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 ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ¬ 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 ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) → ( 𝑆 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) ) )
7	6	simprd	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) → ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) )
8	7	3expia	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) → ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) ) )
9	1 2 3 4 5	islpln2ah	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑌 ∈ 𝑃 ↔ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) )
10	2 3	hlatjcom	⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑆 ) = ( 𝑆 ∨ 𝑄 ) )
11	10	3adant3r2	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑄 ∨ 𝑆 ) = ( 𝑆 ∨ 𝑄 ) )
12	11	breq2d	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑅 ≤ ( 𝑄 ∨ 𝑆 ) ↔ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) ) )
13	12	notbid	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ¬ 𝑅 ≤ ( 𝑄 ∨ 𝑆 ) ↔ ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) ) )
14	8 9 13	3imtr4d	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑌 ∈ 𝑃 → ¬ 𝑅 ≤ ( 𝑄 ∨ 𝑆 ) ) )
15	14	3impia	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ¬ 𝑅 ≤ ( 𝑄 ∨ 𝑆 ) )