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Theorem lkrfval 32732
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 2981 . 2  |-  ( W  e.  X  ->  W  e.  _V )
2 lkrfval.k . . 3  |-  K  =  (LKer `  W )
3 fveq2 5691 . . . . . 6  |-  ( w  =  W  ->  (LFnl `  w )  =  (LFnl `  W ) )
4 lkrfval.f . . . . . 6  |-  F  =  (LFnl `  W )
53, 4syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  (LFnl `  w )  =  F )
6 fveq2 5691 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
7 lkrfval.d . . . . . . . . . 10  |-  D  =  (Scalar `  W )
86, 7syl6eqr 2493 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  w )  =  D )
98fveq2d 5695 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  ( 0g `  D ) )
10 lkrfval.o . . . . . . . 8  |-  .0.  =  ( 0g `  D )
119, 10syl6eqr 2493 . . . . . . 7  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  .0.  )
1211sneqd 3889 . . . . . 6  |-  ( w  =  W  ->  { ( 0g `  (Scalar `  w ) ) }  =  {  .0.  }
)
1312imaeq2d 5169 . . . . 5  |-  ( w  =  W  ->  ( `' f " {
( 0g `  (Scalar `  w ) ) } )  =  ( `' f " {  .0.  } ) )
145, 13mpteq12dv 4370 . . . 4  |-  ( w  =  W  ->  (
f  e.  (LFnl `  w )  |->  ( `' f " { ( 0g `  (Scalar `  w ) ) } ) )  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) ) )
15 df-lkr 32731 . . . 4  |- LKer  =  ( w  e.  _V  |->  ( f  e.  (LFnl `  w )  |->  ( `' f " { ( 0g `  (Scalar `  w ) ) } ) ) )
16 fvex 5701 . . . . . 6  |-  (LFnl `  W )  e.  _V
174, 16eqeltri 2513 . . . . 5  |-  F  e. 
_V
1817mptex 5948 . . . 4  |-  ( f  e.  F  |->  ( `' f " {  .0.  } ) )  e.  _V
1914, 15, 18fvmpt 5774 . . 3  |-  ( W  e.  _V  ->  (LKer `  W )  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) ) )
202, 19syl5eq 2487 . 2  |-  ( W  e.  _V  ->  K  =  ( f  e.  F  |->  ( `' f
" {  .0.  }
) ) )
211, 20syl 16 1  |-  ( W  e.  X  ->  K  =  ( f  e.  F  |->  ( `' f
" {  .0.  }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2972   {csn 3877    e. cmpt 4350   `'ccnv 4839   "cima 4843   ` cfv 5418  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-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-pr 4531
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-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-lkr 32731
This theorem is referenced by:  lkrval  32733
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