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Theorem lkrval 29254
Description: Value of the kernel of a functional. (Contributed by NM, 15-Apr-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
lkrval  |-  ( ( W  e.  X  /\  G  e.  F )  ->  ( K `  G
)  =  ( `' G " {  .0.  } ) )

Proof of Theorem lkrval
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 lkrfval.d . . . 4  |-  D  =  (Scalar `  W )
2 lkrfval.o . . . 4  |-  .0.  =  ( 0g `  D )
3 lkrfval.f . . . 4  |-  F  =  (LFnl `  W )
4 lkrfval.k . . . 4  |-  K  =  (LKer `  W )
51, 2, 3, 4lkrfval 29253 . . 3  |-  ( W  e.  X  ->  K  =  ( f  e.  F  |->  ( `' f
" {  .0.  }
) ) )
65fveq1d 5663 . 2  |-  ( W  e.  X  ->  ( K `  G )  =  ( ( f  e.  F  |->  ( `' f " {  .0.  } ) ) `  G
) )
7 cnvexg 5338 . . . 4  |-  ( G  e.  F  ->  `' G  e.  _V )
8 imaexg 5150 . . . 4  |-  ( `' G  e.  _V  ->  ( `' G " {  .0.  } )  e.  _V )
97, 8syl 16 . . 3  |-  ( G  e.  F  ->  ( `' G " {  .0.  } )  e.  _V )
10 cnveq 4979 . . . . 5  |-  ( f  =  G  ->  `' f  =  `' G
)
1110imaeq1d 5135 . . . 4  |-  ( f  =  G  ->  ( `' f " {  .0.  } )  =  ( `' G " {  .0.  } ) )
12 eqid 2380 . . . 4  |-  ( f  e.  F  |->  ( `' f " {  .0.  } ) )  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) )
1311, 12fvmptg 5736 . . 3  |-  ( ( G  e.  F  /\  ( `' G " {  .0.  } )  e.  _V )  ->  ( ( f  e.  F  |->  ( `' f
" {  .0.  }
) ) `  G
)  =  ( `' G " {  .0.  } ) )
149, 13mpdan 650 . 2  |-  ( G  e.  F  ->  (
( f  e.  F  |->  ( `' f " {  .0.  } ) ) `
 G )  =  ( `' G " {  .0.  } ) )
156, 14sylan9eq 2432 1  |-  ( ( W  e.  X  /\  G  e.  F )  ->  ( K `  G
)  =  ( `' G " {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2892   {csn 3750    e. cmpt 4200   `'ccnv 4810   "cima 4814   ` cfv 5387  Scalarcsca 13452   0gc0g 13643  LFnlclfn 29223  LKerclk 29251
This theorem is referenced by:  ellkr  29255  lkr0f  29260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-lkr 29252
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