Theorem erclwwlkneqlen

Description: If two classes are equivalent regarding .~ , then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-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	erclwwlkneqlen	⊢ ( ( 𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌 ) → ( 𝑇 ∼ 𝑈 → ( ♯ ‘ 𝑇 ) = ( ♯ ‘ 𝑈 ) ) )

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	1 2	erclwwlkneq	⊢ ( ( 𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌 ) → ( 𝑇 ∼ 𝑈 ↔ ( 𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑇 = ( 𝑈 cyclShift 𝑛 ) ) ) )
4		fveq2	⊢ ( 𝑇 = ( 𝑈 cyclShift 𝑛 ) → ( ♯ ‘ 𝑇 ) = ( ♯ ‘ ( 𝑈 cyclShift 𝑛 ) ) )
5		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
6	5	clwwlknwrd	⊢ ( 𝑈 ∈ ( 𝑁 ClWWalksN 𝐺 ) → 𝑈 ∈ Word ( Vtx ‘ 𝐺 ) )
7	6 1	eleq2s	⊢ ( 𝑈 ∈ 𝑊 → 𝑈 ∈ Word ( Vtx ‘ 𝐺 ) )
8	7	adantl	⊢ ( ( 𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ) → 𝑈 ∈ Word ( Vtx ‘ 𝐺 ) )
9		elfzelz	⊢ ( 𝑛 ∈ ( 0 ... 𝑁 ) → 𝑛 ∈ ℤ )
10		cshwlen	⊢ ( ( 𝑈 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑛 ∈ ℤ ) → ( ♯ ‘ ( 𝑈 cyclShift 𝑛 ) ) = ( ♯ ‘ 𝑈 ) )
11	8 9 10	syl2an	⊢ ( ( ( 𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ) ∧ 𝑛 ∈ ( 0 ... 𝑁 ) ) → ( ♯ ‘ ( 𝑈 cyclShift 𝑛 ) ) = ( ♯ ‘ 𝑈 ) )
12	4 11	sylan9eqr	⊢ ( ( ( ( 𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ) ∧ 𝑛 ∈ ( 0 ... 𝑁 ) ) ∧ 𝑇 = ( 𝑈 cyclShift 𝑛 ) ) → ( ♯ ‘ 𝑇 ) = ( ♯ ‘ 𝑈 ) )
13	12	rexlimdva2	⊢ ( ( 𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ) → ( ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑇 = ( 𝑈 cyclShift 𝑛 ) → ( ♯ ‘ 𝑇 ) = ( ♯ ‘ 𝑈 ) ) )
14	13	3impia	⊢ ( ( 𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑇 = ( 𝑈 cyclShift 𝑛 ) ) → ( ♯ ‘ 𝑇 ) = ( ♯ ‘ 𝑈 ) )
15	3 14	syl6bi	⊢ ( ( 𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌 ) → ( 𝑇 ∼ 𝑈 → ( ♯ ‘ 𝑇 ) = ( ♯ ‘ 𝑈 ) ) )