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Theorem cdlemg39 31587
Description: Eliminate  =/= conditions from cdlemg38 31586. TODO: Would this better be done at cdlemg35 31584? TODO: Fix comment. (Contributed by NM, 31-May-2013.)
Hypotheses
Ref Expression
cdlemg35.l  |-  .<_  =  ( le `  K )
cdlemg35.j  |-  .\/  =  ( join `  K )
cdlemg35.m  |-  ./\  =  ( meet `  K )
cdlemg35.a  |-  A  =  ( Atoms `  K )
cdlemg35.h  |-  H  =  ( LHyp `  K
)
cdlemg35.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg35.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg39  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/= 
Q ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )

Proof of Theorem cdlemg39
StepHypRef Expression
1 simpl1 961 . . 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 )  =  ( R `  G ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simpl2l 1011 . . 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 )  =  ( R `  G ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
3 simpl2r 1012 . . 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 )  =  ( R `  G ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
4 simpl31 1039 . . 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 )  =  ( R `  G ) )  ->  F  e.  T )
5 simpl32 1040 . . 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 )  =  ( R `  G ) )  ->  G  e.  T )
6 simpr 449 . . 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 )  =  ( R `  G ) )  -> 
( R `  F
)  =  ( R `
 G ) )
7 cdlemg35.l . . . 4  |-  .<_  =  ( le `  K )
8 cdlemg35.j . . . 4  |-  .\/  =  ( join `  K )
9 cdlemg35.m . . . 4  |-  ./\  =  ( meet `  K )
10 cdlemg35.a . . . 4  |-  A  =  ( Atoms `  K )
11 cdlemg35.h . . . 4  |-  H  =  ( LHyp `  K
)
12 cdlemg35.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
13 cdlemg35.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
147, 8, 9, 10, 11, 12, 13cdlemg15 31527 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
151, 2, 3, 4, 5, 6, 14syl321anc 1207 . 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 )  =  ( R `  G ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
16 simpll1 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 )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  ( K  e.  HL  /\  W  e.  H ) )
17 simpll2 998 . . . 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 )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
18 simpl31 1039 . . . . 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 )  =/=  ( R `  G
) )  ->  F  e.  T )
1918adantr 453 . . . 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 )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  F  e.  T )
20 simpl32 1040 . . . . 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 )  =/=  ( R `  G
) )  ->  G  e.  T )
2120adantr 453 . . . 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 )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  G  e.  T )
22 simpr 449 . . . 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 )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  ( F `  P )  =  P )
237, 8, 9, 10, 11, 12, 13cdlemg14f 31524 . . . 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 ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
2416, 17, 19, 21, 22, 23syl113anc 1197 . . 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 )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  (
( P  .\/  ( F `  ( G `  P ) ) ) 
./\  W )  =  ( ( Q  .\/  ( F `  ( G `
 Q ) ) )  ./\  W )
)
25 simpll1 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 )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  ( K  e.  HL  /\  W  e.  H ) )
26 simpll2 998 . . . 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 )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
2718adantr 453 . . . 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 )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  F  e.  T )
2820adantr 453 . . . 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 )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  G  e.  T )
29 simpr 449 . . . 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 )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  ( G `  P )  =  P )
307, 8, 9, 10, 11, 12, 13cdlemg14g 31525 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( G `  P )  =  P ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
3125, 26, 27, 28, 29, 30syl113anc 1197 . . 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 )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  (
( P  .\/  ( F `  ( G `  P ) ) ) 
./\  W )  =  ( ( Q  .\/  ( F `  ( G `
 Q ) ) )  ./\  W )
)
32 simpll1 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 )  =/=  ( R `  G
) )  /\  (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
33 simpl2l 1011 . . . . 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 )  =/=  ( R `  G
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3433adantr 453 . . . 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 )  =/=  ( R `  G
) )  /\  (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
35 simpl2r 1012 . . . . 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 )  =/=  ( R `  G
) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
3635adantr 453 . . . 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 )  =/=  ( R `  G
) )  /\  (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
37 simpll3 999 . . . 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 )  =/=  ( R `  G
) )  /\  (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P ) )  ->  ( F  e.  T  /\  G  e.  T  /\  P  =/= 
Q ) )
38 simpr 449 . . . 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 )  =/=  ( R `  G
) )  /\  (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P ) )  ->  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P
) )
39 simplr 733 . . . 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 )  =/=  ( R `  G
) )  /\  (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P ) )  ->  ( R `  F )  =/=  ( R `  G )
)
407, 8, 9, 10, 11, 12, 13cdlemg38 31586 . . . 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
)  /\  ( (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P )  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
4132, 34, 36, 37, 38, 39, 40syl312anc 1206 . . 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 )  =/=  ( R `  G
) )  /\  (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P ) )  ->  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
4224, 31, 41pm2.61da2ne 2685 . 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 )  =/=  ( R `  G
) )  ->  (
( P  .\/  ( F `  ( G `  P ) ) ) 
./\  W )  =  ( ( Q  .\/  ( F `  ( G `
 Q ) ) )  ./\  W )
)
4315, 42pm2.61dane 2684 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 ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   lecple 13541   joincjn 14406   meetcmee 14407   Atomscatm 30135   HLchlt 30222   LHypclh 30855   LTrncltrn 30972   trLctrl 31029
This theorem is referenced by:  cdlemg40  31588
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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