Theorem 2llnma2rN

Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 2-May-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	2llnm.l	⊢ ≤ = ( le ‘ 𝐾 )
	2llnm.j	⊢ ∨ = ( join ‘ 𝐾 )
	2llnm.m	⊢ ∧ = ( meet ‘ 𝐾 )
	2llnm.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	2llnma2rN	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ 𝑅 ) ∧ ( 𝑄 ∨ 𝑅 ) ) = 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	2 4	hlatjcom	⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑅 ) = ( 𝑅 ∨ 𝑃 ) )
9	5 6 7 8	syl3anc	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ 𝑅 ) = ( 𝑅 ∨ 𝑃 ) )
10		simp22	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 )
11	2 4	hlatjcom	⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑅 ) = ( 𝑅 ∨ 𝑄 ) )
12	5 10 7 11	syl3anc	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑄 ∨ 𝑅 ) = ( 𝑅 ∨ 𝑄 ) )
13	9 12	oveq12d	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ 𝑅 ) ∧ ( 𝑄 ∨ 𝑅 ) ) = ( ( 𝑅 ∨ 𝑃 ) ∧ ( 𝑅 ∨ 𝑄 ) ) )
14	1 2 3 4	2llnma2	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑅 ∨ 𝑃 ) ∧ ( 𝑅 ∨ 𝑄 ) ) = 𝑅 )
15	13 14	eqtrd	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ 𝑅 ) ∧ ( 𝑄 ∨ 𝑅 ) ) = 𝑅 )