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Metamath Proof Explorer

Theorem wwlknbp

Description: Basic properties of a walk of a fixed length (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 16-Jul-2018) (Revised by AV, 9-Apr-2021) (Proof shortened by AV, 20-May-2021)

Ref Expression
Hypothesis wwlkbp.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion wwlknbp ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉 ) )

Proof

Step Hyp Ref Expression
1 wwlkbp.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2 df-wwlksn ⊢ WWalksN = ( 𝑛 ∈ ℕ0 , 𝑔 ∈ V ↦ { 𝑤 ∈ ( WWalks ‘ 𝑔 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑛 + 1 ) } )
3 2 elmpocl ⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) )
4 simpl ⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) ∧ 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ) → ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) )
5 4 ancomd ⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) ∧ 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ) → ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ) )
6 iswwlksn ⊢ ( 𝑁 ∈ ℕ0 → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ↔ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ) )
7 6 adantr ⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ↔ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ) )
8 1 wwlkbp ⊢ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉 ) )
9 8 simprd ⊢ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) → 𝑊 ∈ Word 𝑉 )
10 9 adantr ⊢ ( ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) → 𝑊 ∈ Word 𝑉 )
11 7 10 biimtrdi ⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → 𝑊 ∈ Word 𝑉 ) )
12 11 imp ⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) ∧ 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ) → 𝑊 ∈ Word 𝑉 )
13 df-3an ⊢ ( ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉 ) ↔ ( ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑊 ∈ Word 𝑉 ) )
14 5 12 13 sylanbrc ⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) ∧ 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ) → ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉 ) )
15 3 14 mpancom ⊢ ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉 ) )