Metamath Proof Explorer


Theorem llni

Description: Condition implying a lattice line. (Contributed by NM, 17-Jun-2012)

Ref Expression
Hypotheses llnset.b 𝐵 = ( Base ‘ 𝐾 )
llnset.c 𝐶 = ( ⋖ ‘ 𝐾 )
llnset.a 𝐴 = ( Atoms ‘ 𝐾 )
llnset.n 𝑁 = ( LLines ‘ 𝐾 )
Assertion llni ( ( ( 𝐾𝐷𝑋𝐵𝑃𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝑋𝑁 )

Proof

Step Hyp Ref Expression
1 llnset.b 𝐵 = ( Base ‘ 𝐾 )
2 llnset.c 𝐶 = ( ⋖ ‘ 𝐾 )
3 llnset.a 𝐴 = ( Atoms ‘ 𝐾 )
4 llnset.n 𝑁 = ( LLines ‘ 𝐾 )
5 simpl2 ( ( ( 𝐾𝐷𝑋𝐵𝑃𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝑋𝐵 )
6 breq1 ( 𝑝 = 𝑃 → ( 𝑝 𝐶 𝑋𝑃 𝐶 𝑋 ) )
7 6 rspcev ( ( 𝑃𝐴𝑃 𝐶 𝑋 ) → ∃ 𝑝𝐴 𝑝 𝐶 𝑋 )
8 7 3ad2antl3 ( ( ( 𝐾𝐷𝑋𝐵𝑃𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → ∃ 𝑝𝐴 𝑝 𝐶 𝑋 )
9 simpl1 ( ( ( 𝐾𝐷𝑋𝐵𝑃𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝐾𝐷 )
10 1 2 3 4 islln ( 𝐾𝐷 → ( 𝑋𝑁 ↔ ( 𝑋𝐵 ∧ ∃ 𝑝𝐴 𝑝 𝐶 𝑋 ) ) )
11 9 10 syl ( ( ( 𝐾𝐷𝑋𝐵𝑃𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → ( 𝑋𝑁 ↔ ( 𝑋𝐵 ∧ ∃ 𝑝𝐴 𝑝 𝐶 𝑋 ) ) )
12 5 8 11 mpbir2and ( ( ( 𝐾𝐷𝑋𝐵𝑃𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝑋𝑁 )