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Metamath Proof Explorer

Theorem wlkcl

Description: A walk has length # ( F ) , which is an integer. Formerly proven for an Eulerian path, see eupthcl . (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by AV, 18-Feb-2021)

Ref Expression
Assertion wlkcl ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ∈ ℕ0 )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
2 1 wlkf ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) )
3 lencl ⊢ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) → ( ♯ ‘ 𝐹 ) ∈ ℕ0 )
4 2 3 syl ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ♯ ‘ 𝐹 ) ∈ ℕ0 )