Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme0.l |
|- .<_ = ( le ` K ) |
2 |
|
cdleme0.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleme0.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdleme0.a |
|- A = ( Atoms ` K ) |
5 |
|
cdleme0.h |
|- H = ( LHyp ` K ) |
6 |
|
cdleme0.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
7 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) ) |
8 |
|
simp2l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> ( P e. A /\ -. P .<_ W ) ) |
9 |
|
simp2r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> Q e. A ) |
10 |
|
simp3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> P =/= Q ) |
11 |
1 2 3 4 5 6
|
lhpat2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A ) |
12 |
7 8 9 10 11
|
syl112anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> U e. A ) |
13 |
|
simp2ll |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> P e. A ) |
14 |
1 2 3 4 5 6
|
cdlemeulpq |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> U .<_ ( P .\/ Q ) ) |
15 |
7 13 9 14
|
syl12anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> U .<_ ( P .\/ Q ) ) |
16 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> K e. HL ) |
17 |
16
|
hllatd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> K e. Lat ) |
18 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
19 |
18 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
20 |
16 13 9 19
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
21 |
|
simp1r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> W e. H ) |
22 |
18 5
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
23 |
21 22
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> W e. ( Base ` K ) ) |
24 |
18 1 3
|
latmle2 |
|- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W ) |
25 |
17 20 23 24
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> ( ( P .\/ Q ) ./\ W ) .<_ W ) |
26 |
6 25
|
eqbrtrid |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> U .<_ W ) |
27 |
|
breq1 |
|- ( u = U -> ( u .<_ ( P .\/ Q ) <-> U .<_ ( P .\/ Q ) ) ) |
28 |
|
breq1 |
|- ( u = U -> ( u .<_ W <-> U .<_ W ) ) |
29 |
27 28
|
anbi12d |
|- ( u = U -> ( ( u .<_ ( P .\/ Q ) /\ u .<_ W ) <-> ( U .<_ ( P .\/ Q ) /\ U .<_ W ) ) ) |
30 |
29
|
rspcev |
|- ( ( U e. A /\ ( U .<_ ( P .\/ Q ) /\ U .<_ W ) ) -> E. u e. A ( u .<_ ( P .\/ Q ) /\ u .<_ W ) ) |
31 |
12 15 26 30
|
syl12anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> E. u e. A ( u .<_ ( P .\/ Q ) /\ u .<_ W ) ) |