Metamath Proof Explorer

Theorem 2wlkond

Description: A walk of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017) (Revised by AV, 30-Jan-2021) (Revised by AV, 24-Mar-2021)

Ref Expression
Hypotheses 2wlkd.p ⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
2wlkd.f ⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
2wlkd.s ⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
2wlkd.n ⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
2wlkd.e ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
2wlkd.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2wlkd.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion 2wlkond ( 𝜑 → 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐶 ) 𝑃 )

Proof

Step Hyp Ref Expression
1 2wlkd.p ⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
2 2wlkd.f ⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
3 2wlkd.s ⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
4 2wlkd.n ⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
5 2wlkd.e ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
6 2wlkd.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
7 2wlkd.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
8 1 2 3 4 5 6 7 2wlkd ⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
9 3 simp1d ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
10 1 fveq1i ⊢ ( 𝑃 ‘ 0 ) = ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 0 )
11 s3fv0 ⊢ ( 𝐴 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 0 ) = 𝐴 )
12 10 11 eqtrid ⊢ ( 𝐴 ∈ 𝑉 → ( 𝑃 ‘ 0 ) = 𝐴 )
13 9 12 syl ⊢ ( 𝜑 → ( 𝑃 ‘ 0 ) = 𝐴 )
14 2 fveq2i ⊢ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ ⟨“ 𝐽 𝐾 ”⟩ )
15 s2len ⊢ ( ♯ ‘ ⟨“ 𝐽 𝐾 ”⟩ ) = 2
16 14 15 eqtri ⊢ ( ♯ ‘ 𝐹 ) = 2
17 1 16 fveq12i ⊢ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 2 )
18 3 simp3d ⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
19 s3fv2 ⊢ ( 𝐶 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 2 ) = 𝐶 )
20 18 19 syl ⊢ ( 𝜑 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 2 ) = 𝐶 )
21 17 20 eqtrid ⊢ ( 𝜑 → ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐶 )
22 3simpb ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
23 3 22 syl ⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
24 s2cli ⊢ ⟨“ 𝐽 𝐾 ”⟩ ∈ Word V
25 2 24 eqeltri ⊢ 𝐹 ∈ Word V
26 s3cli ⊢ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ Word V
27 1 26 eqeltri ⊢ 𝑃 ∈ Word V
28 25 27 pm3.2i ⊢ ( 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V )
29 6 iswlkon ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V ) ) → ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐶 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐶 ) ) )
30 23 28 29 sylancl ⊢ ( 𝜑 → ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐶 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐶 ) ) )
31 8 13 21 30 mpbir3and ⊢ ( 𝜑 → 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐶 ) 𝑃 )