Step |
Hyp |
Ref |
Expression |
1 |
|
lhple.b |
|- B = ( Base ` K ) |
2 |
|
lhple.l |
|- .<_ = ( le ` K ) |
3 |
|
lhple.j |
|- .\/ = ( join ` K ) |
4 |
|
lhple.m |
|- ./\ = ( meet ` K ) |
5 |
|
lhple.a |
|- A = ( Atoms ` K ) |
6 |
|
lhple.h |
|- H = ( LHyp ` K ) |
7 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> K e. HL ) |
8 |
7
|
hllatd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> K e. Lat ) |
9 |
|
simp2l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> P e. A ) |
10 |
1 5
|
atbase |
|- ( P e. A -> P e. B ) |
11 |
9 10
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> P e. B ) |
12 |
|
simp3l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> X e. B ) |
13 |
1 3
|
latjcom |
|- ( ( K e. Lat /\ P e. B /\ X e. B ) -> ( P .\/ X ) = ( X .\/ P ) ) |
14 |
8 11 12 13
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( P .\/ X ) = ( X .\/ P ) ) |
15 |
14
|
oveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( P .\/ X ) ./\ W ) = ( ( X .\/ P ) ./\ W ) ) |
16 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( K e. HL /\ W e. H ) ) |
17 |
|
simp3r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> X .<_ W ) |
18 |
1 2 3 4 6
|
lhpmod6i1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ P e. B ) /\ X .<_ W ) -> ( X .\/ ( P ./\ W ) ) = ( ( X .\/ P ) ./\ W ) ) |
19 |
16 12 11 17 18
|
syl121anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( X .\/ ( P ./\ W ) ) = ( ( X .\/ P ) ./\ W ) ) |
20 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
21 |
2 4 20 5 6
|
lhpmat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ./\ W ) = ( 0. ` K ) ) |
22 |
21
|
3adant3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( P ./\ W ) = ( 0. ` K ) ) |
23 |
22
|
oveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( X .\/ ( P ./\ W ) ) = ( X .\/ ( 0. ` K ) ) ) |
24 |
|
hlol |
|- ( K e. HL -> K e. OL ) |
25 |
7 24
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> K e. OL ) |
26 |
1 3 20
|
olj01 |
|- ( ( K e. OL /\ X e. B ) -> ( X .\/ ( 0. ` K ) ) = X ) |
27 |
25 12 26
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( X .\/ ( 0. ` K ) ) = X ) |
28 |
23 27
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( X .\/ ( P ./\ W ) ) = X ) |
29 |
15 19 28
|
3eqtr2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( P .\/ X ) ./\ W ) = X ) |