Metamath Proof Explorer

Theorem cdlemg12

Description: TODO: FIX COMMENT. (Contributed by NM, 6-May-2013)

Ref Expression
Hypotheses cdlemg12.l 
|- .<_ = ( le ` K )
cdlemg12.j 
|- .\/ = ( join ` K )
cdlemg12.m 
|- ./\ = ( meet ` K )
cdlemg12.a 
|- A = ( Atoms ` K )
cdlemg12.h 
|- H = ( LHyp ` K )
cdlemg12.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg12b.r 
|- R = ( ( trL ` K ) ` W )
Assertion cdlemg12 
|- ( ( ( ( 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 ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )

Proof

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 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 )
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 ) /\ ( ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) /\ ( R ` F ) =/= ( R ` G ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> 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 ) /\ ( ( -. ( 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 ) )
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 ) /\ ( ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) /\ ( R ` F ) =/= ( R ` G ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> F e. T )
13 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 )
14 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 )
15 11 12 13 10 14 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 )
16 eqid 
 |-  ( Base ` K ) = ( Base ` K )
17 16 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ ( F ` ( G ` P ) ) e. A ) -> ( P .\/ ( F ` ( G ` P ) ) ) e. ( Base ` K ) )
18 8 10 15 17 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 ) )
19 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 )
20 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 )
21 11 12 13 19 20 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 )
22 16 2 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ ( F ` ( G ` Q ) ) e. A ) -> ( Q .\/ ( F ` ( G ` Q ) ) ) e. ( Base ` K ) )
23 8 19 21 22 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 ) )
24 16 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 ) ) ) ) )
25 9 18 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 ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ ( P .\/ ( F ` ( G ` P ) ) ) ) )
26 1 2 3 4 5 6 7 cdlemg12g 
 |-  ( ( ( ( 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 ) )
27 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 ) )
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 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 ) /\ ( ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) /\ ( R ` F ) =/= ( R ` G ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> P =/= Q )
30 29 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 ) /\ ( ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) /\ ( R ` F ) =/= ( R ` G ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> Q =/= P )
31 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 ) )
32 2 4 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
33 8 10 19 32 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 .\/ Q ) = ( Q .\/ P ) )
34 33 breq2d 
 |-  ( ( ( ( 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 ) <-> ( R ` F ) .<_ ( Q .\/ P ) ) )
35 31 34 mtbid 
 |-  ( ( ( ( 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 ) .<_ ( Q .\/ P ) )
36 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 ) )
37 33 breq2d 
 |-  ( ( ( ( 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 ) <-> ( R ` G ) .<_ ( Q .\/ P ) ) )
38 36 37 mtbid 
 |-  ( ( ( ( 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 ) .<_ ( Q .\/ P ) )
39 35 38 jca 
 |-  ( ( ( ( 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 ) .<_ ( Q .\/ P ) /\ -. ( R ` G ) .<_ ( Q .\/ P ) ) )
40 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 ) )
41 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 ) )
42 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 ) ) ) )
43 8 15 21 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 ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) = ( ( F ` ( G ` Q ) ) .\/ ( F ` ( G ` P ) ) ) )
44 41 43 33 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 ) /\ ( ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) /\ ( R ` F ) =/= ( R ` G ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` ( G ` Q ) ) .\/ ( F ` ( G ` P ) ) ) =/= ( Q .\/ P ) )
45 1 2 3 4 5 6 7 cdlemg12g 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( F e. T /\ G e. T /\ Q =/= P ) /\ ( ( -. ( R ` F ) .<_ ( Q .\/ P ) /\ -. ( R ` G ) .<_ ( Q .\/ P ) ) /\ ( R ` F ) =/= ( R ` G ) /\ ( ( F ` ( G ` Q ) ) .\/ ( F ` ( G ` P ) ) ) =/= ( Q .\/ P ) ) ) -> ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ ( P .\/ ( F ` ( G ` P ) ) ) ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
46 11 27 28 12 13 30 39 40 44 45 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 ) /\ ( ( -. ( R ` F ) .<_ ( P .\/ Q ) /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) /\ ( R ` F ) =/= ( R ` G ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ ( P .\/ ( F ` ( G ` P ) ) ) ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
47 25 26 46 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 ) /\ ( ( -. ( 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 ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )