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Metamath Proof Explorer

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 ) )