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Metamath Proof Explorer

Theorem lplnriaN

Description: Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012) (New usage is discouraged.)

Ref Expression
Hypotheses islpln2a.l ⊢ ≤ = ( le ‘ 𝐾 )
islpln2a.j ⊢ ∨ = ( join ‘ 𝐾 )
islpln2a.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
islpln2a.p ⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
islpln2a.y ⊢ 𝑌 = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 )
Assertion lplnriaN ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ¬ 𝑄 ≤ ( 𝑅 ∨ 𝑆 ) )

Proof

Step Hyp Ref Expression
1 islpln2a.l ⊢ ≤ = ( le ‘ 𝐾 )
2 islpln2a.j ⊢ ∨ = ( join ‘ 𝐾 )
3 islpln2a.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 islpln2a.p ⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
5 islpln2a.y ⊢ 𝑌 = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 )
6 1 2 3 4 5 islpln2ah ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑌 ∈ 𝑃 ↔ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) )
7 1 2 3 hlatcon3 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) → ¬ 𝑄 ≤ ( 𝑅 ∨ 𝑆 ) )
8 7 3expia ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) → ¬ 𝑄 ≤ ( 𝑅 ∨ 𝑆 ) ) )
9 6 8 sylbid ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑌 ∈ 𝑃 → ¬ 𝑄 ≤ ( 𝑅 ∨ 𝑆 ) ) )
10 9 3impia ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ¬ 𝑄 ≤ ( 𝑅 ∨ 𝑆 ) )