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Theorem lkrfval 34913
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 3118 . 2  |-  ( W  e.  X  ->  W  e.  _V )
2 lkrfval.k . . 3  |-  K  =  (LKer `  W )
3 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  (LFnl `  w )  =  (LFnl `  W ) )
4 lkrfval.f . . . . . 6  |-  F  =  (LFnl `  W )
53, 4syl6eqr 2516 . . . . 5  |-  ( w  =  W  ->  (LFnl `  w )  =  F )
6 fveq2 5872 . . . . . . . . . 10  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
7 lkrfval.d . . . . . . . . . 10  |-  D  =  (Scalar `  W )
86, 7syl6eqr 2516 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  w )  =  D )
98fveq2d 5876 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  ( 0g `  D ) )
10 lkrfval.o . . . . . . . 8  |-  .0.  =  ( 0g `  D )
119, 10syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  ( 0g `  (Scalar `  w
) )  =  .0.  )
1211sneqd 4044 . . . . . 6  |-  ( w  =  W  ->  { ( 0g `  (Scalar `  w ) ) }  =  {  .0.  }
)
1312imaeq2d 5347 . . . . 5  |-  ( w  =  W  ->  ( `' f " {
( 0g `  (Scalar `  w ) ) } )  =  ( `' f " {  .0.  } ) )
145, 13mpteq12dv 4535 . . . 4  |-  ( w  =  W  ->  (
f  e.  (LFnl `  w )  |->  ( `' f " { ( 0g `  (Scalar `  w ) ) } ) )  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) ) )
15 df-lkr 34912 . . . 4  |- LKer  =  ( w  e.  _V  |->  ( f  e.  (LFnl `  w )  |->  ( `' f " { ( 0g `  (Scalar `  w ) ) } ) ) )
16 fvex 5882 . . . . . 6  |-  (LFnl `  W )  e.  _V
174, 16eqeltri 2541 . . . . 5  |-  F  e. 
_V
1817mptex 6144 . . . 4  |-  ( f  e.  F  |->  ( `' f " {  .0.  } ) )  e.  _V
1914, 15, 18fvmpt 5956 . . 3  |-  ( W  e.  _V  ->  (LKer `  W )  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) ) )
202, 19syl5eq 2510 . 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 1395    e. wcel 1819   _Vcvv 3109   {csn 4032    |-> cmpt 4515   `'ccnv 5007   "cima 5011   ` cfv 5594  Scalarcsca 14714   0gc0g 14856  LFnlclfn 34883  LKerclk 34911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-lkr 34912
This theorem is referenced by:  lkrval  34914
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