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Theorem mapdvalc 32501
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 )
mapdvalc.c  |-  C  =  { g  e.  F  |  ( O `  ( O `  ( L `
 g ) ) )  =  ( L `
 g ) }
Assertion
Ref Expression
mapdvalc  |-  ( ph  ->  ( M `  T
)  =  { f  e.  C  |  ( O `  ( L `
 f ) ) 
C_  T } )
Distinct variable groups:    f, K    f, F    f, W    f,
g, F    g, L    g, O    T, f    ph, f
Allowed substitution hints:    ph( g)    C( f, g)    S( f, g)    T( g)    U( f, g)    H( f, g)    K( g)    L( f)    M( f, g)    O( f)    W( g)    X( f, g)

Proof of Theorem mapdvalc
StepHypRef Expression
1 mapdval.h . . 3  |-  H  =  ( LHyp `  K
)
2 mapdval.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
3 mapdval.s . . 3  |-  S  =  ( LSubSp `  U )
4 mapdval.f . . 3  |-  F  =  (LFnl `  U )
5 mapdval.l . . 3  |-  L  =  (LKer `  U )
6 mapdval.o . . 3  |-  O  =  ( ( ocH `  K
) `  W )
7 mapdval.m . . 3  |-  M  =  ( (mapd `  K
) `  W )
8 mapdval.k . . 3  |-  ( ph  ->  ( K  e.  X  /\  W  e.  H
) )
9 mapdval.t . . 3  |-  ( ph  ->  T  e.  S )
101, 2, 3, 4, 5, 6, 7, 8, 9mapdval 32500 . 2  |-  ( ph  ->  ( M `  T
)  =  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) } )
11 anass 632 . . . . 5  |-  ( ( ( f  e.  F  /\  ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f ) )  /\  ( O `  ( L `
 f ) ) 
C_  T )  <->  ( f  e.  F  /\  (
( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) ) )
12 mapdvalc.c . . . . . . . . 9  |-  C  =  { g  e.  F  |  ( O `  ( O `  ( L `
 g ) ) )  =  ( L `
 g ) }
1312lcfl1lem 32363 . . . . . . . 8  |-  ( f  e.  C  <->  ( f  e.  F  /\  ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
) ) )
1413anbi1i 678 . . . . . . 7  |-  ( ( f  e.  C  /\  ( O `  ( L `
 f ) ) 
C_  T )  <->  ( (
f  e.  F  /\  ( O `  ( O `
 ( L `  f ) ) )  =  ( L `  f ) )  /\  ( O `  ( L `
 f ) ) 
C_  T ) )
1514bicomi 195 . . . . . 6  |-  ( ( ( f  e.  F  /\  ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f ) )  /\  ( O `  ( L `
 f ) ) 
C_  T )  <->  ( f  e.  C  /\  ( O `  ( L `  f ) )  C_  T ) )
1615a1i 11 . . . . 5  |-  ( ph  ->  ( ( ( f  e.  F  /\  ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
) )  /\  ( O `  ( L `  f ) )  C_  T )  <->  ( f  e.  C  /\  ( O `  ( L `  f ) )  C_  T ) ) )
1711, 16syl5bbr 252 . . . 4  |-  ( ph  ->  ( ( f  e.  F  /\  ( ( O `  ( O `
 ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) )  <->  ( f  e.  C  /\  ( O `  ( L `  f ) )  C_  T ) ) )
1817abbidv 2552 . . 3  |-  ( ph  ->  { f  |  ( f  e.  F  /\  ( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) ) 
C_  T ) ) }  =  { f  |  ( f  e.  C  /\  ( O `
 ( L `  f ) )  C_  T ) } )
19 df-rab 2716 . . 3  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) }  =  { f  |  ( f  e.  F  /\  ( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) ) 
C_  T ) ) }
20 df-rab 2716 . . 3  |-  { f  e.  C  |  ( O `  ( L `
 f ) ) 
C_  T }  =  { f  |  ( f  e.  C  /\  ( O `  ( L `
 f ) ) 
C_  T ) }
2118, 19, 203eqtr4g 2495 . 2  |-  ( ph  ->  { f  e.  F  |  ( ( O `
 ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) }  =  { f  e.  C  |  ( O `  ( L `  f ) )  C_  T } )
2210, 21eqtrd 2470 1  |-  ( ph  ->  ( M `  T
)  =  { f  e.  C  |  ( O `  ( L `
 f ) ) 
C_  T } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   {crab 2711    C_ wss 3322   ` cfv 5457   LSubSpclss 16013  LFnlclfn 29929  LKerclk 29957   LHypclh 30855   DVecHcdvh 31950   ocHcoch 32219  mapdcmpd 32496
This theorem is referenced by:  mapdval2N  32502  mapdordlem2  32509  mapdrval  32519
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-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-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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-ral 2712  df-rex 2713  df-reu 2714  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-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  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-mapd 32497
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