Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg35.l |
|- .<_ = ( le ` K ) |
2 |
|
cdlemg35.j |
|- .\/ = ( join ` K ) |
3 |
|
cdlemg35.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdlemg35.a |
|- A = ( Atoms ` K ) |
5 |
|
cdlemg35.h |
|- H = ( LHyp ` K ) |
6 |
|
cdlemg35.t |
|- T = ( ( LTrn ` K ) ` W ) |
7 |
|
cdlemg35.r |
|- R = ( ( trL ` K ) ` W ) |
8 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> K e. HL ) |
9 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( K e. HL /\ W e. H ) ) |
10 |
|
simp21 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
11 |
|
simp22 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> F e. T ) |
12 |
|
simp31 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( F ` P ) =/= P ) |
13 |
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 ) |
14 |
9 10 11 12 13
|
syl112anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` F ) e. A ) |
15 |
|
simp23 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> G e. T ) |
16 |
|
simp32 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( G ` P ) =/= P ) |
17 |
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 ) |
18 |
9 10 15 16 17
|
syl112anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` G ) e. A ) |
19 |
|
simp33 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` F ) =/= ( R ` G ) ) |
20 |
1 2 4
|
hlsupr |
|- ( ( ( K e. HL /\ ( R ` F ) e. A /\ ( R ` G ) e. A ) /\ ( R ` F ) =/= ( R ` G ) ) -> E. v e. A ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) |
21 |
8 14 18 19 20
|
syl31anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> E. v e. A ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) |
22 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
23 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> K e. HL ) |
24 |
23
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> K e. Lat ) |
25 |
22 4
|
atbase |
|- ( v e. A -> v e. ( Base ` K ) ) |
26 |
25
|
3ad2ant2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v e. ( Base ` K ) ) |
27 |
|
simp11 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
28 |
|
simp122 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> F e. T ) |
29 |
22 5 6 7
|
trlcl |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) ) |
30 |
27 28 29
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( R ` F ) e. ( Base ` K ) ) |
31 |
|
simp123 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> G e. T ) |
32 |
22 5 6 7
|
trlcl |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) e. ( Base ` K ) ) |
33 |
27 31 32
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( R ` G ) e. ( Base ` K ) ) |
34 |
22 2
|
latjcl |
|- ( ( K e. Lat /\ ( R ` F ) e. ( Base ` K ) /\ ( R ` G ) e. ( Base ` K ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) e. ( Base ` K ) ) |
35 |
24 30 33 34
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) e. ( Base ` K ) ) |
36 |
|
simp11r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> W e. H ) |
37 |
22 5
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
38 |
36 37
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> W e. ( Base ` K ) ) |
39 |
|
simp33 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) |
40 |
1 5 6 7
|
trlle |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W ) |
41 |
27 28 40
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( R ` F ) .<_ W ) |
42 |
1 5 6 7
|
trlle |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) .<_ W ) |
43 |
27 31 42
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( R ` G ) .<_ W ) |
44 |
22 1 2
|
latjle12 |
|- ( ( K e. Lat /\ ( ( R ` F ) e. ( Base ` K ) /\ ( R ` G ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( R ` F ) .<_ W /\ ( R ` G ) .<_ W ) <-> ( ( R ` F ) .\/ ( R ` G ) ) .<_ W ) ) |
45 |
24 30 33 38 44
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( ( ( R ` F ) .<_ W /\ ( R ` G ) .<_ W ) <-> ( ( R ` F ) .\/ ( R ` G ) ) .<_ W ) ) |
46 |
41 43 45
|
mpbi2and |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) .<_ W ) |
47 |
22 1 24 26 35 38 39 46
|
lattrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v .<_ W ) |
48 |
|
simp31 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v =/= ( R ` F ) ) |
49 |
|
simp32 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v =/= ( R ` G ) ) |
50 |
47 48 49
|
jca32 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) |
51 |
50
|
3expia |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A ) -> ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) -> ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) ) |
52 |
51
|
reximdva |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( E. v e. A ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) -> E. v e. A ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) ) |
53 |
21 52
|
mpd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> E. v e. A ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) |