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Theorem lkrval2 32735
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 2981 . 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 32734 . . . 4  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( x  e.  ( K `  G )  <-> 
( x  e.  V  /\  ( G `  x
)  =  .0.  )
) )
87abbi2dv 2558 . . 3  |-  ( ( W  e.  _V  /\  G  e.  F )  ->  ( K `  G
)  =  { x  |  ( x  e.  V  /\  ( G `
 x )  =  .0.  ) } )
9 df-rab 2724 . . 3  |-  { x  e.  V  |  ( G `  x )  =  .0.  }  =  {
x  |  ( x  e.  V  /\  ( G `  x )  =  .0.  ) }
108, 9syl6eqr 2493 . 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 1369    e. wcel 1756   {cab 2429   {crab 2719   _Vcvv 2972   ` cfv 5418   Basecbs 14174  Scalarcsca 14241   0gc0g 14378  LFnlclfn 32702  LKerclk 32730
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-8 1758  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-pow 4470  ax-pr 4531  ax-un 6372
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-pw 3862  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-oprab 6095  df-mpt2 6096  df-map 7216  df-lfl 32703  df-lkr 32731
This theorem is referenced by:  lkrlss  32740
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