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Theorem mapdvalc 34996
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 34995 . 2  |-  ( ph  ->  ( M `  T
)  =  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) } )
11 anass 644 . . . . 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 34858 . . . . . . . 8  |-  ( f  e.  C  <->  ( f  e.  F  /\  ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
) ) )
1413anbi1i 690 . . . . . . 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 2555 . . 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 2722 . . 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 2722 . . 3  |-  { f  e.  C  |  ( O `  ( L `
 f ) ) 
C_  T }  =  { f  |  ( f  e.  C  /\  ( O `  ( L `
 f ) ) 
C_  T ) }
2118, 19, 203eqtr4g 2498 . 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 2473 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 1364    e. wcel 1761   {cab 2427   {crab 2717    C_ wss 3325   ` cfv 5415   LSubSpclss 16991  LFnlclfn 32424  LKerclk 32452   LHypclh 33350   DVecHcdvh 34445   ocHcoch 34714  mapdcmpd 34991
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 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-mapd 34992
This theorem is referenced by:  mapdval2N  34997  mapdordlem2  35004  mapdrval  35014
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