Metamath Proof Explorer


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 = ⟨“ 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
Assertion 3pthd φ F Paths G 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 s4cli ⟨“ ABCD ”⟩ Word V
10 1 9 eqeltri P Word V
11 10 a1i φ P Word V
12 2 fveq2i F = ⟨“ JKL ”⟩
13 s3len ⟨“ JKL ”⟩ = 3
14 12 13 eqtri F = 3
15 4m1e3 4 1 = 3
16 1 fveq2i P = ⟨“ ABCD ”⟩
17 s4len ⟨“ ABCD ”⟩ = 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