Theorem 2wlkd

Description: Construction of a walk from two given edges in a graph. (Contributed by Alexander van der Vekens, 5-Feb-2018) (Revised by AV, 23-Jan-2021) (Proof shortened by AV, 14-Feb-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 )
Assertion	2wlkd	⊢ ( 𝜑 → 𝐹 ( Walks ‘ 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		s3cli	⊢ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ Word V
9	1 8	eqeltri	⊢ 𝑃 ∈ Word V
10	9	a1i	⊢ ( 𝜑 → 𝑃 ∈ Word V )
11		s2cli	⊢ ⟨“ 𝐽 𝐾 ”⟩ ∈ Word V
12	2 11	eqeltri	⊢ 𝐹 ∈ Word V
13	12	a1i	⊢ ( 𝜑 → 𝐹 ∈ Word V )
14	1 2	2wlkdlem1	⊢ ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 )
15	14	a1i	⊢ ( 𝜑 → ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) )
16	1 2 3 4 5	2wlkdlem10	⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) )
17	1 2 3 4	2wlkdlem5	⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )
18	6	1vgrex	⊢ ( 𝐴 ∈ 𝑉 → 𝐺 ∈ V )
19	18	3ad2ant1	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → 𝐺 ∈ V )
20	3 19	syl	⊢ ( 𝜑 → 𝐺 ∈ V )
21	1 2 3	2wlkdlem4	⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ... ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ∈ 𝑉 )
22	10 13 15 16 17 20 6 7 21	wlkd	⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )