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Metamath Proof Explorer

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 ) )