Theorem 2llnneN

Description: Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		2lnne.l	⊢ ≤ = ( le ‘ 𝐾 )
2		2lnne.j	⊢ ∨ = ( join ‘ 𝐾 )
3		2lnne.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		simp1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ HL )
5		simp21	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 )
6		simp23	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑅 ∈ 𝐴 )
7		simp21	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑃 ∈ 𝐴 )
8		simp23	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑅 ∈ 𝐴 )
9		simp22	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑄 ∈ 𝐴 )
10	7 8 9	3jca	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) )
11	1 2 3	hlatexch2	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
12	10 11	syld3an2	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
13	12	con3d	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) → ¬ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ) )
14	13	3exp	⊢ ( 𝐾 ∈ HL → ( ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 → ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) → ¬ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ) ) ) )
15	14	imp4a	⊢ ( 𝐾 ∈ HL → ( ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ¬ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ) ) )
16	15	3imp	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) )
17	1 2 3	2llnne2N	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ¬ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ) → ( 𝑅 ∨ 𝑃 ) ≠ ( 𝑅 ∨ 𝑄 ) )
18	4 5 6 16 17	syl121anc	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ∨ 𝑃 ) ≠ ( 𝑅 ∨ 𝑄 ) )