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. ) |