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 ∧ ( 𝑄𝐴𝑅𝐴𝑆𝐴 ) ∧ 𝑌𝑃 ) → ¬ 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 ∧ ( 𝑄𝐴𝑅𝐴𝑆𝐴 ) ∧ ( 𝑄𝑅 ∧ ¬ 𝑆 ( 𝑄 𝑅 ) ) ) → ( 𝑆𝑄 ∧ ¬ 𝑅 ( 𝑆 𝑄 ) ) )
7 6 simprd ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑆𝐴 ) ∧ ( 𝑄𝑅 ∧ ¬ 𝑆 ( 𝑄 𝑅 ) ) ) → ¬ 𝑅 ( 𝑆 𝑄 ) )
8 7 3expia ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑆𝐴 ) ) → ( ( 𝑄𝑅 ∧ ¬ 𝑆 ( 𝑄 𝑅 ) ) → ¬ 𝑅 ( 𝑆 𝑄 ) ) )
9 1 2 3 4 5 islpln2ah ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑆𝐴 ) ) → ( 𝑌𝑃 ↔ ( 𝑄𝑅 ∧ ¬ 𝑆 ( 𝑄 𝑅 ) ) ) )
10 2 3 hlatjcom ( ( 𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴 ) → ( 𝑄 𝑆 ) = ( 𝑆 𝑄 ) )
11 10 3adant3r2 ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑆𝐴 ) ) → ( 𝑄 𝑆 ) = ( 𝑆 𝑄 ) )
12 11 breq2d ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑆𝐴 ) ) → ( 𝑅 ( 𝑄 𝑆 ) ↔ 𝑅 ( 𝑆 𝑄 ) ) )
13 12 notbid ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑆𝐴 ) ) → ( ¬ 𝑅 ( 𝑄 𝑆 ) ↔ ¬ 𝑅 ( 𝑆 𝑄 ) ) )
14 8 9 13 3imtr4d ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑆𝐴 ) ) → ( 𝑌𝑃 → ¬ 𝑅 ( 𝑄 𝑆 ) ) )
15 14 3impia ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑆𝐴 ) ∧ 𝑌𝑃 ) → ¬ 𝑅 ( 𝑄 𝑆 ) )