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 |
|
cdlemg10.r |
|- R = ( ( trL ` K ) ` W ) |
8 |
|
simp33 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> -. ( R ` G ) .<_ ( P .\/ Q ) ) |
9 |
|
simpl1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( K e. HL /\ W e. H ) ) |
10 |
|
simpl31 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> G e. T ) |
11 |
|
simpl2l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
12 |
1 2 3 4 5 6 7
|
trlval2 |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` G ) = ( ( P .\/ ( G ` P ) ) ./\ W ) ) |
13 |
9 10 11 12
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( R ` G ) = ( ( P .\/ ( G ` P ) ) ./\ W ) ) |
14 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
15 |
|
simpl1l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> K e. HL ) |
16 |
15
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> K e. Lat ) |
17 |
|
simp2ll |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> P e. A ) |
18 |
17
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> P e. A ) |
19 |
14 4
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
20 |
18 19
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> P e. ( Base ` K ) ) |
21 |
14 5 6
|
ltrncl |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ P e. ( Base ` K ) ) -> ( G ` P ) e. ( Base ` K ) ) |
22 |
9 10 20 21
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( G ` P ) e. ( Base ` K ) ) |
23 |
14 2
|
latjcl |
|- ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( G ` P ) e. ( Base ` K ) ) -> ( P .\/ ( G ` P ) ) e. ( Base ` K ) ) |
24 |
16 20 22 23
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( P .\/ ( G ` P ) ) e. ( Base ` K ) ) |
25 |
|
simpl1r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> W e. H ) |
26 |
14 5
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
27 |
25 26
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> W e. ( Base ` K ) ) |
28 |
14 3
|
latmcl |
|- ( ( K e. Lat /\ ( P .\/ ( G ` P ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ ( G ` P ) ) ./\ W ) e. ( Base ` K ) ) |
29 |
16 24 27 28
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ W ) e. ( Base ` K ) ) |
30 |
|
simpl2r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> Q e. A ) |
31 |
14 4
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
32 |
30 31
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> Q e. ( Base ` K ) ) |
33 |
14 2
|
latjcl |
|- ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
34 |
16 20 32 33
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
35 |
14 1 3
|
latmle1 |
|- ( ( K e. Lat /\ ( P .\/ ( G ` P ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ ( G ` P ) ) ./\ W ) .<_ ( P .\/ ( G ` P ) ) ) |
36 |
16 24 27 35
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ W ) .<_ ( P .\/ ( G ` P ) ) ) |
37 |
14 1 2
|
latlej1 |
|- ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> P .<_ ( P .\/ Q ) ) |
38 |
16 20 32 37
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> P .<_ ( P .\/ Q ) ) |
39 |
14 5 6
|
ltrncl |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ Q e. ( Base ` K ) ) -> ( G ` Q ) e. ( Base ` K ) ) |
40 |
9 10 32 39
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( G ` Q ) e. ( Base ` K ) ) |
41 |
14 1 2
|
latlej1 |
|- ( ( K e. Lat /\ ( G ` P ) e. ( Base ` K ) /\ ( G ` Q ) e. ( Base ` K ) ) -> ( G ` P ) .<_ ( ( G ` P ) .\/ ( G ` Q ) ) ) |
42 |
16 22 40 41
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( G ` P ) .<_ ( ( G ` P ) .\/ ( G ` Q ) ) ) |
43 |
|
simpr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) |
44 |
42 43
|
breqtrrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( G ` P ) .<_ ( P .\/ Q ) ) |
45 |
14 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ ( G ` P ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( P .\/ Q ) /\ ( G ` P ) .<_ ( P .\/ Q ) ) <-> ( P .\/ ( G ` P ) ) .<_ ( P .\/ Q ) ) ) |
46 |
16 20 22 34 45
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( ( P .<_ ( P .\/ Q ) /\ ( G ` P ) .<_ ( P .\/ Q ) ) <-> ( P .\/ ( G ` P ) ) .<_ ( P .\/ Q ) ) ) |
47 |
38 44 46
|
mpbi2and |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( P .\/ ( G ` P ) ) .<_ ( P .\/ Q ) ) |
48 |
14 1 16 29 24 34 36 47
|
lattrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ W ) .<_ ( P .\/ Q ) ) |
49 |
13 48
|
eqbrtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( R ` G ) .<_ ( P .\/ Q ) ) |
50 |
49
|
ex |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) -> ( R ` G ) .<_ ( P .\/ Q ) ) ) |
51 |
50
|
necon3bd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( -. ( R ` G ) .<_ ( P .\/ Q ) -> ( P .\/ Q ) =/= ( ( G ` P ) .\/ ( G ` Q ) ) ) ) |
52 |
8 51
|
mpd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) =/= ( ( G ` P ) .\/ ( G ` Q ) ) ) |