Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg12e Structured version   Unicode version

Theorem cdlemg12e 33630
Description: TODO: FIX COMMENT (Contributed by NM, 6-May-2013.)
Hypotheses
Ref Expression
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
Ref Expression
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 of Theorem cdlemg12e
StepHypRef Expression
1 simp33 1033 . 2  |-  ( ( ( ( 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
) )
2 simpl1 998 . . . . . . . 8  |-  ( ( ( ( ( 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 ) ) )
3 simpl21 1073 . . . . . . . 8  |-  ( ( ( ( ( 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 )
4 simpl22 1074 . . . . . . . 8  |-  ( ( ( ( ( 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 )
5 simpl23 1075 . . . . . . . 8  |-  ( ( ( ( ( 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 )
6 simpl31 1076 . . . . . . . 8  |-  ( ( ( ( ( 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
) )
7 simpl32 1077 . . . . . . . 8  |-  ( ( ( ( ( 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
) )
8 cdlemg12.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
9 cdlemg12.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
10 cdlemg12.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
11 cdlemg12.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
12 cdlemg12.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
13 cdlemg12.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
14 cdlemg12b.r . . . . . . . . 9  |-  R  =  ( ( trL `  K
) `  W )
158, 9, 10, 11, 12, 13, 14cdlemg12d 33629 . . . . . . . 8  |-  ( ( ( ( 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 ) ) ) )
162, 3, 4, 5, 6, 7, 15syl123anc 1245 . . . . . . 7  |-  ( ( ( ( ( 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 ) ) ) )
17 simpr 459 . . . . . . . . 9  |-  ( ( ( ( ( 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.  )
1817oveq2d 6248 . . . . . . . 8  |-  ( ( ( ( ( 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.  ) )
19 simp11l 1106 . . . . . . . . . . 11  |-  ( ( ( ( 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 )
2019adantr 463 . . . . . . . . . 10  |-  ( ( ( ( ( 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 )
21 hlol 32343 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OL )
2220, 21syl 17 . . . . . . . . 9  |-  ( ( ( ( ( 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 )
23 simpl11 1070 . . . . . . . . . 10  |-  ( ( ( ( ( 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 ) )
24 eqid 2400 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
2524, 12, 13, 14trlcl 33146 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
2623, 3, 25syl2anc 659 . . . . . . . . 9  |-  ( ( ( ( ( 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 )
)
27 cdlemg12e.z . . . . . . . . . 10  |-  .0.  =  ( 0. `  K )
2824, 9, 27olj01 32207 . . . . . . . . 9  |-  ( ( K  e.  OL  /\  ( R `  F )  e.  ( Base `  K
) )  ->  (
( R `  F
)  .\/  .0.  )  =  ( R `  F ) )
2922, 26, 28syl2anc 659 . . . . . . . 8  |-  ( ( ( ( ( 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 ) )
3018, 29eqtrd 2441 . . . . . . 7  |-  ( ( ( ( ( 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
) )
3116, 30breqtrd 4416 . . . . . 6  |-  ( ( ( ( ( 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 32342 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  AtLat )
3320, 32syl 17 . . . . . . 7  |-  ( ( ( ( ( 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 32344 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
3520, 34syl 17 . . . . . . . . 9  |-  ( ( ( ( ( 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 )
3624, 12, 13, 14trlcl 33146 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  e.  (
Base `  K )
)
3723, 4, 36syl2anc 659 . . . . . . . . 9  |-  ( ( ( ( ( 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 1108 . . . . . . . . . . 11  |-  ( ( ( ( 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 )
3938adantr 463 . . . . . . . . . 10  |-  ( ( ( ( ( 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 1110 . . . . . . . . . . 11  |-  ( ( ( ( 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 )
4140adantr 463 . . . . . . . . . 10  |-  ( ( ( ( ( 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 )
4224, 9, 11hlatjcl 32348 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
4320, 39, 41, 42syl3anc 1228 . . . . . . . . 9  |-  ( ( ( ( ( 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 )
)
4424, 8, 27opnlen0 32170 . . . . . . . . 9  |-  ( ( ( K  e.  OP  /\  ( R `  G
)  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  /\  -.  ( R `  G ) 
.<_  ( P  .\/  Q
) )  ->  ( R `  G )  =/=  .0.  )
4535, 37, 43, 7, 44syl31anc 1231 . . . . . . . 8  |-  ( ( ( ( ( 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 1107 . . . . . . . . . 10  |-  ( ( ( ( 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 )
4746adantr 463 . . . . . . . . 9  |-  ( ( ( ( ( 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 )
4827, 11, 12, 13, 14trlatn0 33154 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( ( R `  G )  e.  A  <->  ( R `  G )  =/=  .0.  ) )
4920, 47, 4, 48syl21anc 1227 . . . . . . . 8  |-  ( ( ( ( ( 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.  ) )
5045, 49mpbird 232 . . . . . . 7  |-  ( ( ( ( ( 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
)
5124, 8, 27opnlen0 32170 . . . . . . . . 9  |-  ( ( ( K  e.  OP  /\  ( R `  F
)  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  /\  -.  ( R `  F ) 
.<_  ( P  .\/  Q
) )  ->  ( R `  F )  =/=  .0.  )
5235, 26, 43, 6, 51syl31anc 1231 . . . . . . . 8  |-  ( ( ( ( ( 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.  )
5327, 11, 12, 13, 14trlatn0 33154 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  <->  ( R `  F )  =/=  .0.  ) )
5420, 47, 3, 53syl21anc 1227 . . . . . . . 8  |-  ( ( ( ( ( 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.  ) )
5552, 54mpbird 232 . . . . . . 7  |-  ( ( ( ( ( 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
)
568, 11atcmp 32293 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  ( R `  G )  e.  A  /\  ( R `  F )  e.  A )  ->  (
( R `  G
)  .<_  ( R `  F )  <->  ( R `  G )  =  ( R `  F ) ) )
5733, 50, 55, 56syl3anc 1228 . . . . . 6  |-  ( ( ( ( ( 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 ) ) )
5831, 57mpbid 210 . . . . 5  |-  ( ( ( ( ( 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 ) )
5958eqcomd 2408 . . . 4  |-  ( ( ( ( ( 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 ) )
6059ex 432 . . 3  |-  ( ( ( ( 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 ) ) )
6160necon3d 2625 . 2  |-  ( ( ( ( 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.  ) )
621, 61mpd 15 1  |-  ( ( ( ( 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.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   meetcmee 15788   0.cp0 15881   OPcops 32154   OLcol 32156   Atomscatm 32245   AtLatcal 32246   HLchlt 32332   LHypclh 32965   LTrncltrn 33082   trLctrl 33140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-riotaBAD 31941
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737  df-undef 6957  df-map 7377  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-p1 15884  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333  df-llines 32479  df-lplanes 32480  df-lvols 32481  df-lines 32482  df-psubsp 32484  df-pmap 32485  df-padd 32777  df-lhyp 32969  df-laut 32970  df-ldil 33085  df-ltrn 33086  df-trl 33141
This theorem is referenced by:  cdlemg12g  33632
  Copyright terms: Public domain W3C validator