Theorem 2pthd

Description: A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017) (Revised by AV, 24-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 )
	2trld.n	⊢ ( 𝜑 → 𝐽 ≠ 𝐾 )
Assertion	2pthd	⊢ ( 𝜑 → 𝐹 ( Paths ‘ 𝐺 ) 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		2trld.n	⊢ ( 𝜑 → 𝐽 ≠ 𝐾 )
9		s3cli	⊢ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ Word V
10	1 9	eqeltri	⊢ 𝑃 ∈ Word V
11	10	a1i	⊢ ( 𝜑 → 𝑃 ∈ Word V )
12	2	fveq2i	⊢ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ ⟨“ 𝐽 𝐾 ”⟩ )
13		s2len	⊢ ( ♯ ‘ ⟨“ 𝐽 𝐾 ”⟩ ) = 2
14	12 13	eqtri	⊢ ( ♯ ‘ 𝐹 ) = 2
15		3m1e2	⊢ ( 3 − 1 ) = 2
16	1	fveq2i	⊢ ( ♯ ‘ 𝑃 ) = ( ♯ ‘ ⟨“ 𝐴 𝐵 𝐶 ”⟩ )
17		s3len	⊢ ( ♯ ‘ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ) = 3
18	16 17	eqtr2i	⊢ 3 = ( ♯ ‘ 𝑃 )
19	18	oveq1i	⊢ ( 3 − 1 ) = ( ( ♯ ‘ 𝑃 ) − 1 )
20	14 15 19	3eqtr2i	⊢ ( ♯ ‘ 𝐹 ) = ( ( ♯ ‘ 𝑃 ) − 1 )
21	1 2 3 4	2pthdlem1	⊢ ( 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	2trld	⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )
24	11 20 21 22 23	pthd	⊢ ( 𝜑 → 𝐹 ( Paths ‘ 𝐺 ) 𝑃 )