Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg12.l |
|- .<_ = ( le ` K ) |
2 |
|
cdlemg12.j |
|- .\/ = ( join ` K ) |
3 |
|
cdlemg12.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdlemg12.a |
|- A = ( Atoms ` K ) |
5 |
|
cdlemg12.h |
|- H = ( LHyp ` K ) |
6 |
|
cdlemg12.t |
|- T = ( ( LTrn ` K ) ` W ) |
7 |
|
cdlemg12b.r |
|- R = ( ( trL ` K ) ` W ) |
8 |
|
simp3l |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) .<_ ( P .\/ Q ) ) |
9 |
|
simp11 |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> K e. HL ) |
10 |
|
simp12 |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> W e. H ) |
11 |
|
simp13 |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> G e. T ) |
12 |
1 5 6 7
|
trlle |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) .<_ W ) |
13 |
9 10 11 12
|
syl21anc |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) .<_ W ) |
14 |
9
|
hllatd |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> K e. Lat ) |
15 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
16 |
15 5 6 7
|
trlcl |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) e. ( Base ` K ) ) |
17 |
9 10 11 16
|
syl21anc |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) e. ( Base ` K ) ) |
18 |
|
simp21l |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> P e. A ) |
19 |
|
simp22 |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> Q e. A ) |
20 |
15 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
21 |
9 18 19 20
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
22 |
15 5
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
23 |
10 22
|
syl |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> W e. ( Base ` K ) ) |
24 |
15 1 3
|
latlem12 |
|- ( ( K e. Lat /\ ( ( R ` G ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` G ) .<_ W ) <-> ( R ` G ) .<_ ( ( P .\/ Q ) ./\ W ) ) ) |
25 |
14 17 21 23 24
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` G ) .<_ W ) <-> ( R ` G ) .<_ ( ( P .\/ Q ) ./\ W ) ) ) |
26 |
8 13 25
|
mpbi2and |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) .<_ ( ( P .\/ Q ) ./\ W ) ) |
27 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
28 |
9 27
|
syl |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> K e. AtLat ) |
29 |
|
simp21 |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( P e. A /\ -. P .<_ W ) ) |
30 |
|
simp3r |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( G ` P ) =/= P ) |
31 |
1 4 5 6 7
|
trlat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( G e. T /\ ( G ` P ) =/= P ) ) -> ( R ` G ) e. A ) |
32 |
9 10 29 11 30 31
|
syl212anc |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) e. A ) |
33 |
|
simp23 |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> P =/= Q ) |
34 |
1 2 3 4 5
|
lhpat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> ( ( P .\/ Q ) ./\ W ) e. A ) |
35 |
9 10 29 19 33 34
|
syl212anc |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( ( P .\/ Q ) ./\ W ) e. A ) |
36 |
1 4
|
atcmp |
|- ( ( K e. AtLat /\ ( R ` G ) e. A /\ ( ( P .\/ Q ) ./\ W ) e. A ) -> ( ( R ` G ) .<_ ( ( P .\/ Q ) ./\ W ) <-> ( R ` G ) = ( ( P .\/ Q ) ./\ W ) ) ) |
37 |
28 32 35 36
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( ( R ` G ) .<_ ( ( P .\/ Q ) ./\ W ) <-> ( R ` G ) = ( ( P .\/ Q ) ./\ W ) ) ) |
38 |
26 37
|
mpbid |
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) = ( ( P .\/ Q ) ./\ W ) ) |