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Metamath Proof Explorer

Theorem lplnribN

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 lplnribN ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ¬ 𝑅 ≤ ( 𝑄 ∨ 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 3noncolr1N ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) → ( 𝑆 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) ) )
7 6 simprd ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) → ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) )
8 7 3expia ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) → ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) ) )
9 1 2 3 4 5 islpln2ah ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑌 ∈ 𝑃 ↔ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) )
10 2 3 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑆 ) = ( 𝑆 ∨ 𝑄 ) )
11 10 3adant3r2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑄 ∨ 𝑆 ) = ( 𝑆 ∨ 𝑄 ) )
12 11 breq2d ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑅 ≤ ( 𝑄 ∨ 𝑆 ) ↔ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) ) )
13 12 notbid ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ¬ 𝑅 ≤ ( 𝑄 ∨ 𝑆 ) ↔ ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) ) )
14 8 9 13 3imtr4d ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑌 ∈ 𝑃 → ¬ 𝑅 ≤ ( 𝑄 ∨ 𝑆 ) ) )
15 14 3impia ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ¬ 𝑅 ≤ ( 𝑄 ∨ 𝑆 ) )