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Theorem lkrval 29823
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 29822 . . 3  |-  ( W  e.  X  ->  K  =  ( f  e.  F  |->  ( `' f
" {  .0.  }
) ) )
65fveq1d 5722 . 2  |-  ( W  e.  X  ->  ( K `  G )  =  ( ( f  e.  F  |->  ( `' f " {  .0.  } ) ) `  G
) )
7 cnvexg 5397 . . . 4  |-  ( G  e.  F  ->  `' G  e.  _V )
8 imaexg 5209 . . . 4  |-  ( `' G  e.  _V  ->  ( `' G " {  .0.  } )  e.  _V )
97, 8syl 16 . . 3  |-  ( G  e.  F  ->  ( `' G " {  .0.  } )  e.  _V )
10 cnveq 5038 . . . . 5  |-  ( f  =  G  ->  `' f  =  `' G
)
1110imaeq1d 5194 . . . 4  |-  ( f  =  G  ->  ( `' f " {  .0.  } )  =  ( `' G " {  .0.  } ) )
12 eqid 2435 . . . 4  |-  ( f  e.  F  |->  ( `' f " {  .0.  } ) )  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) )
1311, 12fvmptg 5796 . . 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 2487 1  |-  ( ( W  e.  X  /\  G  e.  F )  ->  ( K `  G
)  =  ( `' G " {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806    e. cmpt 4258   `'ccnv 4869   "cima 4873   ` cfv 5446  Scalarcsca 13524   0gc0g 13715  LFnlclfn 29792  LKerclk 29820
This theorem is referenced by:  ellkr  29824  lkr0f  29829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-lkr 29821
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