ 𝑤 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛)) ↔ (𝑈 ∈ (ClWWalkS‘𝐺) ∧ 𝑊 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))))
14		erclwwlks.r	. 2 ⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalkS‘𝐺) ∧ 𝑤 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
15	13, 14	brabga 4914	1 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 ↔ (𝑈 ∈ (ClWWalkS‘𝐺) ∧ 𝑊 ∈ (ClWWalkS‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   class class class wbr 4583  {copab 4642  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ...cfz 12197  #chash 12979   cyclShift ccsh 13385  ClWWalkScclwwlks 41183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552

This theorem is referenced by:  erclwwlkseqlen  41240  erclwwlksref  41241  erclwwlkssym  41242  erclwwlkstr  41243