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Theorem lcdval2 35204
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 35203 . 2  |-  ( ph  ->  C  =  ( Ds  { f  e.  F  | 
(  ._|_  `  (  ._|_  `  ( L `  f
) ) )  =  ( L `  f
) } ) )
10 lcdval2.b . . 3  |-  B  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
1110oveq2i 6331 . 2  |-  ( Ds  B )  =  ( Ds  { f  e.  F  | 
(  ._|_  `  (  ._|_  `  ( L `  f
) ) )  =  ( L `  f
) } )
129, 11syl6eqr 2514 1  |-  ( ph  ->  C  =  ( Ds  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898   {crab 2753   ` cfv 5605  (class class class)co 6320   ↾s cress 15177  LFnlclfn 32669  LKerclk 32697  LDualcld 32735   LHypclh 33595   DVecHcdvh 34692   ocHcoch 34961  LCDualclcd 35200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-ov 6323  df-lcdual 35201
This theorem is referenced by:  lcdvbase  35207  lcdlss  35233
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