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 |
|
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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> K e. HL ) |
9 |
|
hlop |
|- ( K e. HL -> K e. OP ) |
10 |
8 9
|
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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> K e. OP ) |
11 |
8
|
hllatd |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> K e. Lat ) |
12 |
|
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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> P e. A ) |
13 |
|
simp11 |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) ) |
14 |
|
simp21 |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> F e. T ) |
15 |
|
simp22 |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> G e. T ) |
16 |
1 4 5 6
|
ltrncoat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> ( F ` ( G ` P ) ) e. A ) |
17 |
13 14 15 12 16
|
syl121anc |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( F ` ( G ` P ) ) e. A ) |
18 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
19 |
18 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ ( F ` ( G ` P ) ) e. A ) -> ( P .\/ ( F ` ( G ` P ) ) ) e. ( Base ` K ) ) |
20 |
8 12 17 19
|
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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( P .\/ ( F ` ( G ` P ) ) ) e. ( Base ` K ) ) |
21 |
|
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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> Q e. A ) |
22 |
1 4 5 6
|
ltrncoat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ Q e. A ) -> ( F ` ( G ` Q ) ) e. A ) |
23 |
13 14 15 21 22
|
syl121anc |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( F ` ( G ` Q ) ) e. A ) |
24 |
18 2 4
|
hlatjcl |
|- ( ( K e. HL /\ Q e. A /\ ( F ` ( G ` Q ) ) e. A ) -> ( Q .\/ ( F ` ( G ` Q ) ) ) e. ( Base ` K ) ) |
25 |
8 21 23 24
|
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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( Q .\/ ( F ` ( G ` Q ) ) ) e. ( Base ` K ) ) |
26 |
18 3
|
latmcl |
|- ( ( K e. Lat /\ ( P .\/ ( F ` ( G ` P ) ) ) e. ( Base ` K ) /\ ( Q .\/ ( F ` ( G ` Q ) ) ) e. ( Base ` K ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) e. ( Base ` K ) ) |
27 |
11 20 25 26
|
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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) e. ( Base ` K ) ) |
28 |
|
simp12 |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
29 |
|
simp13 |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
30 |
|
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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) |
31 |
1 2 3 4 5 6
|
cdlemg11a |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( F ` ( G ` P ) ) =/= P ) |
32 |
31
|
necomd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> P =/= ( F ` ( G ` P ) ) ) |
33 |
13 28 29 14 15 30 32
|
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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> P =/= ( F ` ( G ` P ) ) ) |
34 |
1 2 3 4 5
|
lhpat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` ( G ` P ) ) e. A /\ P =/= ( F ` ( G ` P ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) e. A ) |
35 |
13 28 17 33 34
|
syl112anc |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) e. A ) |
36 |
2 4
|
hlatjcom |
|- ( ( K e. HL /\ P e. A /\ ( F ` ( G ` P ) ) e. A ) -> ( P .\/ ( F ` ( G ` P ) ) ) = ( ( F ` ( G ` P ) ) .\/ P ) ) |
37 |
8 12 17 36
|
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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( P .\/ ( F ` ( G ` P ) ) ) = ( ( F ` ( G ` P ) ) .\/ P ) ) |
38 |
2 4
|
hlatjcom |
|- ( ( K e. HL /\ Q e. A /\ ( F ` ( G ` Q ) ) e. A ) -> ( Q .\/ ( F ` ( G ` Q ) ) ) = ( ( F ` ( G ` Q ) ) .\/ Q ) ) |
39 |
8 21 23 38
|
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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( Q .\/ ( F ` ( G ` Q ) ) ) = ( ( F ` ( G ` Q ) ) .\/ Q ) ) |
40 |
37 39
|
oveq12d |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) = ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) ) |
41 |
|
simp1 |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) |
42 |
|
simp2 |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( F e. T /\ G e. T /\ P =/= Q ) ) |
43 |
|
simp31l |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> -. ( R ` F ) .<_ ( P .\/ Q ) ) |
44 |
|
simp31r |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> -. ( R ` G ) .<_ ( P .\/ Q ) ) |
45 |
|
simp32 |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( R ` F ) =/= ( R ` G ) ) |
46 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
47 |
1 2 3 4 5 6 7 46
|
cdlemg12e |
|- ( ( ( ( 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 ) ) |
48 |
41 42 43 44 45 47
|
syl113anc |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( ( F ` ( G ` P ) ) .\/ P ) ./\ ( ( F ` ( G ` Q ) ) .\/ Q ) ) =/= ( 0. ` K ) ) |
49 |
40 48
|
eqnetrd |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) =/= ( 0. ` K ) ) |
50 |
1 2 3 4 5 6 7
|
cdlemg12f |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) .<_ ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) ) |
51 |
18 1 46 4
|
leat2 |
|- ( ( ( K e. OP /\ ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) e. ( Base ` K ) /\ ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) e. A ) /\ ( ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) =/= ( 0. ` K ) /\ ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) .<_ ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) = ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) ) |
52 |
10 27 35 49 50 51
|
syl32anc |
|- ( ( ( ( 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) = ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) ) |