Metamath Proof Explorer


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. )