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Theorem lcdval2 36788
Description: Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
Hypotheses
Ref Expression
lcdval.h  |-  H  =  ( LHyp `  K
)
lcdval.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
lcdval.c  |-  C  =  ( (LCDual `  K
) `  W )
lcdval.u  |-  U  =  ( ( DVecH `  K
) `  W )
lcdval.f  |-  F  =  (LFnl `  U )
lcdval.l  |-  L  =  (LKer `  U )
lcdval.d  |-  D  =  (LDual `  U )
lcdval.k  |-  ( ph  ->  ( K  e.  X  /\  W  e.  H
) )
lcdval2.b  |-  B  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
Assertion
Ref Expression
lcdval2  |-  ( ph  ->  C  =  ( Ds  B ) )
Distinct variable groups:    f, K    f, F    f, W
Allowed substitution hints:    ph( f)    B( f)    C( f)    D( f)    U( f)    H( f)    L( f)   
._|_ ( f)    X( f)

Proof of Theorem lcdval2
StepHypRef Expression
1 lcdval.h . . 3  |-  H  =  ( LHyp `  K
)
2 lcdval.o . . 3  |-  ._|_  =  ( ( ocH `  K
) `  W )
3 lcdval.c . . 3  |-  C  =  ( (LCDual `  K
) `  W )
4 lcdval.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
5 lcdval.f . . 3  |-  F  =  (LFnl `  U )
6 lcdval.l . . 3  |-  L  =  (LKer `  U )
7 lcdval.d . . 3  |-  D  =  (LDual `  U )
8 lcdval.k . . 3  |-  ( ph  ->  ( K  e.  X  /\  W  e.  H
) )
91, 2, 3, 4, 5, 6, 7, 8lcdval 36787 . 2  |-  ( ph  ->  C  =  ( Ds  { f  e.  F  | 
(  ._|_  `  (  ._|_  `  ( L `  f
) ) )  =  ( L `  f
) } ) )
10 lcdval2.b . . 3  |-  B  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
1110oveq2i 6306 . 2  |-  ( Ds  B )  =  ( Ds  { f  e.  F  | 
(  ._|_  `  (  ._|_  `  ( L `  f
) ) )  =  ( L `  f
) } )
129, 11syl6eqr 2526 1  |-  ( ph  ->  C  =  ( Ds  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821   ` cfv 5594  (class class class)co 6295   ↾s cress 14508  LFnlclfn 34255  LKerclk 34283  LDualcld 34321   LHypclh 35181   DVecHcdvh 36276   ocHcoch 36545  LCDualclcd 36784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-lcdual 36785
This theorem is referenced by:  lcdvbase  36791  lcdlss  36817
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