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Theorem lkrval 29278
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 29277 . . 3  |-  ( W  e.  X  ->  K  =  ( f  e.  F  |->  ( `' f
" {  .0.  }
) ) )
65fveq1d 5527 . 2  |-  ( W  e.  X  ->  ( K `  G )  =  ( ( f  e.  F  |->  ( `' f " {  .0.  } ) ) `  G
) )
7 cnvexg 5208 . . . 4  |-  ( G  e.  F  ->  `' G  e.  _V )
8 imaexg 5026 . . . 4  |-  ( `' G  e.  _V  ->  ( `' G " {  .0.  } )  e.  _V )
97, 8syl 15 . . 3  |-  ( G  e.  F  ->  ( `' G " {  .0.  } )  e.  _V )
10 cnveq 4855 . . . . 5  |-  ( f  =  G  ->  `' f  =  `' G
)
1110imaeq1d 5011 . . . 4  |-  ( f  =  G  ->  ( `' f " {  .0.  } )  =  ( `' G " {  .0.  } ) )
12 eqid 2283 . . . 4  |-  ( f  e.  F  |->  ( `' f " {  .0.  } ) )  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) )
1311, 12fvmptg 5600 . . 3  |-  ( ( G  e.  F  /\  ( `' G " {  .0.  } )  e.  _V )  ->  ( ( f  e.  F  |->  ( `' f
" {  .0.  }
) ) `  G
)  =  ( `' G " {  .0.  } ) )
149, 13mpdan 649 . 2  |-  ( G  e.  F  ->  (
( f  e.  F  |->  ( `' f " {  .0.  } ) ) `
 G )  =  ( `' G " {  .0.  } ) )
156, 14sylan9eq 2335 1  |-  ( ( W  e.  X  /\  G  e.  F )  ->  ( K `  G
)  =  ( `' G " {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = 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:  ellkr  29279  lkr0f  29284
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-pw 3627  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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