Metamath Proof Explorer

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 = ⟨“ ABC ”⟩
2wlkd.f ⊢ F = ⟨“ JK ”⟩
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 = ⟨“ ABC ”⟩
2 2wlkd.f ⊢ F = ⟨“ JK ”⟩
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 ⊢ ⟨“ ABC ”⟩ ∈ Word V
10 1 9 eqeltri ⊢ P ∈ Word V
11 10 a1i ⊢ φ → P ∈ Word V
12 2 fveq2i ⊢ F = ⟨“ JK ”⟩
13 s2len ⊢ ⟨“ JK ”⟩ = 2
14 12 13 eqtri ⊢ F = 2
15 3m1e2 ⊢ 3 − 1 = 2
16 1 fveq2i ⊢ P = ⟨“ ABC ”⟩
17 s3len ⊢ ⟨“ ABC ”⟩ = 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