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Theorem cdlemg13a 35664
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 )
Assertion
Ref Expression
cdlemg13a  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( P  .\/  ( F `  ( G `  P )
) )  =  ( ( G `  P
)  .\/  ( F `  ( G `  P
) ) ) )

Proof of Theorem cdlemg13a
StepHypRef Expression
1 simp11l 1107 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  K  e.  HL )
2 simp12l 1109 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  P  e.  A )
3 simp11 1026 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4 simp2r 1023 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  G  e.  T )
5 cdlemg12.l . . . . . . 7  |-  .<_  =  ( le `  K )
6 cdlemg12.a . . . . . . 7  |-  A  =  ( Atoms `  K )
7 cdlemg12.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
8 cdlemg12.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
95, 6, 7, 8ltrnat 35153 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  A
)  ->  ( G `  P )  e.  A
)
103, 4, 2, 9syl3anc 1228 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( G `  P )  e.  A
)
11 cdlemg12.j . . . . . 6  |-  .\/  =  ( join `  K )
125, 11, 6hlatlej1 34388 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( G `  P )  e.  A )  ->  P  .<_  ( P  .\/  ( G `  P ) ) )
131, 2, 10, 12syl3anc 1228 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  P  .<_  ( P  .\/  ( G `
 P ) ) )
14 simp32 1033 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( R `  F )  =  ( R `  G ) )
15 simp2l 1022 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  F  e.  T )
16 simp12 1027 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
175, 6, 7, 8ltrnel 35152 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
183, 4, 16, 17syl3anc 1228 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
19 cdlemg12.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
20 cdlemg12b.r . . . . . . . . 9  |-  R  =  ( ( trL `  K
) `  W )
215, 11, 19, 6, 7, 8, 20trlval2 35176 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )  ->  ( R `  F )  =  ( ( ( G `  P )  .\/  ( F `  ( G `  P ) ) ) 
./\  W ) )
223, 15, 18, 21syl3anc 1228 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( R `  F )  =  ( ( ( G `  P )  .\/  ( F `  ( G `  P ) ) ) 
./\  W ) )
235, 11, 19, 6, 7, 8, 20trlval2 35176 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  G )  =  ( ( P  .\/  ( G `  P )
)  ./\  W )
)
243, 4, 16, 23syl3anc 1228 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( R `  G )  =  ( ( P  .\/  ( G `  P )
)  ./\  W )
)
2514, 22, 243eqtr3d 2516 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( (
( G `  P
)  .\/  ( F `  ( G `  P
) ) )  ./\  W )  =  ( ( P  .\/  ( G `
 P ) ) 
./\  W ) )
2625oveq2d 6301 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( G `  P )  .\/  ( ( ( G `
 P )  .\/  ( F `  ( G `
 P ) ) )  ./\  W )
)  =  ( ( G `  P ) 
.\/  ( ( P 
.\/  ( G `  P ) )  ./\  W ) ) )
275, 6, 7, 8ltrncoat 35157 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  ( F `  ( G `  P ) )  e.  A )
283, 15, 4, 2, 27syl121anc 1233 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( F `  ( G `  P
) )  e.  A
)
29 eqid 2467 . . . . . . 7  |-  ( ( ( G `  P
)  .\/  ( F `  ( G `  P
) ) )  ./\  W )  =  ( ( ( G `  P
)  .\/  ( F `  ( G `  P
) ) )  ./\  W )
305, 11, 19, 6, 7, 29cdleme0cp 35227 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( G `  P )  e.  A  /\  -.  ( G `  P ) 
.<_  W )  /\  ( F `  ( G `  P ) )  e.  A ) )  -> 
( ( G `  P )  .\/  (
( ( G `  P )  .\/  ( F `  ( G `  P ) ) ) 
./\  W ) )  =  ( ( G `
 P )  .\/  ( F `  ( G `
 P ) ) ) )
313, 18, 28, 30syl12anc 1226 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( G `  P )  .\/  ( ( ( G `
 P )  .\/  ( F `  ( G `
 P ) ) )  ./\  W )
)  =  ( ( G `  P ) 
.\/  ( F `  ( G `  P ) ) ) )
32 eqid 2467 . . . . . . 7  |-  ( ( P  .\/  ( G `
 P ) ) 
./\  W )  =  ( ( P  .\/  ( G `  P ) )  ./\  W )
335, 11, 19, 6, 7, 32cdleme0cq 35228 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( ( G `  P )  e.  A  /\  -.  ( G `  P ) 
.<_  W ) ) )  ->  ( ( G `
 P )  .\/  ( ( P  .\/  ( G `  P ) )  ./\  W )
)  =  ( P 
.\/  ( G `  P ) ) )
343, 2, 18, 33syl12anc 1226 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( G `  P )  .\/  ( ( P  .\/  ( G `  P ) )  ./\  W )
)  =  ( P 
.\/  ( G `  P ) ) )
3526, 31, 343eqtr3rd 2517 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( P  .\/  ( G `  P
