Theorem wwlkswwlksn

Description: A walk of a fixed length as word is a walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 17-Jul-2018) (Revised by AV, 12-Apr-2021)

	Ref	Expression
Assertion	wwlkswwlksn	⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑊 ∈ ( WWalks ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2	1	wwlknbp	⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) )
3		iswwlksn	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ↔ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ) )
4	3	3ad2ant2	⊢ ( ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ↔ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ) )
5		simpl	⊢ ( ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → 𝑊 ∈ ( WWalks ‘ 𝐺 ) )
6	4 5	biimtrdi	⊢ ( ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ) → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑊 ∈ ( WWalks ‘ 𝐺 ) ) )
7	2 6	mpcom	⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑊 ∈ ( WWalks ‘ 𝐺 ) )