Theorem erclwwlkeq 29997

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 2816	. . . 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 2816	. . . 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 6840	. . . . . 6 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
6	5	oveq2d 7385	. . . . 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 7376	. . . . . 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 2745	. . . 4 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (𝑢 = (𝑤 cyclShift 𝑛) ↔ 𝑈 = (𝑊 cyclShift 𝑛)))
12	7, 11	rexeqbidv 3317	. . 3 ⊢ ((𝑢 = 𝑈 ∧ 𝑤 = 𝑊) → (∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0...(♯‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)))
13	2, 4, 12	3anbi123d 1438	. 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 5489	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 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   class class class wbr 5102  {copab 5164  ‘cfv 6499  (class class class)co 7369  0cc0 11044  ...cfz 13444  ♯chash 14271   cyclShift ccsh 14729  ClWWalkscclwwlk 29960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-iota 6452  df-fv 6507  df-ov 7372

This theorem is referenced by:  erclwwlkeqlen  29998  erclwwlkref  29999  erclwwlksym  30000  erclwwlktr  30001