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Theorem cdlemg12g 34602
Description: TODO: FIX COMMENT TODO: Combine with cdlemg12f 34601. (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 )
Assertion
Ref Expression
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 )
)

Proof of Theorem cdlemg12g
StepHypRef Expression
1 simp11l 1099 . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  K  e.  HL )
2 hlop 33316 . . 3  |-  ( K  e.  HL  ->  K  e.  OP )
31, 2syl 16 . 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
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  K  e.  OP )
4 hllat 33317 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
51, 4syl 16 . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  K  e.  Lat )
6 simp12l 1101 . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  P  e.  A )
7 simp11 1018 . . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
8 simp21 1021 . . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  F  e.  T )
9 simp22 1022 . . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  G  e.  T )
10 cdlemg12.l . . . . . 6  |-  .<_  =  ( le `  K )
11 cdlemg12.a . . . . . 6  |-  A  =  ( Atoms `  K )
12 cdlemg12.h . . . . . 6  |-  H  =  ( LHyp `  K
)
13 cdlemg12.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
1410, 11, 12, 13ltrncoat 34097 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  ( F `  ( G `  P ) )  e.  A )
157, 8, 9, 6, 14syl121anc 1224 . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( F `  ( G `  P
) )  e.  A
)
16 eqid 2451 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
17 cdlemg12.j . . . . 5  |-  .\/  =  ( join `  K )
1816, 17, 11hlatjcl 33320 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  ( G `
 P ) )  e.  A )  -> 
( P  .\/  ( F `  ( G `  P ) ) )  e.  ( Base `  K
) )
191, 6, 15, 18syl3anc 1219 . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( P  .\/  ( F `  ( G `  P )
) )  e.  (
Base `  K )
)
20 simp13l 1103 . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  Q  e.  A )
2110, 11, 12, 13ltrncoat 34097 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  Q  e.  A )  ->  ( F `  ( G `  Q ) )  e.  A )
227, 8, 9, 20, 21syl121anc 1224 . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( F `  ( G `  Q
) )  e.  A
)
2316, 17, 11hlatjcl 33320 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  ( G `
 Q ) )  e.  A )  -> 
( Q  .\/  ( F `  ( G `  Q ) ) )  e.  ( Base `  K
) )
241, 20, 22, 23syl3anc 1219 . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( Q  .\/  ( F `  ( G `  Q )
) )  e.  (
Base `  K )
)
25 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
2616, 25latmcl 15333 . . 3  |-  ( ( 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 ) ) ) )  e.  ( Base `  K
) )
275, 19, 24, 26syl3anc 1219 . 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
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  e.  ( Base `  K
) )
28 simp12 1019 . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
29 simp13 1020 . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
30 simp33 1026 . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) )
3110, 17, 25, 11, 12, 13cdlemg11a 34590 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
) ) )  -> 
( F `  ( G `  P )
)  =/=  P )
3231necomd 2719 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
) ) )  ->  P  =/=  ( F `  ( G `  P ) ) )
337, 28, 29, 8, 9, 30, 32syl123anc 1236 . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  P  =/=  ( F `  ( G `
 P ) ) )
3410, 17, 25, 11, 12lhpat 33996 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  ( G `  P )
)  e.  A  /\  P  =/=  ( F `  ( G `  P ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  e.  A )
357, 28, 15, 33, 34syl112anc 1223 . 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
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W
)  e.  A )
3617, 11hlatjcom 33321 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  ( G `
 P ) )  e.  A )  -> 
( P  .\/  ( F `  ( G `  P ) ) )  =  ( ( F `
 ( G `  P ) )  .\/  P ) )
371, 6, 15, 36syl3anc 1219 . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( P  .\/  ( F `  ( G `  P )
) )  =  ( ( F `  ( G `  P )
)  .\/  P )
)
3817, 11hlatjcom 33321 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  ( G `
 Q ) )  e.  A )  -> 
( Q  .\/  ( F `  ( G `  Q ) ) )  =  ( ( F `
 ( G `  Q ) )  .\/  Q ) )
391, 20, 22, 38syl3anc 1219 . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( Q  .\/  ( F `  ( G `  Q )
) )  =  ( ( F `  ( G `  Q )
)  .\/  Q )
)
4037, 39oveq12d 6211 . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  =  ( ( ( F `
 ( G `  P ) )  .\/  P )  ./\  ( ( F `  ( G `  Q ) )  .\/  Q ) ) )
41 simp1 988 . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
42 simp2 989 . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( F  e.  T  /\  G  e.  T  /\  P  =/= 
Q ) )
43 simp31l 1111 . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  -.  ( R `  F )  .<_  ( P  .\/  Q
) )
44 simp31r 1112 . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  -.  ( R `  G )  .<_  ( P  .\/  Q
) )
45 simp32 1025 . . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( R `  F )  =/=  ( R `  G )
)
46 cdlemg12b.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
47 eqid 2451 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
4810, 17, 25, 11, 12, 13, 46, 47cdlemg12e 34600 . . . 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. `  K ) )
4941, 42, 43, 44, 45, 48syl113anc 1231 . . 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( (
( F `  ( G `  P )
)  .\/  P )  ./\  ( ( F `  ( G `  Q ) )  .\/  Q ) )  =/=  ( 0.
`  K ) )
5040, 49eqnetrd 2741 . 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
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  =/=  ( 0. `  K
) )
5110, 17, 25, 11, 12, 13, 46cdlemg12f 34601 . 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
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  .<_  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )
)
5216, 10, 47, 11leat2 33248 . 2  |-  ( ( ( K  e.  OP  /\  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  e.  ( Base `  K
)  /\  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W
)  e.  A )  /\  ( ( ( P  .\/  ( F `
 ( G `  P ) ) ) 
./\  ( Q  .\/  ( F `  ( G `
 Q ) ) ) )  =/=  ( 0. `  K )  /\  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  .<_  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )
) )  ->  (
( P  .\/  ( F `  ( G `  P ) ) ) 
./\  ( Q  .\/  ( F `  ( G `
 Q ) ) ) )  =  ( ( P  .\/  ( F `  ( G `  P ) ) ) 
./\  W ) )
533, 27, 35, 50, 51, 52syl32anc 1227 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 ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  =  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   Basecbs 14285   lecple 14356   joincjn 15225   meetcmee 15226   0.cp0 15318   Latclat 15326   OPcops 33126   Atomscatm 33217   HLchlt 33304   LHypclh 33937   LTrncltrn 34054   trLctrl 34111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-riotaBAD 32913
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-undef 6895  df-map 7319  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-p1 15321  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-llines 33451  df-lplanes 33452  df-lvols 33453  df-lines 33454  df-psubsp 33456  df-pmap 33457  df-padd 33749  df-lhyp 33941  df-laut 33942  df-ldil 34057  df-ltrn 34058  df-trl 34112
This theorem is referenced by:  cdlemg12  34603
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