Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemk3.b |
|- B = ( Base ` K ) |
2 |
|
cdlemk3.l |
|- .<_ = ( le ` K ) |
3 |
|
cdlemk3.j |
|- .\/ = ( join ` K ) |
4 |
|
cdlemk3.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdlemk3.a |
|- A = ( Atoms ` K ) |
6 |
|
cdlemk3.h |
|- H = ( LHyp ` K ) |
7 |
|
cdlemk3.t |
|- T = ( ( LTrn ` K ) ` W ) |
8 |
|
cdlemk3.r |
|- R = ( ( trL ` K ) ` W ) |
9 |
|
cdlemk3.s |
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) |
10 |
|
cdlemk3.u1 |
|- Y = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) ) |
11 |
|
cdlemk3.o2 |
|- Q = ( S ` D ) |
12 |
|
cdlemk3.u2 |
|- Z = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) |
13 |
|
fveq2 |
|- ( d = D -> ( S ` d ) = ( S ` D ) ) |
14 |
13 11
|
eqtr4di |
|- ( d = D -> ( S ` d ) = Q ) |
15 |
14
|
fveq1d |
|- ( d = D -> ( ( S ` d ) ` P ) = ( Q ` P ) ) |
16 |
|
cnveq |
|- ( d = D -> `' d = `' D ) |
17 |
16
|
coeq2d |
|- ( d = D -> ( e o. `' d ) = ( e o. `' D ) ) |
18 |
17
|
fveq2d |
|- ( d = D -> ( R ` ( e o. `' d ) ) = ( R ` ( e o. `' D ) ) ) |
19 |
15 18
|
oveq12d |
|- ( d = D -> ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) = ( ( Q ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) |
20 |
19
|
oveq2d |
|- ( d = D -> ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) |
21 |
20
|
eqeq2d |
|- ( d = D -> ( ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) <-> ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) |
22 |
21
|
riotabidv |
|- ( d = D -> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) = ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) |
23 |
|
fveq2 |
|- ( e = G -> ( R ` e ) = ( R ` G ) ) |
24 |
23
|
oveq2d |
|- ( e = G -> ( P .\/ ( R ` e ) ) = ( P .\/ ( R ` G ) ) ) |
25 |
|
coeq1 |
|- ( e = G -> ( e o. `' D ) = ( G o. `' D ) ) |
26 |
25
|
fveq2d |
|- ( e = G -> ( R ` ( e o. `' D ) ) = ( R ` ( G o. `' D ) ) ) |
27 |
26
|
oveq2d |
|- ( e = G -> ( ( Q ` P ) .\/ ( R ` ( e o. `' D ) ) ) = ( ( Q ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) |
28 |
24 27
|
oveq12d |
|- ( e = G -> ( ( P .\/ ( R ` e ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) |
29 |
28
|
eqeq2d |
|- ( e = G -> ( ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) <-> ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) ) |
30 |
29
|
riotabidv |
|- ( e = G -> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) = ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) ) |
31 |
|
riotaex |
|- ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) e. _V |
32 |
22 30 10 31
|
ovmpo |
|- ( ( D e. T /\ G e. T ) -> ( D Y G ) = ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) ) |
33 |
1 2 3 5 6 7 8 4 12
|
cdlemksv |
|- ( G e. T -> ( Z ` G ) = ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) ) |
34 |
33
|
adantl |
|- ( ( D e. T /\ G e. T ) -> ( Z ` G ) = ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) ) |
35 |
32 34
|
eqtr4d |
|- ( ( D e. T /\ G e. T ) -> ( D Y G ) = ( Z ` G ) ) |