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Metamath Proof Explorer

Theorem 2llnma2

Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 28-May-2012)

Ref Expression
Hypotheses 2llnm.l ⊢ ≤ = ( le ‘ 𝐾 )
2llnm.j ⊢ ∨ = ( join ‘ 𝐾 )
2llnm.m ⊢ ∧ = ( meet ‘ 𝐾 )
2llnm.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion 2llnma2 ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑅 ∨ 𝑃 ) ∧ ( 𝑅 ∨ 𝑄 ) ) = 𝑅 )

Proof

Step Hyp Ref Expression
1 2llnm.l ⊢ ≤ = ( le ‘ 𝐾 )
2 2llnm.j ⊢ ∨ = ( join ‘ 𝐾 )
3 2llnm.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 2llnm.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 simp1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ HL )
6 simp21 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 )
7 simp23 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑅 ∈ 𝐴 )
8 simp22 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 )
9 1 2 4 4atlem0ae ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑅 ) )
10 1 2 3 4 2llnma1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑅 ) ) → ( ( 𝑅 ∨ 𝑃 ) ∧ ( 𝑅 ∨ 𝑄 ) ) = 𝑅 )
11 5 6 7 8 9 10 syl131anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑅 ∨ 𝑃 ) ∧ ( 𝑅 ∨ 𝑄 ) ) = 𝑅 )