Metamath Proof Explorer


Theorem 2llnma1

Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012)

Ref Expression
Hypotheses 2llnm.l = ( le ‘ 𝐾 )
2llnm.j = ( join ‘ 𝐾 )
2llnm.m = ( meet ‘ 𝐾 )
2llnm.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion 2llnma1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑄 𝑃 ) ( 𝑄 𝑅 ) ) = 𝑄 )

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 ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝐾 ∈ HL )
6 simp21 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑃𝐴 )
7 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
8 7 4 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
9 6 8 syl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
10 simp22 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑄𝐴 )
11 simp23 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑅𝐴 )
12 simp3 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ¬ 𝑅 ( 𝑃 𝑄 ) )
13 2 4 hlatjcom ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
14 5 6 10 13 syl3anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
15 14 breq2d ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑅 ( 𝑃 𝑄 ) ↔ 𝑅 ( 𝑄 𝑃 ) ) )
16 12 15 mtbid ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ¬ 𝑅 ( 𝑄 𝑃 ) )
17 7 1 2 3 4 2llnma1b ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑄 𝑃 ) ) → ( ( 𝑄 𝑃 ) ( 𝑄 𝑅 ) ) = 𝑄 )
18 5 9 10 11 16 17 syl131anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑄 𝑃 ) ( 𝑄 𝑅 ) ) = 𝑄 )