Step |
Hyp |
Ref |
Expression |
1 |
|
latdisd.b |
|- B = ( Base ` K ) |
2 |
|
latdisd.j |
|- .\/ = ( join ` K ) |
3 |
|
latdisd.m |
|- ./\ = ( meet ` K ) |
4 |
1 2 3
|
latdisdlem |
|- ( K e. Lat -> ( A. x e. B A. y e. B A. z e. B ( x .\/ ( y ./\ z ) ) = ( ( x .\/ y ) ./\ ( x .\/ z ) ) -> A. u e. B A. v e. B A. w e. B ( u ./\ ( v .\/ w ) ) = ( ( u ./\ v ) .\/ ( u ./\ w ) ) ) ) |
5 |
|
eqid |
|- ( ODual ` K ) = ( ODual ` K ) |
6 |
5
|
odulat |
|- ( K e. Lat -> ( ODual ` K ) e. Lat ) |
7 |
5 1
|
odubas |
|- B = ( Base ` ( ODual ` K ) ) |
8 |
5 3
|
odujoin |
|- ./\ = ( join ` ( ODual ` K ) ) |
9 |
5 2
|
odumeet |
|- .\/ = ( meet ` ( ODual ` K ) ) |
10 |
7 8 9
|
latdisdlem |
|- ( ( ODual ` K ) e. Lat -> ( A. u e. B A. v e. B A. w e. B ( u ./\ ( v .\/ w ) ) = ( ( u ./\ v ) .\/ ( u ./\ w ) ) -> A. x e. B A. y e. B A. z e. B ( x .\/ ( y ./\ z ) ) = ( ( x .\/ y ) ./\ ( x .\/ z ) ) ) ) |
11 |
6 10
|
syl |
|- ( K e. Lat -> ( A. u e. B A. v e. B A. w e. B ( u ./\ ( v .\/ w ) ) = ( ( u ./\ v ) .\/ ( u ./\ w ) ) -> A. x e. B A. y e. B A. z e. B ( x .\/ ( y ./\ z ) ) = ( ( x .\/ y ) ./\ ( x .\/ z ) ) ) ) |
12 |
4 11
|
impbid |
|- ( K e. Lat -> ( A. x e. B A. y e. B A. z e. B ( x .\/ ( y ./\ z ) ) = ( ( x .\/ y ) ./\ ( x .\/ z ) ) <-> A. u e. B A. v e. B A. w e. B ( u ./\ ( v .\/ w ) ) = ( ( u ./\ v ) .\/ ( u ./\ w ) ) ) ) |
13 |
|
oveq1 |
|- ( u = x -> ( u ./\ ( v .\/ w ) ) = ( x ./\ ( v .\/ w ) ) ) |
14 |
|
oveq1 |
|- ( u = x -> ( u ./\ v ) = ( x ./\ v ) ) |
15 |
|
oveq1 |
|- ( u = x -> ( u ./\ w ) = ( x ./\ w ) ) |
16 |
14 15
|
oveq12d |
|- ( u = x -> ( ( u ./\ v ) .\/ ( u ./\ w ) ) = ( ( x ./\ v ) .\/ ( x ./\ w ) ) ) |
17 |
13 16
|
eqeq12d |
|- ( u = x -> ( ( u ./\ ( v .\/ w ) ) = ( ( u ./\ v ) .\/ ( u ./\ w ) ) <-> ( x ./\ ( v .\/ w ) ) = ( ( x ./\ v ) .\/ ( x ./\ w ) ) ) ) |
18 |
|
oveq1 |
|- ( v = y -> ( v .\/ w ) = ( y .\/ w ) ) |
19 |
18
|
oveq2d |
|- ( v = y -> ( x ./\ ( v .\/ w ) ) = ( x ./\ ( y .\/ w ) ) ) |
20 |
|
oveq2 |
|- ( v = y -> ( x ./\ v ) = ( x ./\ y ) ) |
21 |
20
|
oveq1d |
|- ( v = y -> ( ( x ./\ v ) .\/ ( x ./\ w ) ) = ( ( x ./\ y ) .\/ ( x ./\ w ) ) ) |
22 |
19 21
|
eqeq12d |
|- ( v = y -> ( ( x ./\ ( v .\/ w ) ) = ( ( x ./\ v ) .\/ ( x ./\ w ) ) <-> ( x ./\ ( y .\/ w ) ) = ( ( x ./\ y ) .\/ ( x ./\ w ) ) ) ) |
23 |
|
oveq2 |
|- ( w = z -> ( y .\/ w ) = ( y .\/ z ) ) |
24 |
23
|
oveq2d |
|- ( w = z -> ( x ./\ ( y .\/ w ) ) = ( x ./\ ( y .\/ z ) ) ) |
25 |
|
oveq2 |
|- ( w = z -> ( x ./\ w ) = ( x ./\ z ) ) |
26 |
25
|
oveq2d |
|- ( w = z -> ( ( x ./\ y ) .\/ ( x ./\ w ) ) = ( ( x ./\ y ) .\/ ( x ./\ z ) ) ) |
27 |
24 26
|
eqeq12d |
|- ( w = z -> ( ( x ./\ ( y .\/ w ) ) = ( ( x ./\ y ) .\/ ( x ./\ w ) ) <-> ( x ./\ ( y .\/ z ) ) = ( ( x ./\ y ) .\/ ( x ./\ z ) ) ) ) |
28 |
17 22 27
|
cbvral3vw |
|- ( A. u e. B A. v e. B A. w e. B ( u ./\ ( v .\/ w ) ) = ( ( u ./\ v ) .\/ ( u ./\ w ) ) <-> A. x e. B A. y e. B A. z e. B ( x ./\ ( y .\/ z ) ) = ( ( x ./\ y ) .\/ ( x ./\ z ) ) ) |
29 |
12 28
|
bitrdi |
|- ( K e. Lat -> ( A. x e. B A. y e. B A. z e. B ( x .\/ ( y ./\ z ) ) = ( ( x .\/ y ) ./\ ( x .\/ z ) ) <-> A. x e. B A. y e. B A. z e. B ( x ./\ ( y .\/ z ) ) = ( ( x ./\ y ) .\/ ( x ./\ z ) ) ) ) |