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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 ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝑋 ∈ 𝑁 )

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 ⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝑋 ∈ 𝐵 )
6 breq1 ⊢ ( 𝑝 = 𝑃 → ( 𝑝 𝐶 𝑋 ↔ 𝑃 𝐶 𝑋 ) )
7 6 rspcev ⊢ ( ( 𝑃 ∈ 𝐴 ∧ 𝑃 𝐶 𝑋 ) → ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 )
8 7 3ad2antl3 ⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 )
9 simpl1 ⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝐾 ∈ 𝐷 )
10 1 2 3 4 islln ⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) ) )
11 9 10 syl ⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) ) )
12 5 8 11 mpbir2and ⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝑋 ∈ 𝑁 )