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Theorem lkrval2 34288
Description: Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
Hypotheses
Ref Expression
lkrfval2.v  |-  V  =  ( Base `  W
)
lkrfval2.d  |-  D  =  (Scalar `  W )
lkrfval2.o  |-  .0.  =  ( 0g `  D )
lkrfval2.f  |-  F  =  (LFnl `  W )
lkrfval2.k  |-  K  =  (LKer `  W )
Assertion
Ref Expression
lkrval2  |-  ( ( W  e.  X  /\  G  e.  F )  ->  ( K `  G
)  =  { x  e.  V  |  ( G `  x )  =  .0.  } )
Distinct variable groups:    x, F    x, G    x, K    x, W
Allowed substitution hints:    D( x)    V( x)    X( x)    .0. ( x)

Proof of Theorem lkrval2
StepHypRef Expression
1 elex 3127 . 2  |-  ( W  e.  X  ->  W  e.  _V )
2 lkrfval2.v . . . . 5  |-  V  =  ( Base `  W
)
3 lkrfval2.d . . . . 5  |-  D  =  (Scalar `  W )
4 lkrfval2.o . . . . 5  |-  .0.  =  ( 0g `  D )
5 lkrfval2.f . . . . 5  |-  F  =  (LFnl `  W )
6 lkrfval2.k . . . . 5  |-  K  =  (LKer `  W )
72, 3, 4, 5, 6ellkr 34287 . . . 4  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( x  e.  ( K `  G )  <-> 
( x  e.  V  /\  ( G `  x
)  =  .0.  )
) )
87abbi2dv 2604 . . 3  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( K `  G
)  =  { x  |  ( x  e.  V  /\  ( G `
 x )  =  .0.  ) } )
9 df-rab 2826 . . 3  |-  { x  e.  V  |  ( G `  x )  =  .0.  }  =  {
x  |  ( x  e.  V  /\  ( G `  x )  =  .0.  ) }
108, 9syl6eqr 2526 . 2  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( K `  G
)  =  { x  e.  V  |  ( G `  x )  =  .0.  } )
111, 10sylan 471 1  |-  ( ( W  e.  X  /\  G  e.  F )  ->  ( K `  G
)  =  { x  e.  V  |  ( G `  x )  =  .0.  } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   {crab 2821   _Vcvv 3118   ` cfv 5594   Basecbs 14507  Scalarcsca 14575   0gc0g 14712  LFnlclfn 34255  LKerclk 34283
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-8 1769  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-pow 4631  ax-pr 4692  ax-un 6587
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-pw 4018  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-oprab 6299  df-mpt2 6300  df-map 7434  df-lfl 34256  df-lkr 34284
This theorem is referenced by:  lkrlss  34293
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