Step |
Hyp |
Ref |
Expression |
1 |
|
trl0.l |
|- .<_ = ( le ` K ) |
2 |
|
trl0.z |
|- .0. = ( 0. ` K ) |
3 |
|
trl0.a |
|- A = ( Atoms ` K ) |
4 |
|
trl0.h |
|- H = ( LHyp ` K ) |
5 |
|
trl0.t |
|- T = ( ( LTrn ` K ) ` W ) |
6 |
|
trl0.r |
|- R = ( ( trL ` K ) ` W ) |
7 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( K e. HL /\ W e. H ) ) |
8 |
|
simp3l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> F e. T ) |
9 |
|
simp2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P e. A /\ -. P .<_ W ) ) |
10 |
|
eqid |
|- ( join ` K ) = ( join ` K ) |
11 |
|
eqid |
|- ( meet ` K ) = ( meet ` K ) |
12 |
1 10 11 3 4 5 6
|
trlval2 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` F ) = ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) ) |
13 |
7 8 9 12
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( R ` F ) = ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) ) |
14 |
|
simp3r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( F ` P ) = P ) |
15 |
14
|
oveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P ( join ` K ) ( F ` P ) ) = ( P ( join ` K ) P ) ) |
16 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> K e. HL ) |
17 |
|
simp2l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> P e. A ) |
18 |
10 3
|
hlatjidm |
|- ( ( K e. HL /\ P e. A ) -> ( P ( join ` K ) P ) = P ) |
19 |
16 17 18
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P ( join ` K ) P ) = P ) |
20 |
15 19
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P ( join ` K ) ( F ` P ) ) = P ) |
21 |
20
|
oveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( ( P ( join ` K ) ( F ` P ) ) ( meet ` K ) W ) = ( P ( meet ` K ) W ) ) |
22 |
1 11 2 3 4
|
lhpmat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ( meet ` K ) W ) = .0. ) |
23 |
7 9 22
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( P ( meet ` K ) W ) = .0. ) |
24 |
13 21 23
|
3eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) = P ) ) -> ( R ` F ) = .0. ) |