Step |
Hyp |
Ref |
Expression |
1 |
|
lhpelim.b |
|- B = ( Base ` K ) |
2 |
|
lhpelim.l |
|- .<_ = ( le ` K ) |
3 |
|
lhpelim.j |
|- .\/ = ( join ` K ) |
4 |
|
lhpelim.m |
|- ./\ = ( meet ` K ) |
5 |
|
lhpelim.a |
|- A = ( Atoms ` K ) |
6 |
|
lhpelim.h |
|- H = ( LHyp ` K ) |
7 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
8 |
2 4 7 5 6
|
lhpmat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ./\ W ) = ( 0. ` K ) ) |
9 |
8
|
3adant3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( P ./\ W ) = ( 0. ` K ) ) |
10 |
9
|
oveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( ( P ./\ W ) .\/ ( X ./\ W ) ) = ( ( 0. ` K ) .\/ ( X ./\ W ) ) ) |
11 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> K e. HL ) |
12 |
|
simp2l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> P e. A ) |
13 |
11
|
hllatd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> K e. Lat ) |
14 |
|
simp3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> X e. B ) |
15 |
|
simp1r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> W e. H ) |
16 |
1 6
|
lhpbase |
|- ( W e. H -> W e. B ) |
17 |
15 16
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> W e. B ) |
18 |
1 4
|
latmcl |
|- ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) e. B ) |
19 |
13 14 17 18
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( X ./\ W ) e. B ) |
20 |
1 2 4
|
latmle2 |
|- ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) .<_ W ) |
21 |
13 14 17 20
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( X ./\ W ) .<_ W ) |
22 |
1 2 3 4 5
|
atmod4i2 |
|- ( ( K e. HL /\ ( P e. A /\ ( X ./\ W ) e. B /\ W e. B ) /\ ( X ./\ W ) .<_ W ) -> ( ( P ./\ W ) .\/ ( X ./\ W ) ) = ( ( P .\/ ( X ./\ W ) ) ./\ W ) ) |
23 |
11 12 19 17 21 22
|
syl131anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( ( P ./\ W ) .\/ ( X ./\ W ) ) = ( ( P .\/ ( X ./\ W ) ) ./\ W ) ) |
24 |
|
hlol |
|- ( K e. HL -> K e. OL ) |
25 |
11 24
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> K e. OL ) |
26 |
1 3 7
|
olj02 |
|- ( ( K e. OL /\ ( X ./\ W ) e. B ) -> ( ( 0. ` K ) .\/ ( X ./\ W ) ) = ( X ./\ W ) ) |
27 |
25 19 26
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( ( 0. ` K ) .\/ ( X ./\ W ) ) = ( X ./\ W ) ) |
28 |
10 23 27
|
3eqtr3d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( ( P .\/ ( X ./\ W ) ) ./\ W ) = ( X ./\ W ) ) |