Metamath Proof Explorer

Theorem lhpmatb

Description: An element covered by the lattice unit, when conjoined with an atom, equals zero iff the atom is not under it. (Contributed by NM, 15-Jun-2013)

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 lhpmatb 
|- ( ( ( 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 1 2 3 4 5 lhpmat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ./\ W ) = .0. )
7 6 anassrs 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A ) /\ -. P .<_ W ) -> ( P ./\ W ) = .0. )
8 hlatl 
 |-  ( K e. HL -> K e. AtLat )
9 8 ad3antrrr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A ) /\ ( P ./\ W ) = .0. ) -> K e. AtLat )
10 simplr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A ) /\ ( P ./\ W ) = .0. ) -> P e. A )
11 3 4 atn0 
 |-  ( ( K e. AtLat /\ P e. A ) -> P =/= .0. )
12 11 necomd 
 |-  ( ( K e. AtLat /\ P e. A ) -> .0. =/= P )
13 9 10 12 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A ) /\ ( P ./\ W ) = .0. ) -> .0. =/= P )
14 neeq1 
 |-  ( ( P ./\ W ) = .0. -> ( ( P ./\ W ) =/= P <-> .0. =/= P ) )
15 14 adantl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A ) /\ ( P ./\ W ) = .0. ) -> ( ( P ./\ W ) =/= P <-> .0. =/= P ) )
16 13 15 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A ) /\ ( P ./\ W ) = .0. ) -> ( P ./\ W ) =/= P )
17 hllat 
 |-  ( K e. HL -> K e. Lat )
18 17 ad3antrrr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A ) /\ ( P ./\ W ) = .0. ) -> K e. Lat )
19 eqid 
 |-  ( Base ` K ) = ( Base ` K )
20 19 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
21 10 20 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A ) /\ ( P ./\ W ) = .0. ) -> P e. ( Base ` K ) )
22 19 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
23 22 ad3antlr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A ) /\ ( P ./\ W ) = .0. ) -> W e. ( Base ` K ) )
24 19 1 2 latleeqm1 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( P .<_ W <-> ( P ./\ W ) = P ) )
25 18 21 23 24 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A ) /\ ( P ./\ W ) = .0. ) -> ( P .<_ W <-> ( P ./\ W ) = P ) )
26 25 necon3bbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A ) /\ ( P ./\ W ) = .0. ) -> ( -. P .<_ W <-> ( P ./\ W ) =/= P ) )
27 16 26 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A ) /\ ( P ./\ W ) = .0. ) -> -. P .<_ W )
28 7 27 impbida 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A ) -> ( -. P .<_ W <-> ( P ./\ W ) = .0. ) )