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Metamath Proof Explorer

Theorem wlkd

Description: Two words representing a walk in a graph. (Contributed by AV, 7-Feb-2021)

Ref Expression
Hypotheses wlkd.p ⊢ ( 𝜑 → 𝑃 ∈ Word V )
wlkd.f ⊢ ( 𝜑 → 𝐹 ∈ Word V )
wlkd.l ⊢ ( 𝜑 → ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) )
wlkd.e ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) )
wlkd.n ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )
wlkd.g ⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
wlkd.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
wlkd.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
wlkd.a ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 )
Assertion wlkd ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )

Proof

Step Hyp Ref Expression
1 wlkd.p ⊢ ( 𝜑 → 𝑃 ∈ Word V )
2 wlkd.f ⊢ ( 𝜑 → 𝐹 ∈ Word V )
3 wlkd.l ⊢ ( 𝜑 → ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) )
4 wlkd.e ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) )
5 wlkd.n ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )
6 wlkd.g ⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
7 wlkd.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
8 wlkd.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
9 wlkd.a ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 )
10 1 2 3 4 wlkdlem3 ⊢ ( 𝜑 → 𝐹 ∈ Word dom 𝐼 )
11 1 2 3 9 wlkdlem1 ⊢ ( 𝜑 → 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 )
12 1 2 3 4 5 wlkdlem4 ⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) )
13 7 8 iswlk ⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V ) → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )
14 6 2 1 13 syl3anc ⊢ ( 𝜑 → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )
15 10 11 12 14 mpbir3and ⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )