Step |
Hyp |
Ref |
Expression |
1 |
|
isoml.b |
|- B = ( Base ` K ) |
2 |
|
isoml.l |
|- .<_ = ( le ` K ) |
3 |
|
isoml.j |
|- .\/ = ( join ` K ) |
4 |
|
isoml.m |
|- ./\ = ( meet ` K ) |
5 |
|
isoml.o |
|- ._|_ = ( oc ` K ) |
6 |
|
fveq2 |
|- ( k = K -> ( Base ` k ) = ( Base ` K ) ) |
7 |
6 1
|
eqtr4di |
|- ( k = K -> ( Base ` k ) = B ) |
8 |
|
fveq2 |
|- ( k = K -> ( le ` k ) = ( le ` K ) ) |
9 |
8 2
|
eqtr4di |
|- ( k = K -> ( le ` k ) = .<_ ) |
10 |
9
|
breqd |
|- ( k = K -> ( x ( le ` k ) y <-> x .<_ y ) ) |
11 |
|
fveq2 |
|- ( k = K -> ( join ` k ) = ( join ` K ) ) |
12 |
11 3
|
eqtr4di |
|- ( k = K -> ( join ` k ) = .\/ ) |
13 |
|
eqidd |
|- ( k = K -> x = x ) |
14 |
|
fveq2 |
|- ( k = K -> ( meet ` k ) = ( meet ` K ) ) |
15 |
14 4
|
eqtr4di |
|- ( k = K -> ( meet ` k ) = ./\ ) |
16 |
|
eqidd |
|- ( k = K -> y = y ) |
17 |
|
fveq2 |
|- ( k = K -> ( oc ` k ) = ( oc ` K ) ) |
18 |
17 5
|
eqtr4di |
|- ( k = K -> ( oc ` k ) = ._|_ ) |
19 |
18
|
fveq1d |
|- ( k = K -> ( ( oc ` k ) ` x ) = ( ._|_ ` x ) ) |
20 |
15 16 19
|
oveq123d |
|- ( k = K -> ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) = ( y ./\ ( ._|_ ` x ) ) ) |
21 |
12 13 20
|
oveq123d |
|- ( k = K -> ( x ( join ` k ) ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) ) = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) |
22 |
21
|
eqeq2d |
|- ( k = K -> ( y = ( x ( join ` k ) ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) ) <-> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) ) |
23 |
10 22
|
imbi12d |
|- ( k = K -> ( ( x ( le ` k ) y -> y = ( x ( join ` k ) ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) ) ) <-> ( x .<_ y -> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) ) ) |
24 |
7 23
|
raleqbidv |
|- ( k = K -> ( A. y e. ( Base ` k ) ( x ( le ` k ) y -> y = ( x ( join ` k ) ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) ) ) <-> A. y e. B ( x .<_ y -> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) ) ) |
25 |
7 24
|
raleqbidv |
|- ( k = K -> ( A. x e. ( Base ` k ) A. y e. ( Base ` k ) ( x ( le ` k ) y -> y = ( x ( join ` k ) ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) ) ) <-> A. x e. B A. y e. B ( x .<_ y -> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) ) ) |
26 |
|
df-oml |
|- OML = { k e. OL | A. x e. ( Base ` k ) A. y e. ( Base ` k ) ( x ( le ` k ) y -> y = ( x ( join ` k ) ( y ( meet ` k ) ( ( oc ` k ) ` x ) ) ) ) } |
27 |
25 26
|
elrab2 |
|- ( K e. OML <-> ( K e. OL /\ A. x e. B A. y e. B ( x .<_ y -> y = ( x .\/ ( y ./\ ( ._|_ ` x ) ) ) ) ) ) |