Metamath Proof Explorer

Theorem 3spthond

Description: A simple 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 = ⟨“ 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
3trld.n ⊢ φ → J ≠ K ∧ J ≠ L ∧ K ≠ L
3spthd.n ⊢ φ → A ≠ D
Assertion 3spthond ⊢ φ → F A SPathsOn ⁡ 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 3trld.n ⊢ φ → J ≠ K ∧ J ≠ L ∧ K ≠ L
9 3spthd.n ⊢ φ → A ≠ D
10 1 2 3 4 5 6 7 8 3trlond ⊢ φ → F A TrailsOn ⁡ G D P
11 1 2 3 4 5 6 7 8 9 3spthd ⊢ φ → F SPaths ⁡ G P
12 3 simplld ⊢ φ → A ∈ V
13 3 simprrd ⊢ φ → D ∈ V
14 s3cli ⊢ ⟨“ JKL ”⟩ ∈ Word V
15 2 14 eqeltri ⊢ F ∈ Word V
16 s4cli ⊢ ⟨“ ABCD ”⟩ ∈ Word V
17 1 16 eqeltri ⊢ P ∈ Word V
18 15 17 pm3.2i ⊢ F ∈ Word V ∧ P ∈ Word V
19 18 a1i ⊢ φ → F ∈ Word V ∧ P ∈ Word V
20 6 isspthson ⊢ A ∈ V ∧ D ∈ V ∧ F ∈ Word V ∧ P ∈ Word V → F A SPathsOn ⁡ G D P ↔ F A TrailsOn ⁡ G D P ∧ F SPaths ⁡ G P
21 12 13 19 20 syl21anc ⊢ φ → F A SPathsOn ⁡ G D P ↔ F A TrailsOn ⁡ G D P ∧ F SPaths ⁡ G P
22 10 11 21 mpbir2and ⊢ φ → F A SPathsOn ⁡ G D P