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Theorem lkrfval 29277
Description: The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
lkrfval.d  |-  D  =  (Scalar `  W )
lkrfval.o  |-  .0.  =  ( 0g `  D )
lkrfval.f  |-  F  =  (LFnl `  W )
lkrfval.k  |-  K  =  (LKer `  W )
Assertion
Ref Expression
lkrfval  |-  ( W  e.  X  ->  K  =  ( f  e.  F  |->  ( `' f
" {  .0.  }
) ) )
Distinct variable groups:    f, F    f, W
Allowed substitution hints:    D( f)    K( f)    X( f)    .0. ( f)

Proof of Theorem lkrfval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( W  e.  X  ->  W  e.  _V )
2 lkrfval.k . . 3  |-  K  =  (LKer `  W )
3 fveq2 5525 . . . . . 6  |-  ( w  =  W  ->  (LFnl `  w )  =  (LFnl `  W ) )
4 lkrfval.f . . . . . 6  |-  F  =  (LFnl `  W )
53, 4syl6eqr 2333 . . . . 5  |-  ( w  =  W  ->  (LFnl `  w )  =  F )
6 fveq2 5525 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
7 lkrfval.d . . . . . . . . . 10  |-  D  =  (Scalar `  W )
86, 7syl6eqr 2333 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  w )  =  D )
98fveq2d 5529 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  ( 0g `  D ) )
10 lkrfval.o . . . . . . . 8  |-  .0.  =  ( 0g `  D )
119, 10syl6eqr 2333 . . . . . . 7  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  .0.  )
1211sneqd 3653 . . . . . 6  |-  ( w  =  W  ->  { ( 0g `  (Scalar `  w ) ) }  =  {  .0.  }
)
1312imaeq2d 5012 . . . . 5  |-  ( w  =  W  ->  ( `' f " {
( 0g `  (Scalar `  w ) ) } )  =  ( `' f " {  .0.  } ) )
145, 13mpteq12dv 4098 . . . 4  |-  ( w  =  W  ->  (
f  e.  (LFnl `  w )  |->  ( `' f " { ( 0g `  (Scalar `  w ) ) } ) )  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) ) )
15 df-lkr 29276 . . . 4  |- LKer  =  ( w  e.  _V  |->  ( f  e.  (LFnl `  w )  |->  ( `' f " { ( 0g `  (Scalar `  w ) ) } ) ) )
16 fvex 5539 . . . . . 6  |-  (LFnl `  W )  e.  _V
174, 16eqeltri 2353 . . . . 5  |-  F  e. 
_V
1817mptex 5746 . . . 4  |-  ( f  e.  F  |->  ( `' f " {  .0.  } ) )  e.  _V
1914, 15, 18fvmpt 5602 . . 3  |-  ( W  e.  _V  ->  (LKer `  W )  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) ) )
202, 19syl5eq 2327 . 2  |-  ( W  e.  _V  ->  K  =  ( f  e.  F  |->  ( `' f
" {  .0.  }
) ) )
211, 20syl 15 1  |-  ( W  e.  X  ->  K  =  ( f  e.  F  |->  ( `' f
" {  .0.  }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    e. cmpt 4077   `'ccnv 4688   "cima 4692   ` cfv 5255  Scalarcsca 13211   0gc0g 13400  LFnlclfn 29247  LKerclk 29275
This theorem is referenced by:  lkrval  29278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-lkr 29276
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