Theorem 3pthd

Description: A path of length 3 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017) (Revised 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	3pthd	⊢ ( 𝜑 → 𝐹 ( Paths ‘ 𝐺 ) 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		s4cli	⊢ ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ∈ Word V
10	1 9	eqeltri	⊢ 𝑃 ∈ Word V
11	10	a1i	⊢ ( 𝜑 → 𝑃 ∈ Word V )
12	2	fveq2i	⊢ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ ⟨“ 𝐽 𝐾 𝐿 ”⟩ )
13		s3len	⊢ ( ♯ ‘ ⟨“ 𝐽 𝐾 𝐿 ”⟩ ) = 3
14	12 13	eqtri	⊢ ( ♯ ‘ 𝐹 ) = 3
15		4m1e3	⊢ ( 4 − 1 ) = 3
16	1	fveq2i	⊢ ( ♯ ‘ 𝑃 ) = ( ♯ ‘ ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ )
17		s4len	⊢ ( ♯ ‘ ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ) = 4
18	16 17	eqtr2i	⊢ 4 = ( ♯ ‘ 𝑃 )
19	18	oveq1i	⊢ ( 4 − 1 ) = ( ( ♯ ‘ 𝑃 ) − 1 )
20	14 15 19	3eqtr2i	⊢ ( ♯ ‘ 𝐹 ) = ( ( ♯ ‘ 𝑃 ) − 1 )
21	1 2 3 4	3pthdlem1	⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ∀ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) )
22		eqid	⊢ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ 𝐹 )
23	1 2 3 4 5 6 7 8	3trld	⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )
24	11 20 21 22 23	pthd	⊢ ( 𝜑 → 𝐹 ( Paths ‘ 𝐺 ) 𝑃 )