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Theorem mapdval 35241
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 35240 . . . 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 17 . . 3  |-  ( ph  ->  M  =  ( s  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  s ) } ) )
1110fveq1d 5890 . 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 5898 . . . . 5  |-  (LFnl `  U )  e.  _V
145, 13eqeltri 2536 . . . 4  |-  F  e. 
_V
1514rabex 4568 . . 3  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) }  e.  _V
16 sseq2 3466 . . . . . 6  |-  ( s  =  T  ->  (
( O `  ( L `  f )
)  C_  s  <->  ( O `  ( L `  f
) )  C_  T
) )
1716anbi2d 715 . . . . 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 3048 . . . 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 2462 . . . 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 5969 . . 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 673 . 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 2496 1  |-  ( ph  ->  ( M `  T
)  =  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898   {crab 2753   _Vcvv 3057    C_ wss 3416    |-> cmpt 4475   ` cfv 5601   LSubSpclss 18204  LFnlclfn 32668  LKerclk 32696   LHypclh 33594   DVecHcdvh 34691   ocHcoch 34960  mapdcmpd 35237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-mapd 35238
This theorem is referenced by:  mapdvalc  35242  mapddlssN  35253  mapdsn  35254  mapd1o  35261  mapd0  35278
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