Metamath Proof Explorer

Theorem 3trlond

Description: A trail of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021) (Revised by AV, 24-Mar-2021)

Ref Expression
Hypotheses 3wlkd.p ⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
3wlkd.f ⊢ 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
3wlkd.s ⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) )
3wlkd.n ⊢ ( 𝜑 → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) )
3wlkd.e ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
3wlkd.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
3wlkd.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3trld.n ⊢ ( 𝜑 → ( 𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿 ) )
Assertion 3trlond ( 𝜑 → 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐷 ) 𝑃 )

Proof

Step Hyp Ref Expression
1 3wlkd.p ⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
2 3wlkd.f ⊢ 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
3 3wlkd.s ⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) )
4 3wlkd.n ⊢ ( 𝜑 → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) )
5 3wlkd.e ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
6 3wlkd.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
7 3wlkd.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
8 3trld.n ⊢ ( 𝜑 → ( 𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿 ) )
9 1 2 3 4 5 6 7 3wlkond ⊢ ( 𝜑 → 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐷 ) 𝑃 )
10 1 2 3 4 5 6 7 8 3trld ⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )
11 3 simplld ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
12 3 simprrd ⊢ ( 𝜑 → 𝐷 ∈ 𝑉 )
13 s3cli ⊢ ⟨“ 𝐽 𝐾 𝐿 ”⟩ ∈ Word V
14 2 13 eqeltri ⊢ 𝐹 ∈ Word V
15 s4cli ⊢ ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ∈ Word V
16 1 15 eqeltri ⊢ 𝑃 ∈ Word V
17 14 16 pm3.2i ⊢ ( 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V )
18 17 a1i ⊢ ( 𝜑 → ( 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V ) )
19 6 istrlson ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ∧ ( 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V ) ) → ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐷 ) 𝑃 ↔ ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐷 ) 𝑃 ∧ 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ) ) )
20 11 12 18 19 syl21anc ⊢ ( 𝜑 → ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐷 ) 𝑃 ↔ ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐷 ) 𝑃 ∧ 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ) ) )
21 9 10 20 mpbir2and ⊢ ( 𝜑 → 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐷 ) 𝑃 )