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