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 → ( ( 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	3pthond	⊢ ( 𝜑 → 𝐹 ( 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 → ( ( 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 8	3trlond	⊢ ( 𝜑 → 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐷 ) 𝑃 )
10	1 2 3 4 5 6 7 8	3pthd	⊢ ( 𝜑 → 𝐹 ( Paths ‘ 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	ispthson	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ∧ ( 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V ) ) → ( 𝐹 ( 𝐴 ( PathsOn ‘ 𝐺 ) 𝐷 ) 𝑃 ↔ ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐷 ) 𝑃 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ) )
20	11 12 18 19	syl21anc	⊢ ( 𝜑 → ( 𝐹 ( 𝐴 ( PathsOn ‘ 𝐺 ) 𝐷 ) 𝑃 ↔ ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐷 ) 𝑃 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ) )
21	9 10 20	mpbir2and	⊢ ( 𝜑 → 𝐹 ( 𝐴 ( PathsOn ‘ 𝐺 ) 𝐷 ) 𝑃 )