Metamath Proof Explorer


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 ⟩ ∣ ( 𝑡𝑊𝑢𝑊 ∧ ∃ 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 ⟩ ∣ ( 𝑡𝑊𝑢𝑊 ∧ ∃ 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 ) )