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 |
|
simp22 |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> D e. T ) |
12 |
|
simp21 |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T ) |
13 |
|
eqid |
|- ( S ` D ) = ( S ` D ) |
14 |
|
eqid |
|- ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) |
15 |
1 2 3 4 5 6 7 8 9 10 13 14
|
cdlemkuu |
|- ( ( D e. T /\ F e. T ) -> ( D Y F ) = ( ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) ` F ) ) |
16 |
11 12 15
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( D Y F ) = ( ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) ` F ) ) |
17 |
16
|
fveq1d |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( D Y F ) ` P ) = ( ( ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) ` F ) ` P ) ) |
18 |
1 2 3 4 5 6 7 8 9 13 14
|
cdlemk18-2N |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( N ` P ) = ( ( ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) ` F ) ` P ) ) |
19 |
17 18
|
eqtr4d |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( D Y F ) ` P ) = ( N ` P ) ) |