Metamath Proof Explorer


Theorem 3pthond

Description: A path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-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 → ( ( 𝐴𝑉𝐵𝑉 ) ∧ ( 𝐶𝑉𝐷𝑉 ) ) )
3wlkd.n ( 𝜑 → ( ( 𝐴𝐵𝐴𝐶 ) ∧ ( 𝐵𝐶𝐵𝐷 ) ∧ 𝐶𝐷 ) )
3wlkd.e ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼𝐿 ) ) )
3wlkd.v 𝑉 = ( Vtx ‘ 𝐺 )
3wlkd.i 𝐼 = ( iEdg ‘ 𝐺 )
3trld.n ( 𝜑 → ( 𝐽𝐾𝐽𝐿𝐾𝐿 ) )
Assertion 3pthond ( 𝜑𝐹 ( 𝐴 ( PathsOn ‘ 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 → ( ( 𝐴𝑉𝐵𝑉 ) ∧ ( 𝐶𝑉𝐷𝑉 ) ) )
4 3wlkd.n ( 𝜑 → ( ( 𝐴𝐵𝐴𝐶 ) ∧ ( 𝐵𝐶𝐵𝐷 ) ∧ 𝐶𝐷 ) )
5 3wlkd.e ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼𝐿 ) ) )
6 3wlkd.v 𝑉 = ( Vtx ‘ 𝐺 )
7 3wlkd.i 𝐼 = ( iEdg ‘ 𝐺 )
8 3trld.n ( 𝜑 → ( 𝐽𝐾𝐽𝐿𝐾𝐿 ) )
9 1 2 3 4 5 6 7 8 3trlond ( 𝜑𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐷 ) 𝑃 )
10 1 2 3 4 5 6 7 8 3pthd ( 𝜑𝐹 ( Paths ‘ 𝐺 ) 𝑃 )
11 3 simplld ( 𝜑𝐴𝑉 )
12 3 simprrd ( 𝜑𝐷𝑉 )
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 ispthson ( ( ( 𝐴𝑉𝐷𝑉 ) ∧ ( 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V ) ) → ( 𝐹 ( 𝐴 ( PathsOn ‘ 𝐺 ) 𝐷 ) 𝑃 ↔ ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐷 ) 𝑃𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ) )
20 11 12 18 19 syl21anc ( 𝜑 → ( 𝐹 ( 𝐴 ( PathsOn ‘ 𝐺 ) 𝐷 ) 𝑃 ↔ ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐷 ) 𝑃𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ) )
21 9 10 20 mpbir2and ( 𝜑𝐹 ( 𝐴 ( PathsOn ‘ 𝐺 ) 𝐷 ) 𝑃 )