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Metamath Proof Explorer

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 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. )