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Metamath Proof Explorer

Theorem hlatcon3

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 hlatcon3 ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) )

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 3noncolr2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ) )
5 4 simprd ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) )