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Metamath Proof Explorer

Theorem hlatcon2

Description: Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012)

Ref Expression
Hypotheses 3noncol.l ⊢ ≤ = ( le ‘ 𝐾 )
3noncol.j ⊢ ∨ = ( join ‘ 𝐾 )
3noncol.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion hlatcon2 ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) )

Proof

Step Hyp Ref Expression
1 3noncol.l ⊢ ≤ = ( le ‘ 𝐾 )
2 3noncol.j ⊢ ∨ = ( join ‘ 𝐾 )
3 3noncol.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 1 2 3 hlatcon3 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) )
5 simp1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ HL )
6 simp22 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 )
7 simp23 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑅 ∈ 𝐴 )
8 2 3 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑅 ) = ( 𝑅 ∨ 𝑄 ) )
9 5 6 7 8 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑄 ∨ 𝑅 ) = ( 𝑅 ∨ 𝑄 ) )
10 9 breq2d ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ↔ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ) )
11 4 10 mtbid ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) )