Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme23.b |
|- B = ( Base ` K ) |
2 |
|
cdleme23.l |
|- .<_ = ( le ` K ) |
3 |
|
cdleme23.j |
|- .\/ = ( join ` K ) |
4 |
|
cdleme23.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdleme23.a |
|- A = ( Atoms ` K ) |
6 |
|
cdleme23.h |
|- H = ( LHyp ` K ) |
7 |
|
cdleme23.v |
|- V = ( ( S .\/ T ) ./\ ( X ./\ W ) ) |
8 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> K e. HL ) |
9 |
8
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> K e. Lat ) |
10 |
|
simp12l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> S e. A ) |
11 |
|
simp13l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> T e. A ) |
12 |
1 3 5
|
hlatjcl |
|- ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. B ) |
13 |
8 10 11 12
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( S .\/ T ) e. B ) |
14 |
|
simp2l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> X e. B ) |
15 |
|
simp11r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> W e. H ) |
16 |
1 6
|
lhpbase |
|- ( W e. H -> W e. B ) |
17 |
15 16
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> W e. B ) |
18 |
1 4
|
latmcl |
|- ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) e. B ) |
19 |
9 14 17 18
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( X ./\ W ) e. B ) |
20 |
1 4
|
latmcl |
|- ( ( K e. Lat /\ ( S .\/ T ) e. B /\ ( X ./\ W ) e. B ) -> ( ( S .\/ T ) ./\ ( X ./\ W ) ) e. B ) |
21 |
9 13 19 20
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( S .\/ T ) ./\ ( X ./\ W ) ) e. B ) |
22 |
1 2 4
|
latmle2 |
|- ( ( K e. Lat /\ ( S .\/ T ) e. B /\ ( X ./\ W ) e. B ) -> ( ( S .\/ T ) ./\ ( X ./\ W ) ) .<_ ( X ./\ W ) ) |
23 |
9 13 19 22
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( S .\/ T ) ./\ ( X ./\ W ) ) .<_ ( X ./\ W ) ) |
24 |
1 2 4
|
latmle2 |
|- ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) .<_ W ) |
25 |
9 14 17 24
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( X ./\ W ) .<_ W ) |
26 |
1 2 9 21 19 17 23 25
|
lattrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> ( ( S .\/ T ) ./\ ( X ./\ W ) ) .<_ W ) |
27 |
7 26
|
eqbrtrid |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( X e. B /\ -. X .<_ W ) /\ ( S =/= T /\ ( S .\/ ( X ./\ W ) ) = X /\ ( T .\/ ( X ./\ W ) ) = X ) ) -> V .<_ W ) |