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Metamath Proof Explorer

Theorem cdlemg12e

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 )
cdlemg12e.z 
|- .0. = ( 0. ` K )
Assertion 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. )

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 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. )