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 |
|
cdlemg18b.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
9 |
|
simp33 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) |
10 |
|
simp3r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> P .<_ ( U .\/ ( F ` Q ) ) ) |
11 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> K e. HL ) |
12 |
|
simp1r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> W e. H ) |
13 |
|
simp21 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
14 |
|
simp22l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> Q e. A ) |
15 |
|
simp3l1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> P =/= Q ) |
16 |
1 2 3 4 5 8
|
cdleme0a |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A ) |
17 |
11 12 13 14 15 16
|
syl212anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> U e. A ) |
18 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
19 |
|
simp23 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> F e. T ) |
20 |
1 4 5 6
|
ltrnat |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ Q e. A ) -> ( F ` Q ) e. A ) |
21 |
18 19 14 20
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( F ` Q ) e. A ) |
22 |
1 2 4
|
hlatlej1 |
|- ( ( K e. HL /\ U e. A /\ ( F ` Q ) e. A ) -> U .<_ ( U .\/ ( F ` Q ) ) ) |
23 |
11 17 21 22
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> U .<_ ( U .\/ ( F ` Q ) ) ) |
24 |
11
|
hllatd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> K e. Lat ) |
25 |
|
simp21l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> P e. A ) |
26 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
27 |
26 4
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
28 |
25 27
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> P e. ( Base ` K ) ) |
29 |
26 4
|
atbase |
|- ( U e. A -> U e. ( Base ` K ) ) |
30 |
17 29
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> U e. ( Base ` K ) ) |
31 |
26 2 4
|
hlatjcl |
|- ( ( K e. HL /\ U e. A /\ ( F ` Q ) e. A ) -> ( U .\/ ( F ` Q ) ) e. ( Base ` K ) ) |
32 |
11 17 21 31
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( U .\/ ( F ` Q ) ) e. ( Base ` K ) ) |
33 |
26 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ U e. ( Base ` K ) /\ ( U .\/ ( F ` Q ) ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( U .\/ ( F ` Q ) ) /\ U .<_ ( U .\/ ( F ` Q ) ) ) <-> ( P .\/ U ) .<_ ( U .\/ ( F ` Q ) ) ) ) |
34 |
24 28 30 32 33
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( ( P .<_ ( U .\/ ( F ` Q ) ) /\ U .<_ ( U .\/ ( F ` Q ) ) ) <-> ( P .\/ U ) .<_ ( U .\/ ( F ` Q ) ) ) ) |
35 |
10 23 34
|
mpbi2and |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( P .\/ U ) .<_ ( U .\/ ( F ` Q ) ) ) |
36 |
1 2 3 4 5 8
|
cdleme0cp |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> ( P .\/ U ) = ( P .\/ Q ) ) |
37 |
11 12 13 14 36
|
syl22anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( P .\/ U ) = ( P .\/ Q ) ) |
38 |
|
simp22 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
39 |
5 6 1 2 4 3 8
|
cdlemg2kq |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` Q ) .\/ U ) ) |
40 |
18 13 38 19 39
|
syl121anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` Q ) .\/ U ) ) |
41 |
2 4
|
hlatjcom |
|- ( ( K e. HL /\ ( F ` Q ) e. A /\ U e. A ) -> ( ( F ` Q ) .\/ U ) = ( U .\/ ( F ` Q ) ) ) |
42 |
11 21 17 41
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( ( F ` Q ) .\/ U ) = ( U .\/ ( F ` Q ) ) ) |
43 |
40 42
|
eqtr2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( U .\/ ( F ` Q ) ) = ( ( F ` P ) .\/ ( F ` Q ) ) ) |
44 |
35 37 43
|
3brtr3d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( P .\/ Q ) .<_ ( ( F ` P ) .\/ ( F ` Q ) ) ) |
45 |
1 4 5 6
|
ltrnat |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( F ` P ) e. A ) |
46 |
18 19 25 45
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( F ` P ) e. A ) |
47 |
1 2 4
|
ps-1 |
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( ( F ` P ) e. A /\ ( F ` Q ) e. A ) ) -> ( ( P .\/ Q ) .<_ ( ( F ` P ) .\/ ( F ` Q ) ) <-> ( P .\/ Q ) = ( ( F ` P ) .\/ ( F ` Q ) ) ) ) |
48 |
11 25 14 15 46 21 47
|
syl132anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( ( P .\/ Q ) .<_ ( ( F ` P ) .\/ ( F ` Q ) ) <-> ( P .\/ Q ) = ( ( F ` P ) .\/ ( F ` Q ) ) ) ) |
49 |
44 48
|
mpbid |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( P .\/ Q ) = ( ( F ` P ) .\/ ( F ` Q ) ) ) |
50 |
2 4
|
hlatjcom |
|- ( ( K e. HL /\ ( F ` P ) e. A /\ ( F ` Q ) e. A ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` Q ) .\/ ( F ` P ) ) ) |
51 |
11 46 21 50
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` Q ) .\/ ( F ` P ) ) ) |
52 |
49 51
|
eqtr2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) = ( P .\/ Q ) ) |
53 |
52
|
3exp |
|- ( ( K e. HL /\ W e. H ) -> ( ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) -> ( ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) = ( P .\/ Q ) ) ) ) |
54 |
53
|
exp4a |
|- ( ( K e. HL /\ W e. H ) -> ( ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) -> ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) -> ( P .<_ ( U .\/ ( F ` Q ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) = ( P .\/ Q ) ) ) ) ) |
55 |
54
|
3imp |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( P .<_ ( U .\/ ( F ` Q ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) = ( P .\/ Q ) ) ) |
56 |
55
|
necon3ad |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) -> -. P .<_ ( U .\/ ( F ` Q ) ) ) ) |
57 |
9 56
|
mpd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> -. P .<_ ( U .\/ ( F ` Q ) ) ) |