Theorem hashclwwlkn0

Description: The number of closed walks (defined as words) with a fixed length is the sum of the sizes of all equivalence classes according to .~ . (Contributed by Alexander van der Vekens, 10-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	hashclwwlkn0	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( ♯ ‘ 𝑊 ) = Σ 𝑥 ∈ ( 𝑊 / ∼ ) ( ♯ ‘ 𝑥 ) )

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	erclwwlkn	⊢ ∼ Er 𝑊
4	3	a1i	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ∼ Er 𝑊 )
5		clwwlknfi	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑁 ClWWalksN 𝐺 ) ∈ Fin )
6	1 5	eqeltrid	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → 𝑊 ∈ Fin )
7	4 6	qshash	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( ♯ ‘ 𝑊 ) = Σ 𝑥 ∈ ( 𝑊 / ∼ ) ( ♯ ‘ 𝑥 ) )