Theorem lplnri1

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

	Ref	Expression
Hypotheses	lplnri1.j	⊢ ∨ = ( join ‘ 𝐾 )
	lplnri1.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	lplnri1.p	⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
	lplnri1.y	⊢ 𝑌 = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 )
Assertion	lplnri1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝑄 ≠ 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		lplnri1.j	⊢ ∨ = ( join ‘ 𝐾 )
2		lplnri1.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		lplnri1.p	⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
4		lplnri1.y	⊢ 𝑌 = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 )
5		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
6	5 1 2 3 4	islpln2ah	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑌 ∈ 𝑃 ↔ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ( le ‘ 𝐾 ) ( 𝑄 ∨ 𝑅 ) ) ) )
7	6	biimp3a	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ( le ‘ 𝐾 ) ( 𝑄 ∨ 𝑅 ) ) )
8	7	simpld	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝑄 ≠ 𝑅 )