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Theorem mapdvalc 32124
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 32123 . 2  |-  ( ph  ->  ( M `  T
)  =  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) } )
11 anass 631 . . . . 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 31986 . . . . . . . 8  |-  ( f  e.  C  <->  ( f  e.  F  /\  ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
) ) )
1413anbi1i 677 . . . . . . 7  |-  ( ( f  e.  C  /\  ( O `  ( L `
 f ) ) 
C_  T )  <->  ( (
f  e.  F  /\  ( O `  ( O `
 ( L `  f ) ) )  =  ( L `  f ) )  /\  ( O `  ( L `
 f ) ) 
C_  T ) )
1514bicomi 194 . . . . . 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 251 . . . 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 2526 . . 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 2683 . . 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 2683 . . 3  |-  { f  e.  C  |  ( O `  ( L `
 f ) ) 
C_  T }  =  { f  |  ( f  e.  C  /\  ( O `  ( L `
 f ) ) 
C_  T ) }
2118, 19, 203eqtr4g 2469 . 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 2444 1  |-  ( ph  ->  ( M `  T
)  =  { f  e.  C  |  ( O `  ( L `
 f ) ) 
C_  T } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2398   {crab 2678    C_ wss 3288   ` cfv 5421   LSubSpclss 15971  LFnlclfn 29552  LKerclk 29580   LHypclh 30478   DVecHcdvh 31573   ocHcoch 31842  mapdcmpd 32119
This theorem is referenced by:  mapdval2N  32125  mapdordlem2  32132  mapdrval  32142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-mapd 32120
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