Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme21.l |
|- .<_ = ( le ` K ) |
2 |
|
cdleme21.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleme21.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdleme21.a |
|- A = ( Atoms ` K ) |
5 |
|
cdleme21.h |
|- H = ( LHyp ` K ) |
6 |
|
cdleme21.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
7 |
|
simp23 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> -. S .<_ ( P .\/ Q ) ) |
8 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> K e. HL ) |
9 |
|
hlcvl |
|- ( K e. HL -> K e. CvLat ) |
10 |
8 9
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> K e. CvLat ) |
11 |
|
simp12l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> P e. A ) |
12 |
|
simp21 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> S e. A ) |
13 |
|
simp3l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> z e. A ) |
14 |
|
simp13 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> Q e. A ) |
15 |
1 2 4
|
atnlej1 |
|- ( ( K e. HL /\ ( S e. A /\ P e. A /\ Q e. A ) /\ -. S .<_ ( P .\/ Q ) ) -> S =/= P ) |
16 |
15
|
necomd |
|- ( ( K e. HL /\ ( S e. A /\ P e. A /\ Q e. A ) /\ -. S .<_ ( P .\/ Q ) ) -> P =/= S ) |
17 |
8 12 11 14 7 16
|
syl131anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> P =/= S ) |
18 |
|
simp3r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( P .\/ z ) = ( S .\/ z ) ) |
19 |
4 2
|
cvlsupr7 |
|- ( ( K e. CvLat /\ ( P e. A /\ S e. A /\ z e. A ) /\ ( P =/= S /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( P .\/ S ) = ( z .\/ S ) ) |
20 |
10 11 12 13 17 18 19
|
syl132anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( P .\/ S ) = ( z .\/ S ) ) |
21 |
2 4
|
hlatjcom |
|- ( ( K e. HL /\ z e. A /\ S e. A ) -> ( z .\/ S ) = ( S .\/ z ) ) |
22 |
8 13 12 21
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( z .\/ S ) = ( S .\/ z ) ) |
23 |
20 22
|
eqtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( P .\/ S ) = ( S .\/ z ) ) |
24 |
23
|
breq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( U .<_ ( P .\/ S ) <-> U .<_ ( S .\/ z ) ) ) |
25 |
|
simp11r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> W e. H ) |
26 |
|
simp12r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> -. P .<_ W ) |
27 |
|
simp22 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> P =/= Q ) |
28 |
1 2 3 4 5 6
|
cdleme0a |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A ) |
29 |
8 25 11 26 14 27 28
|
syl222anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> U e. A ) |
30 |
8
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> K e. Lat ) |
31 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
32 |
31 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
33 |
8 11 14 32
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
34 |
31 5
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
35 |
25 34
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> W e. ( Base ` K ) ) |
36 |
31 1 3
|
latmle2 |
|- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W ) |
37 |
30 33 35 36
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W ) |
38 |
6 37
|
eqbrtrid |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> U .<_ W ) |
39 |
|
nbrne2 |
|- ( ( U .<_ W /\ -. P .<_ W ) -> U =/= P ) |
40 |
38 26 39
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> U =/= P ) |
41 |
1 2 4
|
cvlatexch1 |
|- ( ( K e. CvLat /\ ( U e. A /\ S e. A /\ P e. A ) /\ U =/= P ) -> ( U .<_ ( P .\/ S ) -> S .<_ ( P .\/ U ) ) ) |
42 |
10 29 12 11 40 41
|
syl131anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( U .<_ ( P .\/ S ) -> S .<_ ( P .\/ U ) ) ) |
43 |
1 2 4
|
hlatlej1 |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> P .<_ ( P .\/ Q ) ) |
44 |
8 11 14 43
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> P .<_ ( P .\/ Q ) ) |
45 |
1 2 3 4 5 6
|
cdlemeulpq |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> U .<_ ( P .\/ Q ) ) |
46 |
8 25 11 14 45
|
syl22anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> U .<_ ( P .\/ Q ) ) |
47 |
31 4
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
48 |
11 47
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> P e. ( Base ` K ) ) |
49 |
31 4
|
atbase |
|- ( U e. A -> U e. ( Base ` K ) ) |
50 |
29 49
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> U e. ( Base ` K ) ) |
51 |
31 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ U e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( P .\/ Q ) /\ U .<_ ( P .\/ Q ) ) <-> ( P .\/ U ) .<_ ( P .\/ Q ) ) ) |
52 |
30 48 50 33 51
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( ( P .<_ ( P .\/ Q ) /\ U .<_ ( P .\/ Q ) ) <-> ( P .\/ U ) .<_ ( P .\/ Q ) ) ) |
53 |
44 46 52
|
mpbi2and |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( P .\/ U ) .<_ ( P .\/ Q ) ) |
54 |
31 4
|
atbase |
|- ( S e. A -> S e. ( Base ` K ) ) |
55 |
12 54
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> S e. ( Base ` K ) ) |
56 |
31 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ U e. A ) -> ( P .\/ U ) e. ( Base ` K ) ) |
57 |
8 11 29 56
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( P .\/ U ) e. ( Base ` K ) ) |
58 |
31 1
|
lattr |
|- ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ ( P .\/ U ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( S .<_ ( P .\/ U ) /\ ( P .\/ U ) .<_ ( P .\/ Q ) ) -> S .<_ ( P .\/ Q ) ) ) |
59 |
30 55 57 33 58
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( ( S .<_ ( P .\/ U ) /\ ( P .\/ U ) .<_ ( P .\/ Q ) ) -> S .<_ ( P .\/ Q ) ) ) |
60 |
53 59
|
mpan2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( S .<_ ( P .\/ U ) -> S .<_ ( P .\/ Q ) ) ) |
61 |
42 60
|
syld |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( U .<_ ( P .\/ S ) -> S .<_ ( P .\/ Q ) ) ) |
62 |
24 61
|
sylbird |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( U .<_ ( S .\/ z ) -> S .<_ ( P .\/ Q ) ) ) |
63 |
7 62
|
mtod |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> -. U .<_ ( S .\/ z ) ) |