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 |
|
cdlemg12e.z |
|- .0. = ( 0. ` K ) |
9 |
|
simp33 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` F ) =/= ( R ` G ) ) |
10 |
|
simpl1 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) |
11 |
|
simpl21 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> F e. T ) |
12 |
|
simpl22 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> G e. T ) |
13 |
|
simpl23 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> P =/= Q ) |
14 |
|
simpl31 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> -. ( R ` F ) .<_ ( P .\/ Q ) ) |
15 |
|
simpl32 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> -. ( R ` G ) .<_ ( P .\/ Q ) ) |
16 |
1 2 3 4 5 6 7
|
cdlemg12d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) /\ ( P =/= Q /\ -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( R ` G ) .<_ ( ( R ` F ) .\/ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) ) ) |
17 |
10 11 12 13 14 15 16
|
syl123anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( R ` G ) .<_ ( ( R ` F ) .\/ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) ) ) |
18 |
|
simpr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) |
19 |
18
|
oveq2d |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( ( R ` F ) .\/ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) ) = ( ( R ` F ) .\/ .0. ) ) |
20 |
|
simp11l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> K e. HL ) |
21 |
20
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> K e. HL ) |
22 |
|
hlol |
|- ( K e. HL -> K e. OL ) |
23 |
21 22
|
syl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> K e. OL ) |
24 |
|
simpl11 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( K e. HL /\ W e. H ) ) |
25 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
26 |
25 5 6 7
|
trlcl |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) ) |
27 |
24 11 26
|
syl2anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( R ` F ) e. ( Base ` K ) ) |
28 |
25 2 8
|
olj01 |
|- ( ( K e. OL /\ ( R ` F ) e. ( Base ` K ) ) -> ( ( R ` F ) .\/ .0. ) = ( R ` F ) ) |
29 |
23 27 28
|
syl2anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( ( R ` F ) .\/ .0. ) = ( R ` F ) ) |
30 |
19 29
|
eqtrd |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( ( R ` F ) .\/ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) ) = ( R ` F ) ) |
31 |
17 30
|
breqtrd |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( R ` G ) .<_ ( R ` F ) ) |
32 |
|
hlatl |
|- ( K e. HL -> K e. AtLat ) |
33 |
21 32
|
syl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> K e. AtLat ) |
34 |
|
hlop |
|- ( K e. HL -> K e. OP ) |
35 |
21 34
|
syl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> K e. OP ) |
36 |
25 5 6 7
|
trlcl |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) e. ( Base ` K ) ) |
37 |
24 12 36
|
syl2anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( R ` G ) e. ( Base ` K ) ) |
38 |
|
simp12l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> P e. A ) |
39 |
38
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> P e. A ) |
40 |
|
simp13l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> Q e. A ) |
41 |
40
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> Q e. A ) |
42 |
25 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
43 |
21 39 41 42
|
syl3anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( P .\/ Q ) e. ( Base ` K ) ) |
44 |
25 1 8
|
opnlen0 |
|- ( ( ( K e. OP /\ ( R ` G ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) -> ( R ` G ) =/= .0. ) |
45 |
35 37 43 15 44
|
syl31anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( R ` G ) =/= .0. ) |
46 |
|
simp11r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> W e. H ) |
47 |
46
|
adantr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> W e. H ) |
48 |
8 4 5 6 7
|
trlatn0 |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( ( R ` G ) e. A <-> ( R ` G ) =/= .0. ) ) |
49 |
21 47 12 48
|
syl21anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( ( R ` G ) e. A <-> ( R ` G ) =/= .0. ) ) |
50 |
45 49
|
mpbird |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( R ` G ) e. A ) |
51 |
25 1 8
|
opnlen0 |
|- ( ( ( K e. OP /\ ( R ` F ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) /\ -. ( R ` F ) .<_ ( P .\/ Q ) ) -> ( R ` F ) =/= .0. ) |
52 |
35 27 43 14 51
|
syl31anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( R ` F ) =/= .0. ) |
53 |
8 4 5 6 7
|
trlatn0 |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( ( R ` F ) e. A <-> ( R ` F ) =/= .0. ) ) |
54 |
21 47 11 53
|
syl21anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( ( R ` F ) e. A <-> ( R ` F ) =/= .0. ) ) |
55 |
52 54
|
mpbird |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( R ` F ) e. A ) |
56 |
1 4
|
atcmp |
|- ( ( K e. AtLat /\ ( R ` G ) e. A /\ ( R ` F ) e. A ) -> ( ( R ` G ) .<_ ( R ` F ) <-> ( R ` G ) = ( R ` F ) ) ) |
57 |
33 50 55 56
|
syl3anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( ( R ` G ) .<_ ( R ` F ) <-> ( R ` G ) = ( R ` F ) ) ) |
58 |
31 57
|
mpbid |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( R ` G ) = ( R ` F ) ) |
59 |
58
|
eqcomd |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. ) -> ( R ` F ) = ( R ` G ) ) |
60 |
59
|
ex |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) = .0. -> ( R ` F ) = ( R ` G ) ) ) |
61 |
60
|
necon3d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( R ` F ) =/= ( R ` G ) -> ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) =/= .0. ) ) |
62 |
9 61
|
mpd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) =/= .0. ) |