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Theorem cdlemg19a 30924
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 1066 . . . . 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 29605 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 15 . . . 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 1068 . . . . 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 985 . . . . . 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 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 ) ) ) )  -> 
( 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 30385 . . . . . 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 1182 . . . . 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 2358 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
14 cdlemg12.j . . . . . 6  |-  .\/  =  ( join `  K )
1513, 14, 8hlatjcl 29608 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  ( G `
 P ) )  e.  A )  -> 
( P  .\/  ( F `  ( G `  P ) ) )  e.  ( Base `  K
) )
161, 4, 12, 15syl3anc 1182 . . . 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 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 ) ) ) )  ->  Q  e.  A )
187, 8, 9, 10ltrncoat 30385 . . . . . 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 1182 . . . . 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 29608 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( F `  ( G `
 Q ) )  e.  A )  -> 
( Q  .\/  ( F `  ( G `  Q ) ) )  e.  ( Base `  K
) )
211, 17, 19, 20syl3anc 1182 . . . 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 14275 . . . 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 1182 . . 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 30923 . . 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 30922 . . . . 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 29531 . . . . 5  |-  ( ( ( P  .\/  ( F `  ( G `  P ) ) ) 
./\  ( Q  .\/  ( F `  ( G `
 Q ) ) ) )  e.  A  ->  ( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  e.  ( Base `  K
) )
2927, 28syl 15 . . . 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 1067 . . . . 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 30239 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3230, 31syl 15 . . . 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 14277 . . . 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 1184 . . 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 887 . 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 29602 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
371, 36syl 15 . . 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 986 . . . 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 987 . . . . . 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 1072 . . . . . 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 1073 . . . . . 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 992 . . . . . 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 30878 . . . . . 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 1199 . . . . 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 2604 . . . 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 30284 . . . 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 1186 . . 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 29553 . . 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 1182 . 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 201 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   Basecbs 13239   lecple 13306   joincjn 14171   meetcmee 14172   Latclat 14244   Atomscatm 29505   AtLatcal 29506   HLchlt 29592   LHypclh 30225   LTrncltrn 30342   trLctrl 30399
This theorem is referenced by:  cdlemg19  30925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-undef 6382  df-riota 6388  df-map 6859  df-poset 14173  df-plt 14185  df-lub 14201  df-glb 14202  df-join 14203  df-meet 14204  df-p0 14238  df-p1 14239  df-lat 14245  df-clat 14307  df-oposet 29418  df-ol 29420  df-oml 29421  df-covers 29508  df-ats 29509  df-atl 29540  df-cvlat 29564  df-hlat 29593  df-llines 29739  df-lplanes 29740  df-lvols 29741  df-lines 29742  df-psubsp 29744  df-pmap 29745  df-padd 30037  df-lhyp 30229  df-laut 30230  df-ldil 30345  df-ltrn 30346  df-trl 30400
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