) )  =  ( ( G `  P
)  .\/  ( F `  ( G `  P
) ) ) )
3613, 35breqtrd 4471 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  P  .<_  ( ( G `  P
)  .\/  ( F `  ( G `  P
) ) ) )
375, 11, 6hlatlej2 34389 . . . 4  |-  ( ( K  e.  HL  /\  ( G `  P )  e.  A  /\  ( F `  ( G `  P ) )  e.  A )  ->  ( F `  ( G `  P ) )  .<_  ( ( G `  P )  .\/  ( F `  ( G `  P ) ) ) )
381, 10, 28, 37syl3anc 1228 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( F `  ( G `  P
) )  .<_  ( ( G `  P ) 
.\/  ( F `  ( G `  P ) ) ) )
39 hllat 34377 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
401, 39syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  K  e.  Lat )
41 eqid 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
4241, 6atbase 34303 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
432, 42syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  P  e.  ( Base `  K )
)
4441, 6atbase 34303 . . . . 5  |-  ( ( F `  ( G `
 P ) )  e.  A  ->  ( F `  ( G `  P ) )  e.  ( Base `  K
) )
4528, 44syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( F `  ( G `  P
) )  e.  (
Base `  K )
)
4641, 11, 6hlatjcl 34380 . . . . 5  |-  ( ( K  e.  HL  /\  ( G `  P )  e.  A  /\  ( F `  ( G `  P ) )  e.  A )  ->  (
( G `  P
)  .\/  ( F `  ( G `  P
) ) )  e.  ( Base `  K
) )
471, 10, 28, 46syl3anc 1228 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( G `  P )  .\/  ( F `  ( G `  P )
) )  e.  (
Base `  K )
)
4841, 5, 11latjle12 15552 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  ( F `  ( G `
 P ) )  e.  ( Base `  K
)  /\  ( ( G `  P )  .\/  ( F `  ( G `  P )
) )  e.  (
Base `  K )
) )  ->  (
( P  .<_  ( ( G `  P ) 
.\/  ( F `  ( G `  P ) ) )  /\  ( F `  ( G `  P ) )  .<_  ( ( G `  P )  .\/  ( F `  ( G `  P ) ) ) )  <->  ( P  .\/  ( F `  ( G `
 P ) ) )  .<_  ( ( G `  P )  .\/  ( F `  ( G `  P )
) ) ) )
4940, 43, 45, 47, 48syl13anc 1230 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( P  .<_  ( ( G `
 P )  .\/  ( F `  ( G `
 P ) ) )  /\  ( F `
 ( G `  P ) )  .<_  ( ( G `  P )  .\/  ( F `  ( G `  P ) ) ) )  <->  ( P  .\/  ( F `  ( G `
 P ) ) )  .<_  ( ( G `  P )  .\/  ( F `  ( G `  P )
) ) ) )
5036, 38, 49mpbi2and 919 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( P  .\/  ( F `  ( G `  P )
) )  .<_  ( ( G `  P ) 
.\/  ( F `  ( G `  P ) ) ) )
51 simp13 1028 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
52 simp33 1034 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) )
535, 11, 19, 6, 7, 8cdlemg11a 35650 . . . . 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 )
543, 16, 51, 15, 4, 52, 53syl123anc 1245 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( F `  ( G `  P
) )  =/=  P
)
5554necomd 2738 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  P  =/=  ( F `  ( G `
 P ) ) )
565, 11, 6ps-1 34490 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( F `  ( G `  P )
)  e.  A  /\  P  =/=  ( F `  ( G `  P ) ) )  /\  (
( G `  P
)  e.  A  /\  ( F `  ( G `
 P ) )  e.  A ) )  ->  ( ( P 
.\/  ( F `  ( G `  P ) ) )  .<_  ( ( G `  P ) 
.\/  ( F `  ( G `  P ) ) )  <->  ( P  .\/  ( F `  ( G `  P )
) )  =  ( ( G `  P
)  .\/  ( F `  ( G `  P
) ) ) ) )
571, 2, 28, 55, 10, 28, 56syl132anc 1246 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  .<_  ( ( G `  P ) 
.\/  ( F `  ( G `  P ) ) )  <->  ( P  .\/  ( F `  ( G `  P )
) )  =  ( ( G `  P
)  .\/  ( F `  ( G `  P
) ) ) ) )
5850, 57mpbid 210 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( R `
 F )  =  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( P  .\/  ( F `  ( G `  P )
) )  =  ( ( G `  P
)  .\/  ( F `  ( G `  P
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434   meetcmee 15435   Latclat 15535   Atomscatm 34277   HLchlt 34364   LHypclh 34997   LTrncltrn 35114   trLctrl 35171
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 6577  ax-riotaBAD 33973
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-undef 7003  df-map 7423  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-p1 15530  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-llines 34511  df-lplanes 34512  df-lvols 34513  df-lines 34514  df-psubsp 34516  df-pmap 34517  df-padd 34809  df-lhyp 35001  df-laut 35002  df-ldil 35117  df-ltrn 35118  df-trl 35172
This theorem is referenced by:  cdlemg13  35665
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