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Theorem lkrfval 29899
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 2809 . 2  |-  ( W  e.  X  ->  W  e.  _V )
2 lkrfval.k . . 3  |-  K  =  (LKer `  W )
3 fveq2 5541 . . . . . 6  |-  ( w  =  W  ->  (LFnl `  w )  =  (LFnl `  W ) )
4 lkrfval.f . . . . . 6  |-  F  =  (LFnl `  W )
53, 4syl6eqr 2346 . . . . 5  |-  ( w  =  W  ->  (LFnl `  w )  =  F )
6 fveq2 5541 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
7 lkrfval.d . . . . . . . . . 10  |-  D  =  (Scalar `  W )
86, 7syl6eqr 2346 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  w )  =  D )
98fveq2d 5545 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  ( 0g `  D ) )
10 lkrfval.o . . . . . . . 8  |-  .0.  =  ( 0g `  D )
119, 10syl6eqr 2346 . . . . . . 7  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  .0.  )
1211sneqd 3666 . . . . . 6  |-  ( w  =  W  ->  { ( 0g `  (Scalar `  w ) ) }  =  {  .0.  }
)
1312imaeq2d 5028 . . . . 5  |-  ( w  =  W  ->  ( `' f " {
( 0g `  (Scalar `  w ) ) } )  =  ( `' f " {  .0.  } ) )
145, 13mpteq12dv 4114 . . . 4  |-  ( w  =  W  ->  (
f  e.  (LFnl `  w )  |->  ( `' f " { ( 0g `  (Scalar `  w ) ) } ) )  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) ) )
15 df-lkr 29898 . . . 4  |- LKer  =  ( w  e.  _V  |->  ( f  e.  (LFnl `  w )  |->  ( `' f " { ( 0g `  (Scalar `  w ) ) } ) ) )
16 fvex 5555 . . . . . 6  |-  (LFnl `  W )  e.  _V
174, 16eqeltri 2366 . . . . 5  |-  F  e. 
_V
1817mptex 5762 . . . 4  |-  ( f  e.  F  |->  ( `' f " {  .0.  } ) )  e.  _V
1914, 15, 18fvmpt 5618 . . 3  |-  ( W  e.  _V  ->  (LKer `  W )  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) ) )
202, 19syl5eq 2340 . 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 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    e. cmpt 4093   `'ccnv 4704   "cima 4708   ` cfv 5271  Scalarcsca 13227   0gc0g 13416  LFnlclfn 29869  LKerclk 29897
This theorem is referenced by:  lkrval  29900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-lkr 29898
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