Theorem erclwwlkeq 30097

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
erclwwlk.r	⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}

Assertion

Ref	Expression
erclwwlkeq	⊢ ((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 erclwwlkeq

Step	Hyp	Ref	Expression
1		eleq1 2825	. . . 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 2825	. . . 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 6835	. . . . . 6 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
6	5	oveq2d 7376	. . . . 5 ⊢ (𝑤 = 𝑊 → (0...(♯‘𝑤)) = (0...(♯‘𝑊)))
7	6	adantl 481	. . . 4 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (0...(♯‘𝑤)) = (0...(♯‘𝑊)))
8		simpl 482	. . . . 5 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → 𝑢 = 𝑈)
9		oveq1 7367	. . . . . 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 2753	. . . 4 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (𝑢 = (𝑤 cyclShift 𝑛) ↔ 𝑈 = (𝑊 cyclShift 𝑛)))
12	7, 11	rexeqbidv 3318	. . 3 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(♯‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)))
13	2, 4, 12	3anbi123d 1439	. 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		erclwwlk.r	. 2 ⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
15	13, 14	brabga 5483	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   class class class wbr 5099  {copab 5161  ‘cfv 6493  (class class class)co 7360  0cc0 11030  ...cfz 13427  ♯chash 14257   cyclShift ccsh 14715  ClWWalkscclwwlk 30060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-iota 6449  df-fv 6501  df-ov 7363

This theorem is referenced by:  erclwwlkeqlen  30098  erclwwlkref  30099  erclwwlksym  30100  erclwwlktr  30101