Metamath Proof Explorer

Theorem 3wlkond

Description: A walk of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021) (Revised by AV, 24-Mar-2021)

Ref Expression
Hypotheses 3wlkd.p ⊢ P = ⟨“ ABCD ”⟩
3wlkd.f ⊢ F = ⟨“ JKL ”⟩
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
Assertion 3wlkond ⊢ φ → F A WalksOn ⁡ G D P

Proof

Step Hyp Ref Expression
1 3wlkd.p ⊢ P = ⟨“ ABCD ”⟩
2 3wlkd.f ⊢ F = ⟨“ JKL ”⟩
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 1 2 3 4 5 6 7 3wlkd ⊢ φ → F Walks ⁡ G P
9 8 wlkonwlk1l ⊢ φ → F P ⁡ 0 WalksOn ⁡ G lastS ⁡ P P
10 1 2 3 3wlkdlem3 ⊢ φ → P ⁡ 0 = A ∧ P ⁡ 1 = B ∧ P ⁡ 2 = C ∧ P ⁡ 3 = D
11 simpll ⊢ P ⁡ 0 = A ∧ P ⁡ 1 = B ∧ P ⁡ 2 = C ∧ P ⁡ 3 = D → P ⁡ 0 = A
12 11 eqcomd ⊢ P ⁡ 0 = A ∧ P ⁡ 1 = B ∧ P ⁡ 2 = C ∧ P ⁡ 3 = D → A = P ⁡ 0
13 10 12 syl ⊢ φ → A = P ⁡ 0
14 1 fveq2i ⊢ lastS ⁡ P = lastS ⁡ ⟨“ ABCD ”⟩
15 fvex ⊢ P ⁡ 3 ∈ V
16 eleq1 ⊢ P ⁡ 3 = D → P ⁡ 3 ∈ V ↔ D ∈ V
17 15 16 mpbii ⊢ P ⁡ 3 = D → D ∈ V
18 lsws4 ⊢ D ∈ V → lastS ⁡ ⟨“ ABCD ”⟩ = D
19 17 18 syl ⊢ P ⁡ 3 = D → lastS ⁡ ⟨“ ABCD ”⟩ = D
20 14 19 eqtr2id ⊢ P ⁡ 3 = D → D = lastS ⁡ P
21 20 ad2antll ⊢ P ⁡ 0 = A ∧ P ⁡ 1 = B ∧ P ⁡ 2 = C ∧ P ⁡ 3 = D → D = lastS ⁡ P
22 10 21 syl ⊢ φ → D = lastS ⁡ P
23 13 22 oveq12d ⊢ φ → A WalksOn ⁡ G D = P ⁡ 0 WalksOn ⁡ G lastS ⁡ P
24 23 breqd ⊢ φ → F A WalksOn ⁡ G D P ↔ F P ⁡ 0 WalksOn ⁡ G lastS ⁡ P P
25 9 24 mpbird ⊢ φ → F A WalksOn ⁡ G D P