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 ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑌 ∈ 𝑃 ↔ ( 𝑄 ≠ 𝑅 ∧ ¬ 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	⊢ ( 𝑌 ∈ 𝑃 ↔ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ∈ 𝑃 )
7	1 2 3 4	islpln2a	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ∈ 𝑃 ↔ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) )
8	6 7	syl5bb	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑌 ∈ 𝑃 ↔ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) )