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	$$⊢F\mathrm{Walks}\left(G\right)P\to \left|F\right|\in {\mathbb{N}}_0$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\mathrm{iEdg}\left(G\right)=\mathrm{iEdg}\left(G\right)$$
2	1	wlkf	$$⊢F\mathrm{Walks}\left(G\right)P\to F\in \mathrm{Word}\mathrm{dom}\mathrm{iEdg}\left(G\right)$$
3		lencl	$$⊢F\in \mathrm{Word}\mathrm{dom}\mathrm{iEdg}\left(G\right)\to \left|F\right|\in {\mathbb{N}}_0$$
4	2 3	syl	$$⊢F\mathrm{Walks}\left(G\right)P\to \left|F\right|\in {\mathbb{N}}_0$$