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Theorem mapdval 31818
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 31817 . . . 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 5527 . 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 5539 . . . . 5  |-  (LFnl `  U )  e.  _V
145, 13eqeltri 2353 . . . 4  |-  F  e. 
_V
1514rabex 4165 . . 3  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) }  e.  _V
16 sseq2 3200 . . . . . 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 2780 . . . 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 2283 . . . 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 5600 . . 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 2315 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 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152    e. cmpt 4077   ` cfv 5255   LSubSpclss 15689  LFnlclfn 29247  LKerclk 29275   LHypclh 30173   DVecHcdvh 31268   ocHcoch 31537  mapdcmpd 31814
This theorem is referenced by:  mapdvalc  31819  mapddlssN  31830  mapdsn  31831  mapd1o  31838  mapd0  31855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-mapd 31815
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