Theorem lplnri3N

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	lplnri1.j	⊢ ∨ = ( join ‘ 𝐾 )
	lplnri1.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	lplnri1.p	⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
	lplnri1.y	⊢ 𝑌 = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 )
Assertion	lplnri3N	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝑅 ≠ 𝑆 )

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		simp1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝐾 ∈ HL )
6		simp22	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝑅 ∈ 𝐴 )
7		simp21	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝑄 ∈ 𝐴 )
8		simp23	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝑆 ∈ 𝐴 )
9		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
10	9 1 2 3 4	lplnribN	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ¬ 𝑅 ( le ‘ 𝐾 ) ( 𝑄 ∨ 𝑆 ) )
11	9 1 2	atnlej2	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ¬ 𝑅 ( le ‘ 𝐾 ) ( 𝑄 ∨ 𝑆 ) ) → 𝑅 ≠ 𝑆 )
12	5 6 7 8 10 11	syl131anc	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝑅 ≠ 𝑆 )