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Theorem lkrval2 29585
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 2932 . 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 29584 . . . 4  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( x  e.  ( K `  G )  <-> 
( x  e.  V  /\  ( G `  x
)  =  .0.  )
) )
87abbi2dv 2527 . . 3  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( K `  G
)  =  { x  |  ( x  e.  V  /\  ( G `
 x )  =  .0.  ) } )
9 df-rab 2683 . . 3  |-  { x  e.  V  |  ( G `  x )  =  .0.  }  =  {
x  |  ( x  e.  V  /\  ( G `  x )  =  .0.  ) }
108, 9syl6eqr 2462 . 2  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( K `  G
)  =  { x  e.  V  |  ( G `  x )  =  .0.  } )
111, 10sylan 458 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 359    = wceq 1649    e. wcel 1721   {cab 2398   {crab 2678   _Vcvv 2924   ` cfv 5421   Basecbs 13432  Scalarcsca 13495   0gc0g 13686  LFnlclfn 29552  LKerclk 29580
This theorem is referenced by:  lkrlss  29590
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-lfl 29553  df-lkr 29581
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