Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg4.l |
|- .<_ = ( le ` K ) |
2 |
|
cdlemg4.a |
|- A = ( Atoms ` K ) |
3 |
|
cdlemg4.h |
|- H = ( LHyp ` K ) |
4 |
|
cdlemg4.t |
|- T = ( ( LTrn ` K ) ` W ) |
5 |
|
cdlemg4.r |
|- R = ( ( trL ` K ) ` W ) |
6 |
|
cdlemg4.j |
|- .\/ = ( join ` K ) |
7 |
|
cdlemg4b.v |
|- V = ( R ` G ) |
8 |
|
cdlemg4.m |
|- ./\ = ( meet ` K ) |
9 |
1 2 3 4 5 6 7 8
|
cdlemg4f |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` Q ) ) = ( ( Q .\/ V ) ./\ ( P .\/ ( ( P .\/ Q ) ./\ W ) ) ) ) |
10 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> K e. HL ) |
11 |
|
simp1r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> W e. H ) |
12 |
|
simp21 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( P e. A /\ -. P .<_ W ) ) |
13 |
|
simp22l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> Q e. A ) |
14 |
|
eqid |
|- ( ( P .\/ Q ) ./\ W ) = ( ( P .\/ Q ) ./\ W ) |
15 |
1 6 8 2 3 14
|
cdleme0cp |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> ( P .\/ ( ( P .\/ Q ) ./\ W ) ) = ( P .\/ Q ) ) |
16 |
10 11 12 13 15
|
syl22anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( P .\/ ( ( P .\/ Q ) ./\ W ) ) = ( P .\/ Q ) ) |
17 |
16
|
oveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( ( Q .\/ V ) ./\ ( P .\/ ( ( P .\/ Q ) ./\ W ) ) ) = ( ( Q .\/ V ) ./\ ( P .\/ Q ) ) ) |
18 |
9 17
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` Q ) ) = ( ( Q .\/ V ) ./\ ( P .\/ Q ) ) ) |