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Theorem cdlemg12g 35445
Description: TODO: FIX COMMENT TODO: Combine with cdlemg12f 35444. (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 1107 . . 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 34159 . . 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 34160 . . . 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 1109 . . . 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 1026 . . . . 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 1029 . . . . 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 1030 . . . . 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 34940 . . . . 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 1233 . . . 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 2467 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
17 cdlemg12.j . . . . 5  |-  .\/  =  ( join `  K )
1816, 17, 11hlatjcl 34163 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  ( G `
 P ) )  e.  A )  -> 
( P  .\/  ( F `  ( G `  P ) ) )  e.  ( Base `  K
) )
191, 6, 15, 18syl3anc 1228 . . 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 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 ) ) )  ->  Q  e.  A )
2110, 11, 12, 13ltrncoat 34940 . . . . 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 1233 . . . 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 34163 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  ( G `
 Q ) )  e.  A )  -> 
( Q  .\/  ( F `  ( G `  Q ) ) )  e.  ( Base `  K
) )
241, 20, 22, 23syl3anc 1228 . . 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 15535 . . 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 1228 . 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 1027 . . 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 1028 . . . 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 1034 . . . 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 35433 . . . . 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 2738 . . . 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 1245 . . 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 34839 . . 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 1232 . 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 34164 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  ( G `
 P ) )  e.  A )  -> 
( P  .\/  ( F `  ( G `  P ) ) )  =  ( ( F `
 ( G `  P ) )  .\/  P ) )
371, 6, 15, 36syl3anc 1228 . . . 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 34164 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  ( G `
 Q ) )  e.  A )  -> 
( Q  .\/  ( F `  ( G `  Q ) ) )  =  ( ( F `
 ( G `  Q ) )  .\/  Q ) )
391, 20, 22, 38syl3anc 1228 . . . 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 6300 . . 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 996 . . . 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 997 . . . 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 1119 . . . 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 1120 . . . 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 1033 . . . 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 2467 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
4810, 17, 25, 11, 12, 13, 46, 47cdlemg12e 35443 . . . 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 1240 . . 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 2760 . 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 35444 . 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 34091 . 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 1236 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   lecple 14558   joincjn 15427   meetcmee 15428   0.cp0 15520   Latclat 15528   OPcops 33969   Atomscatm 34060   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   trLctrl 34954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-undef 6999  df-map 7419  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955
This theorem is referenced by:  cdlemg12  35446
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