Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg8.l |
|- .<_ = ( le ` K ) |
2 |
|
cdlemg8.j |
|- .\/ = ( join ` K ) |
3 |
|
cdlemg8.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdlemg8.a |
|- A = ( Atoms ` K ) |
5 |
|
cdlemg8.h |
|- H = ( LHyp ` K ) |
6 |
|
cdlemg8.t |
|- T = ( ( LTrn ` K ) ` W ) |
7 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( K e. HL /\ W e. H ) ) |
8 |
|
simp2r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
9 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
10 |
1 3 9 4 5
|
lhpmat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Q ./\ W ) = ( 0. ` K ) ) |
11 |
7 8 10
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( Q ./\ W ) = ( 0. ` K ) ) |
12 |
1 4 5 6
|
cdlemg6 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` Q ) ) = Q ) |
13 |
12
|
oveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( Q .\/ ( F ` ( G ` Q ) ) ) = ( Q .\/ Q ) ) |
14 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> K e. HL ) |
15 |
|
simp2rl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> Q e. A ) |
16 |
2 4
|
hlatjidm |
|- ( ( K e. HL /\ Q e. A ) -> ( Q .\/ Q ) = Q ) |
17 |
14 15 16
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( Q .\/ Q ) = Q ) |
18 |
13 17
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( Q .\/ ( F ` ( G ` Q ) ) ) = Q ) |
19 |
18
|
oveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) = ( Q ./\ W ) ) |
20 |
|
simp33 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` P ) ) = P ) |
21 |
20
|
oveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( P .\/ ( F ` ( G ` P ) ) ) = ( P .\/ P ) ) |
22 |
|
simp2ll |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> P e. A ) |
23 |
2 4
|
hlatjidm |
|- ( ( K e. HL /\ P e. A ) -> ( P .\/ P ) = P ) |
24 |
14 22 23
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( P .\/ P ) = P ) |
25 |
21 24
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( P .\/ ( F ` ( G ` P ) ) ) = P ) |
26 |
25
|
oveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( P ./\ W ) ) |
27 |
|
simp2l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( P e. A /\ -. P .<_ W ) ) |
28 |
1 3 9 4 5
|
lhpmat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ./\ W ) = ( 0. ` K ) ) |
29 |
7 27 28
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( P ./\ W ) = ( 0. ` K ) ) |
30 |
26 29
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( 0. ` K ) ) |
31 |
11 19 30
|
3eqtr4rd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` ( G ` P ) ) = P ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) ) |