Step |
Hyp |
Ref |
Expression |
1 |
|
lhp2atnle.l |
|- .<_ = ( le ` K ) |
2 |
|
lhp2atnle.j |
|- .\/ = ( join ` K ) |
3 |
|
lhp2atnle.a |
|- A = ( Atoms ` K ) |
4 |
|
lhp2atnle.h |
|- H = ( LHyp ` K ) |
5 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> K e. HL ) |
6 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
7 |
5 6
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> K e. AtLat ) |
8 |
|
simp3l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> V e. A ) |
9 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
10 |
9 3
|
atn0 |
|- ( ( K e. AtLat /\ V e. A ) -> V =/= ( 0. ` K ) ) |
11 |
7 8 10
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> V =/= ( 0. ` K ) ) |
12 |
5
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> K e. Lat ) |
13 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
14 |
13 3
|
atbase |
|- ( V e. A -> V e. ( Base ` K ) ) |
15 |
8 14
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> V e. ( Base ` K ) ) |
16 |
|
simp12l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> P e. A ) |
17 |
|
simp2l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> U e. A ) |
18 |
13 2 3
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ U e. A ) -> ( P .\/ U ) e. ( Base ` K ) ) |
19 |
5 16 17 18
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( P .\/ U ) e. ( Base ` K ) ) |
20 |
|
eqid |
|- ( meet ` K ) = ( meet ` K ) |
21 |
13 1 20
|
latleeqm2 |
|- ( ( K e. Lat /\ V e. ( Base ` K ) /\ ( P .\/ U ) e. ( Base ` K ) ) -> ( V .<_ ( P .\/ U ) <-> ( ( P .\/ U ) ( meet ` K ) V ) = V ) ) |
22 |
12 15 19 21
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( V .<_ ( P .\/ U ) <-> ( ( P .\/ U ) ( meet ` K ) V ) = V ) ) |
23 |
1 2 20 9 3 4
|
lhp2at0 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( ( P .\/ U ) ( meet ` K ) V ) = ( 0. ` K ) ) |
24 |
|
eqeq1 |
|- ( ( ( P .\/ U ) ( meet ` K ) V ) = V -> ( ( ( P .\/ U ) ( meet ` K ) V ) = ( 0. ` K ) <-> V = ( 0. ` K ) ) ) |
25 |
23 24
|
syl5ibcom |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( ( ( P .\/ U ) ( meet ` K ) V ) = V -> V = ( 0. ` K ) ) ) |
26 |
22 25
|
sylbid |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( V .<_ ( P .\/ U ) -> V = ( 0. ` K ) ) ) |
27 |
26
|
necon3ad |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> ( V =/= ( 0. ` K ) -> -. V .<_ ( P .\/ U ) ) ) |
28 |
11 27
|
mpd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ U =/= V ) /\ ( U e. A /\ U .<_ W ) /\ ( V e. A /\ V .<_ W ) ) -> -. V .<_ ( P .\/ U ) ) |