Theorem lhpmat

Description: An element covered by the lattice unit, 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 e. HL /\ W e. H ) /\ ( P e. 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 e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> -. P .<_ W )
7		hlatl	|- ( K e. HL -> K e. AtLat )
8	7	ad2antrr	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> K e. AtLat )
9		simprl	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> P e. A )
10		eqid	|- ( Base ` K ) = ( Base ` K )
11	10 5	lhpbase	|- ( W e. H -> W e. ( Base ` K ) )
12	11	ad2antlr	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> W e. ( Base ` K ) )
13	10 1 2 3 4	atnle	|- ( ( K e. AtLat /\ P e. A /\ W e. ( Base ` K ) ) -> ( -. P .<_ W <-> ( P ./\ W ) = .0. ) )
14	8 9 12 13	syl3anc	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( -. P .<_ W <-> ( P ./\ W ) = .0. ) )
15	6 14	mpbid	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ./\ W ) = .0. )