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Theorem mapdvalc 31890
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 31889 . 2  |-  ( ph  ->  ( M `  T
)  =  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  T ) } )
11 anass 630 . . . . 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 31752 . . . . . . . 8  |-  ( f  e.  C  <->  ( f  e.  F  /\  ( O `  ( O `  ( L `  f
) ) )  =  ( L `  f
) ) )
1413anbi1i 676 . . . . . . 7  |-  ( ( f  e.  C  /\  ( O `  ( L `
 f ) ) 
C_  T )  <->  ( (
f  e.  F  /\  ( O `  ( O `
 ( L `  f ) ) )  =  ( L `  f ) )  /\  ( O `  ( L `
 f ) ) 
C_  T ) )
1514bicomi 193 . . . . . 6  |-  ( ( ( f  e.  F  /\  ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f ) )  /\  ( O `  ( L `
 f ) ) 
C_  T )  <->  ( f  e.  C  /\  ( O `  ( L `  f ) )  C_  T ) )
1615a1i 10 . . . . 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 250 . . . 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 2480 . . 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 2637 . . 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 2637 . . 3  |-  { f  e.  C  |  ( O `  ( L `
 f ) ) 
C_  T }  =  { f  |  ( f  e.  C  /\  ( O `  ( L `
 f ) ) 
C_  T ) }
2118, 19, 203eqtr4g 2423 . 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 2398 1  |-  ( ph  ->  ( M `  T
)  =  { f  e.  C  |  ( O `  ( L `
 f ) ) 
C_  T } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   {cab 2352   {crab 2632    C_ wss 3238   ` cfv 5358   LSubSpclss 15899  LFnlclfn 29318  LKerclk 29346   LHypclh 30244   DVecHcdvh 31339   ocHcoch 31608  mapdcmpd 31885
This theorem is referenced by:  mapdval2N  31891  mapdordlem2  31898  mapdrval  31908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-mapd 31886
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