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 )