Metamath Proof Explorer


Theorem erclwwlkn

Description: .~ is an equivalence relation over the set of closed walks (defined as words) with a fixed length. (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 erclwwlkn Er 𝑊

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 erclwwlknrel Rel
4 1 2 erclwwlknsym ( 𝑥 𝑦𝑦 𝑥 )
5 1 2 erclwwlkntr ( ( 𝑥 𝑦𝑦 𝑧 ) → 𝑥 𝑧 )
6 1 2 erclwwlknref ( 𝑥𝑊𝑥 𝑥 )
7 3 4 5 6 iseri Er 𝑊