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Theorem mapdval 32440
Description: Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.)
Hypotheses
Ref Expression
mapdval.h  |-  H  =  ( LHyp `  K
)
mapdval.u  |-  U  =  ( ( DVecH `  K
) `  W )
mapdval.s  |-  S  =  ( LSubSp `  U )
mapdval.f  |-  F  =  (LFnl `  U )
mapdval.l  |-  L  =  (LKer `  U )
mapdval.o  |-  O  =  ( ( ocH `  K
) `  W )
mapdval.m  |-  M  =  ( (mapd `  K
) `  W )
mapdval.k  |-  ( ph  ->  ( K  e.  X  /\  W  e.  H
) )
mapdval.t  |-  ( ph  ->  T  e.  S )
Assertion
Ref Expression
mapdval  |-  ( ph  ->  ( M `  T
)  =  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) } )
Distinct variable groups:    f, K    f, F    f, W    T, f
Allowed substitution hints:    ph( f)    S( f)    U( f)    H( f)    L( f)    M( f)    O( f)    X( f)

Proof of Theorem mapdval
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 mapdval.k . . . 4  |-  ( ph  ->  ( K  e.  X  /\  W  e.  H
) )
2 mapdval.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 mapdval.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
4 mapdval.s . . . . 5  |-  S  =  ( LSubSp `  U )
5 mapdval.f . . . . 5  |-  F  =  (LFnl `  U )
6 mapdval.l . . . . 5  |-  L  =  (LKer `  U )
7 mapdval.o . . . . 5  |-  O  =  ( ( ocH `  K
) `  W )
8 mapdval.m . . . . 5  |-  M  =  ( (mapd `  K
) `  W )
92, 3, 4, 5, 6, 7, 8mapdfval 32439 . . . 4  |-  ( ( K  e.  X  /\  W  e.  H )  ->  M  =  ( s  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  s ) } ) )
101, 9syl 15 . . 3  |-  ( ph  ->  M  =  ( s  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  s ) } ) )
1110fveq1d 5543 . 2  |-  ( ph  ->  ( M `  T
)  =  ( ( s  e.  S  |->  { f  e.  F  | 
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) ) 
C_  s ) } ) `  T ) )
12 mapdval.t . . 3  |-  ( ph  ->  T  e.  S )
13 fvex 5555 . . . . 5  |-  (LFnl `  U )  e.  _V
145, 13eqeltri 2366 . . . 4  |-  F  e. 
_V
1514rabex 4181 . . 3  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) }  e.  _V
16 sseq2 3213 . . . . . 6  |-  ( s  =  T  ->  (
( O `  ( L `  f )
)  C_  s  <->  ( O `  ( L `  f
) )  C_  T
) )
1716anbi2d 684 . . . . 5  |-  ( s  =  T  ->  (
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) ) 
C_  s )  <->  ( ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
)  /\  ( O `  ( L `  f
) )  C_  T
) ) )
1817rabbidv 2793 . . . 4  |-  ( s  =  T  ->  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  s ) }  =  { f  e.  F  |  ( ( O `
 ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) } )
19 eqid 2296 . . . 4  |-  ( s  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  s ) } )  =  ( s  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `
 ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  s ) } )
2018, 19fvmptg 5616 . . 3  |-  ( ( T  e.  S  /\  { f  e.  F  | 
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) ) 
C_  T ) }  e.  _V )  -> 
( ( s  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `
 ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  s ) } ) `
 T )  =  { f  e.  F  |  ( ( O `
 ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) } )
2112, 15, 20sylancl 643 . 2  |-  ( ph  ->  ( ( s  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `
 ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  s ) } ) `
 T )  =  { f  e.  F  |  ( ( O `
 ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) } )
2211, 21eqtrd 2328 1  |-  ( ph  ->  ( M `  T
)  =  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165    e. cmpt 4093   ` cfv 5271   LSubSpclss 15705  LFnlclfn 29869  LKerclk 29897   LHypclh 30795   DVecHcdvh 31890   ocHcoch 32159  mapdcmpd 32436
This theorem is referenced by:  mapdvalc  32441  mapddlssN  32452  mapdsn  32453  mapd1o  32460  mapd0  32477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-mapd 32437
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