Metamath Proof Explorer


Theorem islpln2ah

Description: The predicate "is a lattice plane" for join of atoms. Version of islpln2a expressed with an abbreviation hypothesis. (Contributed by NM, 30-Jul-2012)

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 islpln2ah ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑆𝐴 ) ) → ( 𝑌𝑃 ↔ ( 𝑄𝑅 ∧ ¬ 𝑆 ( 𝑄 𝑅 ) ) ) )

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 5 eleq1i ( 𝑌𝑃 ↔ ( ( 𝑄 𝑅 ) 𝑆 ) ∈ 𝑃 )
7 1 2 3 4 islpln2a ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑆𝐴 ) ) → ( ( ( 𝑄 𝑅 ) 𝑆 ) ∈ 𝑃 ↔ ( 𝑄𝑅 ∧ ¬ 𝑆 ( 𝑄 𝑅 ) ) ) )
8 6 7 syl5bb ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑆𝐴 ) ) → ( 𝑌𝑃 ↔ ( 𝑄𝑅 ∧ ¬ 𝑆 ( 𝑄 𝑅 ) ) ) )