Step |
Hyp |
Ref |
Expression |
1 |
|
trlle.l |
|- .<_ = ( le ` K ) |
2 |
|
trlle.h |
|- H = ( LHyp ` K ) |
3 |
|
trlle.t |
|- T = ( ( LTrn ` K ) ` W ) |
4 |
|
trlle.r |
|- R = ( ( trL ` K ) ` W ) |
5 |
|
eqid |
|- ( oc ` K ) = ( oc ` K ) |
6 |
|
eqid |
|- ( Atoms ` K ) = ( Atoms ` K ) |
7 |
1 5 6 2
|
lhpocnel |
|- ( ( K e. HL /\ W e. H ) -> ( ( ( oc ` K ) ` W ) e. ( Atoms ` K ) /\ -. ( ( oc ` K ) ` W ) .<_ W ) ) |
8 |
7
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( ( ( oc ` K ) ` W ) e. ( Atoms ` K ) /\ -. ( ( oc ` K ) ` W ) .<_ W ) ) |
9 |
|
eqid |
|- ( join ` K ) = ( join ` K ) |
10 |
|
eqid |
|- ( meet ` K ) = ( meet ` K ) |
11 |
1 9 10 6 2 3 4
|
trlval2 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( ( oc ` K ) ` W ) e. ( Atoms ` K ) /\ -. ( ( oc ` K ) ` W ) .<_ W ) ) -> ( R ` F ) = ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( F ` ( ( oc ` K ) ` W ) ) ) ( meet ` K ) W ) ) |
12 |
8 11
|
mpd3an3 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) = ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( F ` ( ( oc ` K ) ` W ) ) ) ( meet ` K ) W ) ) |
13 |
|
hllat |
|- ( K e. HL -> K e. Lat ) |
14 |
13
|
ad2antrr |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> K e. Lat ) |
15 |
|
hlop |
|- ( K e. HL -> K e. OP ) |
16 |
15
|
ad2antrr |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> K e. OP ) |
17 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
18 |
17 2
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
19 |
18
|
ad2antlr |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> W e. ( Base ` K ) ) |
20 |
17 5
|
opoccl |
|- ( ( K e. OP /\ W e. ( Base ` K ) ) -> ( ( oc ` K ) ` W ) e. ( Base ` K ) ) |
21 |
16 19 20
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( ( oc ` K ) ` W ) e. ( Base ` K ) ) |
22 |
17 2 3
|
ltrncl |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( oc ` K ) ` W ) e. ( Base ` K ) ) -> ( F ` ( ( oc ` K ) ` W ) ) e. ( Base ` K ) ) |
23 |
21 22
|
mpd3an3 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( F ` ( ( oc ` K ) ` W ) ) e. ( Base ` K ) ) |
24 |
17 9
|
latjcl |
|- ( ( K e. Lat /\ ( ( oc ` K ) ` W ) e. ( Base ` K ) /\ ( F ` ( ( oc ` K ) ` W ) ) e. ( Base ` K ) ) -> ( ( ( oc ` K ) ` W ) ( join ` K ) ( F ` ( ( oc ` K ) ` W ) ) ) e. ( Base ` K ) ) |
25 |
14 21 23 24
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( ( ( oc ` K ) ` W ) ( join ` K ) ( F ` ( ( oc ` K ) ` W ) ) ) e. ( Base ` K ) ) |
26 |
17 1 10
|
latmle2 |
|- ( ( K e. Lat /\ ( ( ( oc ` K ) ` W ) ( join ` K ) ( F ` ( ( oc ` K ) ` W ) ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( F ` ( ( oc ` K ) ` W ) ) ) ( meet ` K ) W ) .<_ W ) |
27 |
14 25 19 26
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( ( ( ( oc ` K ) ` W ) ( join ` K ) ( F ` ( ( oc ` K ) ` W ) ) ) ( meet ` K ) W ) .<_ W ) |
28 |
12 27
|
eqbrtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W ) |