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Theorem lkrval2 29962
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 2966 . 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 29961 . . . 4  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( x  e.  ( K `  G )  <-> 
( x  e.  V  /\  ( G `  x
)  =  .0.  )
) )
87abbi2dv 2553 . . 3  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( K `  G
)  =  { x  |  ( x  e.  V  /\  ( G `
 x )  =  .0.  ) } )
9 df-rab 2716 . . 3  |-  { x  e.  V  |  ( G `  x )  =  .0.  }  =  {
x  |  ( x  e.  V  /\  ( G `  x )  =  .0.  ) }
108, 9syl6eqr 2488 . 2  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( K `  G
)  =  { x  e.  V  |  ( G `  x )  =  .0.  } )
111, 10sylan 459 1  |-  ( ( W  e.  X  /\  G  e.  F )  ->  ( K `  G
)  =  { x  e.  V  |  ( G `  x )  =  .0.  } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   {crab 2711   _Vcvv 2958   ` cfv 5457   Basecbs 13474  Scalarcsca 13537   0gc0g 13728  LFnlclfn 29929  LKerclk 29957
This theorem is referenced by:  lkrlss  29967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-lfl 29930  df-lkr 29958
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