Theorem erclwwlkneq

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)

	Ref	Expression
Hypotheses	erclwwlkn.w	⊢ 𝑊 = ( 𝑁 ClWWalksN 𝐺 )
	erclwwlkn.r	⊢ ∼ = { ⟨ 𝑡 , 𝑢 ⟩ ∣ ( 𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑡 = ( 𝑢 cyclShift 𝑛 ) ) }
Assertion	erclwwlkneq	⊢ ( ( 𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌 ) → ( 𝑇 ∼ 𝑈 ↔ ( 𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑇 = ( 𝑈 cyclShift 𝑛 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		erclwwlkn.w	⊢ 𝑊 = ( 𝑁 ClWWalksN 𝐺 )
2		erclwwlkn.r	⊢ ∼ = { ⟨ 𝑡 , 𝑢 ⟩ ∣ ( 𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑡 = ( 𝑢 cyclShift 𝑛 ) ) }
3		eleq1	⊢ ( 𝑡 = 𝑇 → ( 𝑡 ∈ 𝑊 ↔ 𝑇 ∈ 𝑊 ) )
4	3	adantr	⊢ ( ( 𝑡 = 𝑇 ∧ 𝑢 = 𝑈 ) → ( 𝑡 ∈ 𝑊 ↔ 𝑇 ∈ 𝑊 ) )
5		eleq1	⊢ ( 𝑢 = 𝑈 → ( 𝑢 ∈ 𝑊 ↔ 𝑈 ∈ 𝑊 ) )
6	5	adantl	⊢ ( ( 𝑡 = 𝑇 ∧ 𝑢 = 𝑈 ) → ( 𝑢 ∈ 𝑊 ↔ 𝑈 ∈ 𝑊 ) )
7		simpl	⊢ ( ( 𝑡 = 𝑇 ∧ 𝑢 = 𝑈 ) → 𝑡 = 𝑇 )
8		oveq1	⊢ ( 𝑢 = 𝑈 → ( 𝑢 cyclShift 𝑛 ) = ( 𝑈 cyclShift 𝑛 ) )
9	8	adantl	⊢ ( ( 𝑡 = 𝑇 ∧ 𝑢 = 𝑈 ) → ( 𝑢 cyclShift 𝑛 ) = ( 𝑈 cyclShift 𝑛 ) )
10	7 9	eqeq12d	⊢ ( ( 𝑡 = 𝑇 ∧ 𝑢 = 𝑈 ) → ( 𝑡 = ( 𝑢 cyclShift 𝑛 ) ↔ 𝑇 = ( 𝑈 cyclShift 𝑛 ) ) )
11	10	rexbidv	⊢ ( ( 𝑡 = 𝑇 ∧ 𝑢 = 𝑈 ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑡 = ( 𝑢 cyclShift 𝑛 ) ↔ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑇 = ( 𝑈 cyclShift 𝑛 ) ) )
12	4 6 11	3anbi123d	⊢ ( ( 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 ) ) ) )
13	12 2	brabga	⊢ ( ( 𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌 ) → ( 𝑇 ∼ 𝑈 ↔ ( 𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑇 = ( 𝑈 cyclShift 𝑛 ) ) ) )