Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemk5.b |
|- B = ( Base ` K ) |
2 |
|
cdlemk5.l |
|- .<_ = ( le ` K ) |
3 |
|
cdlemk5.j |
|- .\/ = ( join ` K ) |
4 |
|
cdlemk5.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdlemk5.a |
|- A = ( Atoms ` K ) |
6 |
|
cdlemk5.h |
|- H = ( LHyp ` K ) |
7 |
|
cdlemk5.t |
|- T = ( ( LTrn ` K ) ` W ) |
8 |
|
cdlemk5.r |
|- R = ( ( trL ` K ) ` W ) |
9 |
|
cdlemk5.z |
|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) |
10 |
|
simp1r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ b e. T ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F = N ) |
11 |
10
|
fveq1d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ b e. T ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( F ` P ) = ( N ` P ) ) |
12 |
11
|
oveq1d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ b e. T ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( F ` P ) .\/ ( R ` ( b o. `' F ) ) ) = ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) |
13 |
12
|
oveq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ b e. T ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` b ) ) ./\ ( ( F ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) ) |
14 |
1 2 3 4 5 6 7 8
|
cdlemkfid1N |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ b e. T ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` b ) ) ./\ ( ( F ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) = ( b ` P ) ) |
15 |
14
|
3adant1r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ b e. T ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` b ) ) ./\ ( ( F ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) = ( b ` P ) ) |
16 |
13 15
|
eqtr3d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ b e. T ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) = ( b ` P ) ) |
17 |
9 16
|
syl5eq |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F = N ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ b e. T ) /\ ( ( R ` b ) =/= ( R ` F ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> Z = ( b ` P ) ) |