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Theorem lcdval2 32388
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 32387 . 2  |-  ( ph  ->  C  =  ( Ds  { f  e.  F  | 
(  ._|_  `  (  ._|_  `  ( L `  f
) ) )  =  ( L `  f
) } ) )
10 lcdval2.b . . 3  |-  B  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
1110oveq2i 6092 . 2  |-  ( Ds  B )  =  ( Ds  { f  e.  F  | 
(  ._|_  `  (  ._|_  `  ( L `  f
) ) )  =  ( L `  f
) } )
129, 11syl6eqr 2486 1  |-  ( ph  ->  C  =  ( Ds  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709   ` cfv 5454  (class class class)co 6081   ↾s cress 13470  LFnlclfn 29855  LKerclk 29883  LDualcld 29921   LHypclh 30781   DVecHcdvh 31876   ocHcoch 32145  LCDualclcd 32384
This theorem is referenced by:  lcdvbase  32391  lcdlss  32417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-lcdual 32385
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