Metamath Proof Explorer


Theorem 3cyclpd

Description: Construction of a 3-cycle from three given edges in a graph, containing an endpoint of one of these edges. (Contributed by Alexander van der Vekens, 17-Nov-2017) (Revised by AV, 10-Feb-2021) (Revised by AV, 24-Mar-2021)

Ref Expression
Hypotheses 3wlkd.p 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
3wlkd.f 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
3wlkd.s ( 𝜑 → ( ( 𝐴𝑉𝐵𝑉 ) ∧ ( 𝐶𝑉𝐷𝑉 ) ) )
3wlkd.n ( 𝜑 → ( ( 𝐴𝐵𝐴𝐶 ) ∧ ( 𝐵𝐶𝐵𝐷 ) ∧ 𝐶𝐷 ) )
3wlkd.e ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼𝐿 ) ) )
3wlkd.v 𝑉 = ( Vtx ‘ 𝐺 )
3wlkd.i 𝐼 = ( iEdg ‘ 𝐺 )
3trld.n ( 𝜑 → ( 𝐽𝐾𝐽𝐿𝐾𝐿 ) )
3cycld.e ( 𝜑𝐴 = 𝐷 )
Assertion 3cyclpd ( 𝜑 → ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝐹 ) = 3 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) )

Proof

Step Hyp Ref Expression
1 3wlkd.p 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
2 3wlkd.f 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
3 3wlkd.s ( 𝜑 → ( ( 𝐴𝑉𝐵𝑉 ) ∧ ( 𝐶𝑉𝐷𝑉 ) ) )
4 3wlkd.n ( 𝜑 → ( ( 𝐴𝐵𝐴𝐶 ) ∧ ( 𝐵𝐶𝐵𝐷 ) ∧ 𝐶𝐷 ) )
5 3wlkd.e ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼𝐿 ) ) )
6 3wlkd.v 𝑉 = ( Vtx ‘ 𝐺 )
7 3wlkd.i 𝐼 = ( iEdg ‘ 𝐺 )
8 3trld.n ( 𝜑 → ( 𝐽𝐾𝐽𝐿𝐾𝐿 ) )
9 3cycld.e ( 𝜑𝐴 = 𝐷 )
10 1 2 3 4 5 6 7 8 9 3cycld ( 𝜑𝐹 ( Cycles ‘ 𝐺 ) 𝑃 )
11 2 fveq2i ( ♯ ‘ 𝐹 ) = ( ♯ ‘ ⟨“ 𝐽 𝐾 𝐿 ”⟩ )
12 s3len ( ♯ ‘ ⟨“ 𝐽 𝐾 𝐿 ”⟩ ) = 3
13 11 12 eqtri ( ♯ ‘ 𝐹 ) = 3
14 13 a1i ( 𝜑 → ( ♯ ‘ 𝐹 ) = 3 )
15 1 fveq1i ( 𝑃 ‘ 0 ) = ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 0 )
16 s4fv0 ( 𝐴𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 0 ) = 𝐴 )
17 15 16 eqtrid ( 𝐴𝑉 → ( 𝑃 ‘ 0 ) = 𝐴 )
18 17 ad2antrr ( ( ( 𝐴𝑉𝐵𝑉 ) ∧ ( 𝐶𝑉𝐷𝑉 ) ) → ( 𝑃 ‘ 0 ) = 𝐴 )
19 3 18 syl ( 𝜑 → ( 𝑃 ‘ 0 ) = 𝐴 )
20 10 14 19 3jca ( 𝜑 → ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝐹 ) = 3 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ) )