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 |
|
simp11 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( K e. HL /\ W e. H ) ) |
9 |
|
simp12 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( P e. A /\ -. P .<_ W ) ) |
10 |
|
simp31 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> v =/= ( R ` F ) ) |
11 |
|
simp13 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( v e. A /\ v .<_ W ) ) |
12 |
|
simp2r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> F e. T ) |
13 |
|
simp33 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( F ` P ) =/= P ) |
14 |
1 4 5 6 7
|
trlat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A ) |
15 |
8 9 12 13 14
|
syl112anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A ) |
16 |
1 5 6 7
|
trlle |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W ) |
17 |
8 12 16
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( R ` F ) .<_ W ) |
18 |
1 2 4 5
|
lhp2atnle |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ v =/= ( R ` F ) ) /\ ( v e. A /\ v .<_ W ) /\ ( ( R ` F ) e. A /\ ( R ` F ) .<_ W ) ) -> -. ( R ` F ) .<_ ( P .\/ v ) ) |
19 |
8 9 10 11 15 17 18
|
syl312anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. ( R ` F ) .<_ ( P .\/ v ) ) |
20 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> K e. HL ) |
21 |
|
simp12l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> P e. A ) |
22 |
|
simp13l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> v e. A ) |
23 |
1 2 4
|
hlatlej1 |
|- ( ( K e. HL /\ P e. A /\ v e. A ) -> P .<_ ( P .\/ v ) ) |
24 |
20 21 22 23
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> P .<_ ( P .\/ v ) ) |
25 |
|
simp32 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> z .<_ ( P .\/ v ) ) |
26 |
20
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> K e. Lat ) |
27 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
28 |
27 4
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
29 |
21 28
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> P e. ( Base ` K ) ) |
30 |
|
simp2l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> z e. A ) |
31 |
27 4
|
atbase |
|- ( z e. A -> z e. ( Base ` K ) ) |
32 |
30 31
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> z e. ( Base ` K ) ) |
33 |
27 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ v e. A ) -> ( P .\/ v ) e. ( Base ` K ) ) |
34 |
20 21 22 33
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( P .\/ v ) e. ( Base ` K ) ) |
35 |
27 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ z e. ( Base ` K ) /\ ( P .\/ v ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( P .\/ v ) /\ z .<_ ( P .\/ v ) ) <-> ( P .\/ z ) .<_ ( P .\/ v ) ) ) |
36 |
26 29 32 34 35
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( ( P .<_ ( P .\/ v ) /\ z .<_ ( P .\/ v ) ) <-> ( P .\/ z ) .<_ ( P .\/ v ) ) ) |
37 |
24 25 36
|
mpbi2and |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( P .\/ z ) .<_ ( P .\/ v ) ) |
38 |
27 4
|
atbase |
|- ( ( R ` F ) e. A -> ( R ` F ) e. ( Base ` K ) ) |
39 |
15 38
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. ( Base ` K ) ) |
40 |
27 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ z e. A ) -> ( P .\/ z ) e. ( Base ` K ) ) |
41 |
20 21 30 40
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( P .\/ z ) e. ( Base ` K ) ) |
42 |
27 1
|
lattr |
|- ( ( K e. Lat /\ ( ( R ` F ) e. ( Base ` K ) /\ ( P .\/ z ) e. ( Base ` K ) /\ ( P .\/ v ) e. ( Base ` K ) ) ) -> ( ( ( R ` F ) .<_ ( P .\/ z ) /\ ( P .\/ z ) .<_ ( P .\/ v ) ) -> ( R ` F ) .<_ ( P .\/ v ) ) ) |
43 |
26 39 41 34 42
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( ( ( R ` F ) .<_ ( P .\/ z ) /\ ( P .\/ z ) .<_ ( P .\/ v ) ) -> ( R ` F ) .<_ ( P .\/ v ) ) ) |
44 |
37 43
|
mpan2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( ( R ` F ) .<_ ( P .\/ z ) -> ( R ` F ) .<_ ( P .\/ v ) ) ) |
45 |
19 44
|
mtod |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( z e. A /\ F e. T ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. ( R ` F ) .<_ ( P .\/ z ) ) |