Theorem lhpmat

Description: An element covered by the lattice unity, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012)

	Ref	Expression
Hypotheses	lhpmat.l	⊢ ≤ = ( le ‘ 𝐾 )
	lhpmat.m	⊢ ∧ = ( meet ‘ 𝐾 )
	lhpmat.z	⊢ 0 = ( 0. ‘ 𝐾 )
	lhpmat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	lhpmat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	lhpmat	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∧ 𝑊 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		lhpmat.l	⊢ ≤ = ( le ‘ 𝐾 )
2		lhpmat.m	⊢ ∧ = ( meet ‘ 𝐾 )
3		lhpmat.z	⊢ 0 = ( 0. ‘ 𝐾 )
4		lhpmat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		lhpmat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		simprr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ¬ 𝑃 ≤ 𝑊 )
7		hlatl	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
8	7	ad2antrr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ AtLat )
9		simprl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ 𝐴 )
10		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
11	10 5	lhpbase	⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
12	11	ad2antlr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
13	10 1 2 3 4	atnle	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ¬ 𝑃 ≤ 𝑊 ↔ ( 𝑃 ∧ 𝑊 ) = 0 ) )
14	8 9 12 13	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ¬ 𝑃 ≤ 𝑊 ↔ ( 𝑃 ∧ 𝑊 ) = 0 ) )
15	6 14	mpbid	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∧ 𝑊 ) = 0 )