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Theorem cdlemg19a 31554
Description: Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-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
cdlemg19a  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  =  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )
)
Distinct variable groups:    A, r    G, r    .\/ , r    .<_ , r    P, r    Q, r    W, r    F, r
Allowed substitution hints:    R( r)    T( r)    H( r)    K( r)    ./\ ( r)

Proof of Theorem cdlemg19a
StepHypRef Expression
1 simp11l 1069 . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  K  e.  HL )
2 hllat 30235 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  K  e.  Lat )
4 simp12l 1071 . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  P  e.  A )
5 simp11 988 . . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
6 simp21 991 . . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( F  e.  T  /\  G  e.  T
) )
7 cdlemg12.l . . . . . . 7  |-  .<_  =  ( le `  K )
8 cdlemg12.a . . . . . . 7  |-  A  =  ( Atoms `  K )
9 cdlemg12.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
10 cdlemg12.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
117, 8, 9, 10ltrncoat 31015 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  ( F `  ( G `  P ) )  e.  A )
125, 6, 4, 11syl3anc 1185 . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( F `  ( G `  P )
)  e.  A )
13 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
14 cdlemg12.j . . . . . 6  |-  .\/  =  ( join `  K )
1513, 14, 8hlatjcl 30238 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  ( G `
 P ) )  e.  A )  -> 
( P  .\/  ( F `  ( G `  P ) ) )  e.  ( Base `  K
) )
161, 4, 12, 15syl3anc 1185 . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( P  .\/  ( F `  ( G `  P ) ) )  e.  ( Base `  K
) )
17 simp13l 1073 . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  Q  e.  A )
187, 8, 9, 10ltrncoat 31015 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  Q  e.  A )  ->  ( F `  ( G `  Q ) )  e.  A )
195, 6, 17, 18syl3anc 1185 . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( F `  ( G `  Q )
)  e.  A )
2013, 14, 8hlatjcl 30238 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  ( G `
 Q ) )  e.  A )  -> 
( Q  .\/  ( F `  ( G `  Q ) ) )  e.  ( Base `  K
) )
211, 17, 19, 20syl3anc 1185 . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( Q  .\/  ( F `  ( G `  Q ) ) )  e.  ( Base `  K
) )
22 cdlemg12.m . . . . 5  |-  ./\  =  ( meet `  K )
2313, 7, 22latmle1 14510 . . . 4  |-  ( ( 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 ) ) ) )  .<_  ( P  .\/  ( F `
 ( G `  P ) ) ) )
243, 16, 21, 23syl3anc 1185 . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  .<_  ( P  .\/  ( F `
 ( G `  P ) ) ) )
25 cdlemg12b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
267, 14, 22, 8, 9, 10, 25cdlemg18 31553 . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  .<_  W )
277, 14, 22, 8, 9, 10, 25cdlemg18d 31552 . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  e.  A )
2813, 8atbase 30161 . . . . 5  |-  ( ( ( P  .\/  ( F `  ( G `  P ) ) ) 
./\  ( Q  .\/  ( F `  ( G `
 Q ) ) ) )  e.  A  ->  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  e.  ( Base `  K
) )
2927, 28syl 16 . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  e.  ( Base `  K
) )
30 simp11r 1070 . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  W  e.  H )
3113, 9lhpbase 30869 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3230, 31syl 16 . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  W  e.  ( Base `  K ) )
3313, 7, 22latlem12 14512 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  e.  ( Base `  K
)  /\  ( P  .\/  ( F `  ( G `  P )
) )  e.  (
Base `  K )  /\  W  e.  ( Base `  K ) ) )  ->  ( (
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  .<_  ( P  .\/  ( F `
 ( G `  P ) ) )  /\  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  .<_  W )  <->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q ) ) ) )  .<_  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )
) )
343, 29, 16, 32, 33syl13anc 1187 . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( ( ( P  .\/  ( F `
 ( G `  P ) ) ) 
./\  ( Q  .\/  ( F `  ( G `
 Q ) ) ) )  .<_  ( P 
.\/  ( F `  ( G `  P ) ) )  /\  (
( P  .\/  ( F `  ( G `  P ) ) ) 
./\  ( Q  .\/  ( F `  ( G `
 Q ) ) ) )  .<_  W )  <-> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  .<_  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )
) )
3524, 26, 34mpbi2and 889 . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  .<_  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )
)
36 hlatl 30232 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
371, 36syl 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  K  e.  AtLat )
38 simp12 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
39 simp13 990 . . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
40 simp21l 1075 . . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  F  e.  T )
41 simp21r 1076 . . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  G  e.  T )
42 simp32 995 . . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) )  =/=  ( P  .\/  Q ) )
437, 14, 22, 8, 9, 10cdlemg11a 31508 . . . . . 6  |-  ( ( ( 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 )
445, 38, 39, 40, 41, 42, 43syl123anc 1202 . . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( F `  ( G `  P )
)  =/=  P )
4544necomd 2689 . . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  P  =/=  ( F `  ( G `  P ) ) )
467, 14, 22, 8, 9lhpat 30914 . . . 4  |-  ( ( ( 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 )
475, 38, 12, 45, 46syl112anc 1189 . . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  e.  A )
487, 8atcmp 30183 . . 3  |-  ( ( K  e.  AtLat  /\  (
( P  .\/  ( F `  ( G `  P ) ) ) 
./\  ( Q  .\/  ( F `  ( G `
 Q ) ) ) )  e.  A  /\  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  e.  A )  ->  (
( ( 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 ) ) )
4937, 27, 47, 48syl3anc 1185 . 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( ( 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 ) ) )
5035, 49mpbid 203 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  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  =  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   meetcmee 14407   Latclat 14479   Atomscatm 30135   AtLatcal 30136   HLchlt 30222   LHypclh 30855   LTrncltrn 30972   trLctrl 31029
This theorem is referenced by:  cdlemg19  31555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-map 7023  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lvols 30371  df-lines 30372  df-psubsp 30374  df-pmap 30375  df-padd 30667  df-lhyp 30859  df-laut 30860  df-ldil 30975  df-ltrn 30976  df-trl 31030
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