Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg35.l |
|- .<_ = ( le ` K ) |
2 |
|
cdlemg35.j |
|- .\/ = ( join ` K ) |
3 |
|
cdlemg35.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdlemg35.a |
|- A = ( Atoms ` K ) |
5 |
|
cdlemg35.h |
|- H = ( LHyp ` K ) |
6 |
|
cdlemg35.t |
|- T = ( ( LTrn ` K ) ` W ) |
7 |
|
id |
|- ( P = Q -> P = Q ) |
8 |
|
2fveq3 |
|- ( P = Q -> ( F ` ( G ` P ) ) = ( F ` ( G ` Q ) ) ) |
9 |
7 8
|
oveq12d |
|- ( P = Q -> ( P .\/ ( F ` ( G ` P ) ) ) = ( Q .\/ ( F ` ( G ` Q ) ) ) ) |
10 |
9
|
oveq1d |
|- ( P = Q -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) ) |
11 |
10
|
adantl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P = Q ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) ) |
12 |
|
simpl1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) ) |
13 |
|
simpl2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P =/= Q ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) |
14 |
|
simpl3l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P =/= Q ) -> F e. T ) |
15 |
|
simpl3r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P =/= Q ) -> G e. T ) |
16 |
|
simpr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P =/= Q ) -> P =/= Q ) |
17 |
|
eqid |
|- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W ) |
18 |
1 2 3 4 5 6 17
|
cdlemg39 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) ) |
19 |
12 13 14 15 16 18
|
syl113anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) /\ P =/= Q ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) ) |
20 |
11 19
|
pm2.61dane |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) ) |