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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> K e. HL ) |
9 |
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> K e. Lat ) |
10 |
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> P e. A ) |
11 |
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
12 |
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( F e. T /\ G e. T ) ) |
13 |
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 ) |
14 |
11 12 10 13
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( F ` ( G ` P ) ) e. A ) |
15 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
16 |
15 2 4
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ ( F ` ( G ` P ) ) e. A ) -> ( P .\/ ( F ` ( G ` P ) ) ) e. ( Base ` K ) ) |
17 |
8 10 14 16
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( P .\/ ( F ` ( G ` P ) ) ) e. ( Base ` K ) ) |
18 |
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> Q e. A ) |
19 |
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 ) |
20 |
11 12 18 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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( F ` ( G ` Q ) ) e. A ) |
21 |
15 2 4
|
hlatjcl |
|- ( ( K e. HL /\ Q e. A /\ ( F ` ( G ` Q ) ) e. A ) -> ( Q .\/ ( F ` ( G ` Q ) ) ) e. ( Base ` K ) ) |
22 |
8 18 20 21
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( Q .\/ ( F ` ( G ` Q ) ) ) e. ( Base ` K ) ) |
23 |
15 3
|
latmcom |
|- ( ( 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 ) ) ) ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ ( P .\/ ( F ` ( G ` P ) ) ) ) ) |
24 |
9 17 22 23
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ ( P .\/ ( F ` ( G ` P ) ) ) ) ) |
25 |
1 2 3 4 5 6 7
|
cdlemg19a |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) = ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) ) |
26 |
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
27 |
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
28 |
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> P =/= Q ) |
29 |
28
|
necomd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> Q =/= P ) |
30 |
|
simp21r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> G e. T ) |
31 |
|
simp23 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` P ) =/= P ) |
32 |
1 4 5 6
|
ltrnatneq |
|- ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( G ` P ) =/= P ) -> ( G ` Q ) =/= Q ) |
33 |
11 30 27 26 31 32
|
syl131anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` Q ) =/= Q ) |
34 |
|
simp31 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( R ` G ) .<_ ( P .\/ Q ) ) |
35 |
2 4
|
hlatjcom |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) ) |
36 |
8 10 18 35
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( P .\/ Q ) = ( Q .\/ P ) ) |
37 |
34 36
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( R ` G ) .<_ ( Q .\/ P ) ) |
38 |
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) |
39 |
2 4
|
hlatjcom |
|- ( ( K e. HL /\ ( F ` ( G ` P ) ) e. A /\ ( F ` ( G ` Q ) ) e. A ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) = ( ( F ` ( G ` Q ) ) .\/ ( F ` ( G ` P ) ) ) ) |
40 |
8 14 20 39
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) = ( ( F ` ( G ` Q ) ) .\/ ( F ` ( G ` P ) ) ) ) |
41 |
38 40 36
|
3netr3d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( F ` ( G ` Q ) ) .\/ ( F ` ( G ` P ) ) ) =/= ( Q .\/ P ) ) |
42 |
|
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 /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) |
43 |
|
eqcom |
|- ( ( P .\/ r ) = ( Q .\/ r ) <-> ( Q .\/ r ) = ( P .\/ r ) ) |
44 |
43
|
anbi2i |
|- ( ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) <-> ( -. r .<_ W /\ ( Q .\/ r ) = ( P .\/ r ) ) ) |
45 |
44
|
rexbii |
|- ( E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) <-> E. r e. A ( -. r .<_ W /\ ( Q .\/ r ) = ( P .\/ r ) ) ) |
46 |
42 45
|
sylnib |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> -. E. r e. A ( -. r .<_ W /\ ( Q .\/ r ) = ( P .\/ r ) ) ) |
47 |
1 2 3 4 5 6 7
|
cdlemg19a |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ Q =/= P /\ ( G ` Q ) =/= Q ) /\ ( ( R ` G ) .<_ ( Q .\/ P ) /\ ( ( F ` ( G ` Q ) ) .\/ ( F ` ( G ` P ) ) ) =/= ( Q .\/ P ) /\ -. E. r e. A ( -. r .<_ W /\ ( Q .\/ r ) = ( P .\/ r ) ) ) ) -> ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ ( P .\/ ( F ` ( G ` P ) ) ) ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) ) |
48 |
11 26 27 12 29 33 37 41 46 47
|
syl333anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ ( P .\/ ( F ` ( G ` P ) ) ) ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) ) |
49 |
24 25 48
|
3eqtr3d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( F e. T /\ G e. T ) /\ P =/= Q /\ ( G ` P ) =/= P ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) ) |