Metamath Proof Explorer


Theorem cdlemg19

Description: Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-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 cdlemg19
|- ( ( ( ( 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 ) )

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 /\ ( 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 ) )