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Theorem lkrval2 35228
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 3043 . 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 35227 . . . 4  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( x  e.  ( K `  G )  <-> 
( x  e.  V  /\  ( G `  x
)  =  .0.  )
) )
87abbi2dv 2519 . . 3  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( K `  G
)  =  { x  |  ( x  e.  V  /\  ( G `
 x )  =  .0.  ) } )
9 df-rab 2741 . . 3  |-  { x  e.  V  |  ( G `  x )  =  .0.  }  =  {
x  |  ( x  e.  V  /\  ( G `  x )  =  .0.  ) }
108, 9syl6eqr 2441 . 2  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( K `  G
)  =  { x  e.  V  |  ( G `  x )  =  .0.  } )
111, 10sylan 469 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 367    = wceq 1399    e. wcel 1826   {cab 2367   {crab 2736   _Vcvv 3034   ` cfv 5496   Basecbs 14634  Scalarcsca 14705   0gc0g 14847  LFnlclfn 35195  LKerclk 35223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-map 7340  df-lfl 35196  df-lkr 35224
This theorem is referenced by:  lkrlss  35233
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