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Theorem lcdval2 35235
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 35234 . 2  |-  ( ph  ->  C  =  ( Ds  { f  e.  F  | 
(  ._|_  `  (  ._|_  `  ( L `  f
) ) )  =  ( L `  f
) } ) )
10 lcdval2.b . . 3  |-  B  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
1110oveq2i 6102 . 2  |-  ( Ds  B )  =  ( Ds  { f  e.  F  | 
(  ._|_  `  (  ._|_  `  ( L `  f
) ) )  =  ( L `  f
) } )
129, 11syl6eqr 2493 1  |-  ( ph  ->  C  =  ( Ds  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2719   ` cfv 5418  (class class class)co 6091   ↾s cress 14175  LFnlclfn 32702  LKerclk 32730  LDualcld 32768   LHypclh 33628   DVecHcdvh 34723   ocHcoch 34992  LCDualclcd 35231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-lcdual 35232
This theorem is referenced by:  lcdvbase  35238  lcdlss  35264
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