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Metamath Proof Explorer

Theorem 2llnne2N

Description: Condition implying that two intersecting lines are different. (Contributed by NM, 13-Jun-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 2lnne.l ⊢ ≤ = ( le ‘ 𝐾 )
2lnne.j ⊢ ∨ = ( join ‘ 𝐾 )
2lnne.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion 2llnne2N ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ¬ 𝑃 ≤ ( 𝑅 ∨ 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 simpl ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝐾 ∈ HL )
5 simprr ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑅 ∈ 𝐴 )
6 simprl ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐴 )
7 1 2 3 hlatlej2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ≤ ( 𝑅 ∨ 𝑃 ) )
8 4 5 6 7 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑃 ≤ ( 𝑅 ∨ 𝑃 ) )
9 breq2 ⊢ ( ( 𝑅 ∨ 𝑃 ) = ( 𝑅 ∨ 𝑄 ) → ( 𝑃 ≤ ( 𝑅 ∨ 𝑃 ) ↔ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ) )
10 8 9 syl5ibcom ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑅 ∨ 𝑃 ) = ( 𝑅 ∨ 𝑄 ) → 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ) )
11 10 necon3bd ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ¬ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) → ( 𝑅 ∨ 𝑃 ) ≠ ( 𝑅 ∨ 𝑄 ) ) )
12 11 3impia ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ¬ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ) → ( 𝑅 ∨ 𝑃 ) ≠ ( 𝑅 ∨ 𝑄 ) )