Theorem 3cycld

Description: Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017) (Revised by AV, 10-Feb-2021) (Revised by AV, 24-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Hypotheses	3wlkd.p	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
	3wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
	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 ) )
	3cycld.e	⊢ ( 𝜑 → 𝐴 = 𝐷 )
Assertion	3cycld	⊢ ( 𝜑 → 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		3wlkd.p	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
2		3wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
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		3cycld.e	⊢ ( 𝜑 → 𝐴 = 𝐷 )
10	1 2 3 4 5 6 7 8	3pthd	⊢ ( 𝜑 → 𝐹 ( Paths ‘ 𝐺 ) 𝑃 )
11	1	fveq1i	⊢ ( 𝑃 ‘ 0 ) = ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 0 )
12		s4fv0	⊢ ( 𝐴 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 0 ) = 𝐴 )
13	11 12	syl5eq	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑃 ‘ 0 ) = 𝐴 )
14	13	ad3antrrr	⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) ∧ 𝐴 = 𝐷 ) → ( 𝑃 ‘ 0 ) = 𝐴 )
15		simpr	⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) ∧ 𝐴 = 𝐷 ) → 𝐴 = 𝐷 )
16	2	fveq2i	⊢ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ ⟨“ 𝐽 𝐾 𝐿 ”⟩ )
17		s3len	⊢ ( ♯ ‘ ⟨“ 𝐽 𝐾 𝐿 ”⟩ ) = 3
18	16 17	eqtri	⊢ ( ♯ ‘ 𝐹 ) = 3
19	1 18	fveq12i	⊢ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 3 )
20		s4fv3	⊢ ( 𝐷 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 3 ) = 𝐷 )
21	19 20	syl5req	⊢ ( 𝐷 ∈ 𝑉 → 𝐷 = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) )
22	21	adantl	⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) → 𝐷 = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) )
23	22	ad2antlr	⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) ∧ 𝐴 = 𝐷 ) → 𝐷 = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) )
24	14 15 23	3eqtrd	⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) ∧ 𝐴 = 𝐷 ) → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) )
25	3 9 24	syl2anc	⊢ ( 𝜑 → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) )
26		iscycl	⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) )
27	10 25 26	sylanbrc	⊢ ( 𝜑 → 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 )