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Theorem mapdvalc 36303
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 36302 . 2  |-  ( ph  ->  ( M `  T
)  =  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) } )
11 anass 649 . . . . 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 36165 . . . . . . . 8  |-  ( f  e.  C  <->  ( f  e.  F  /\  ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
) ) )
1413anbi1i 695 . . . . . . 7  |-  ( ( f  e.  C  /\  ( O `  ( L `
 f ) ) 
C_  T )  <->  ( (
f  e.  F  /\  ( O `  ( O `
 ( L `  f ) ) )  =  ( L `  f ) )  /\  ( O `  ( L `
 f ) ) 
C_  T ) )
1514bicomi 202 . . . . . 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 259 . . . 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 2598 . . 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 2818 . . 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 2818 . . 3  |-  { f  e.  C  |  ( O `  ( L `
 f ) ) 
C_  T }  =  { f  |  ( f  e.  C  /\  ( O `  ( L `
 f ) ) 
C_  T ) }
2118, 19, 203eqtr4g 2528 . 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 2503 1  |-  ( ph  ->  ( M `  T
)  =  { f  e.  C  |  ( O `  ( L `
 f ) ) 
C_  T } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   {cab 2447   {crab 2813    C_ wss 3471   ` cfv 5581   LSubSpclss 17356  LFnlclfn 33731  LKerclk 33759   LHypclh 34657   DVecHcdvh 35752   ocHcoch 36021  mapdcmpd 36298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-mapd 36299
This theorem is referenced by:  mapdval2N  36304  mapdordlem2  36311  mapdrval  36321
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