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 |
|
hlol |
|- ( K e. HL -> K e. OL ) |
10 |
8 9
|
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 ) ) -> K e. OL ) |
11 |
|
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 ) |
12 |
|
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 ) |
13 |
1 3 5
|
hlatjcl |
|- ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. B ) |
14 |
8 11 12 13
|
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 ) |
15 |
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 ) |
16 |
|
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 ) |
17 |
|
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 ) |
18 |
1 6
|
lhpbase |
|- ( W e. H -> W e. B ) |
19 |
17 18
|
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 ) |
20 |
1 4
|
latmcl |
|- ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) e. B ) |
21 |
15 16 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 ) ) -> ( X ./\ W ) e. B ) |
22 |
1 3
|
latjcl |
|- ( ( K e. Lat /\ ( S .\/ T ) e. B /\ ( X ./\ W ) e. B ) -> ( ( S .\/ T ) .\/ ( X ./\ W ) ) e. B ) |
23 |
15 14 21 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 ) ) e. B ) |
24 |
1 4
|
latmassOLD |
|- ( ( K e. OL /\ ( ( S .\/ T ) e. B /\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) e. B /\ W e. B ) ) -> ( ( ( S .\/ T ) ./\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) ./\ W ) = ( ( S .\/ T ) ./\ ( ( ( S .\/ T ) .\/ ( X ./\ W ) ) ./\ W ) ) ) |
25 |
10 14 23 19 24
|
syl13anc |
|- ( ( ( ( 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 ) ./\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) ./\ W ) = ( ( S .\/ T ) ./\ ( ( ( S .\/ T ) .\/ ( X ./\ W ) ) ./\ W ) ) ) |
26 |
1 2 3
|
latlej1 |
|- ( ( K e. Lat /\ ( S .\/ T ) e. B /\ ( X ./\ W ) e. B ) -> ( S .\/ T ) .<_ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) |
27 |
15 14 21 26
|
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 ) .<_ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) |
28 |
1 2 4
|
latleeqm1 |
|- ( ( K e. Lat /\ ( S .\/ T ) e. B /\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) e. B ) -> ( ( S .\/ T ) .<_ ( ( S .\/ T ) .\/ ( X ./\ W ) ) <-> ( ( S .\/ T ) ./\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) = ( S .\/ T ) ) ) |
29 |
15 14 23 28
|
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 ) .<_ ( ( S .\/ T ) .\/ ( X ./\ W ) ) <-> ( ( S .\/ T ) ./\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) = ( S .\/ T ) ) ) |
30 |
27 29
|
mpbid |
|- ( ( ( ( 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 ) ./\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) = ( S .\/ T ) ) |
31 |
30
|
oveq1d |
|- ( ( ( ( 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 ) ./\ ( ( S .\/ T ) .\/ ( X ./\ W ) ) ) ./\ W ) = ( ( S .\/ T ) ./\ W ) ) |
32 |
1 5
|
atbase |
|- ( S e. A -> S e. B ) |
33 |
11 32
|
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 ) ) -> S e. B ) |
34 |
1 5
|
atbase |
|- ( T e. A -> T e. B ) |
35 |
12 34
|
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 ) ) -> T e. B ) |
36 |
1 3
|
latjjdir |
|- ( ( K e. Lat /\ ( S e. B /\ T e. B /\ ( X ./\ W ) e. B ) ) -> ( ( S .\/ T ) .\/ ( X ./\ W ) ) = ( ( S .\/ ( X ./\ W ) ) .\/ ( T .\/ ( X ./\ W ) ) ) ) |
37 |
15 33 35 21 36
|
syl13anc |
|- ( ( ( ( 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 ) ) = ( ( S .\/ ( X ./\ W ) ) .\/ ( T .\/ ( X ./\ W ) ) ) ) |
38 |
|
simp32 |
|- ( ( ( ( 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 .\/ ( X ./\ W ) ) = X ) |
39 |
|
simp33 |
|- ( ( ( ( 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 .\/ ( X ./\ W ) ) = X ) |
40 |
38 39
|
oveq12d |
|- ( ( ( ( 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 .\/ ( X ./\ W ) ) .\/ ( T .\/ ( X ./\ W ) ) ) = ( X .\/ X ) ) |
41 |
1 3
|
latjidm |
|- ( ( K e. Lat /\ X e. B ) -> ( X .\/ X ) = X ) |
42 |
15 16 41
|
syl2anc |
|- ( ( ( ( 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 .\/ X ) = X ) |
43 |
37 40 42
|
3eqtrd |
|- ( ( ( ( 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 ) |
44 |
43
|
oveq1d |
|- ( ( ( ( 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 ) = ( X ./\ W ) ) |
45 |
44
|
oveq2d |
|- ( ( ( ( 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 ) ./\ ( ( ( S .\/ T ) .\/ ( X ./\ W ) ) ./\ W ) ) = ( ( S .\/ T ) ./\ ( X ./\ W ) ) ) |
46 |
25 31 45
|
3eqtr3d |
|- ( ( ( ( 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 ) ./\ W ) = ( ( S .\/ T ) ./\ ( X ./\ W ) ) ) |
47 |
|
simp12r |
|- ( ( ( ( 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 .<_ W ) |
48 |
|
simp31 |
|- ( ( ( ( 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 ) |
49 |
2 3 4 5 6
|
lhpat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ S =/= T ) ) -> ( ( S .\/ T ) ./\ W ) e. A ) |
50 |
8 17 11 47 12 48 49
|
syl222anc |
|- ( ( ( ( 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 ) ./\ W ) e. A ) |
51 |
46 50
|
eqeltrrd |
|- ( ( ( ( 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. A ) |
52 |
7 51
|
eqeltrid |
|- ( ( ( ( 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 e. A ) |