Theorem lplnnleat

Description: A lattice plane cannot majorize an atom. (Contributed by NM, 14-Jul-2012)

	Ref	Expression
Hypotheses	lplnnleat.l	⊢ ≤ = ( le ‘ 𝐾 )
	lplnnleat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	lplnnleat.p	⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
Assertion	lplnnleat	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑋 ≤ 𝑄 )

Proof

Step	Hyp	Ref	Expression
1		lplnnleat.l	⊢ ≤ = ( le ‘ 𝐾 )
2		lplnnleat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		lplnnleat.p	⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
4		simp1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → 𝐾 ∈ HL )
5		simp2	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → 𝑋 ∈ 𝑃 )
6		simp3	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → 𝑄 ∈ 𝐴 )
7		eqid	⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
8	1 7 2 3	lplnnle2at	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ¬ 𝑋 ≤ ( 𝑄 ( join ‘ 𝐾 ) 𝑄 ) )
9	4 5 6 6 8	syl13anc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑋 ≤ ( 𝑄 ( join ‘ 𝐾 ) 𝑄 ) )
10	7 2	hlatjidm	⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ) → ( 𝑄 ( join ‘ 𝐾 ) 𝑄 ) = 𝑄 )
11	10	3adant2	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑄 ( join ‘ 𝐾 ) 𝑄 ) = 𝑄 )
12	11	breq2d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑋 ≤ ( 𝑄 ( join ‘ 𝐾 ) 𝑄 ) ↔ 𝑋 ≤ 𝑄 ) )
13	9 12	mtbid	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑋 ≤ 𝑄 )