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Theorem lkrval 35229
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 35228 . . 3  |-  ( W  e.  X  ->  K  =  ( f  e.  F  |->  ( `' f
" {  .0.  }
) ) )
65fveq1d 5850 . 2  |-  ( W  e.  X  ->  ( K `  G )  =  ( ( f  e.  F  |->  ( `' f " {  .0.  } ) ) `  G
) )
7 cnvexg 6719 . . . 4  |-  ( G  e.  F  ->  `' G  e.  _V )
8 imaexg 6710 . . . 4  |-  ( `' G  e.  _V  ->  ( `' G " {  .0.  } )  e.  _V )
97, 8syl 16 . . 3  |-  ( G  e.  F  ->  ( `' G " {  .0.  } )  e.  _V )
10 cnveq 5165 . . . . 5  |-  ( f  =  G  ->  `' f  =  `' G
)
1110imaeq1d 5324 . . . 4  |-  ( f  =  G  ->  ( `' f " {  .0.  } )  =  ( `' G " {  .0.  } ) )
12 eqid 2454 . . . 4  |-  ( f  e.  F  |->  ( `' f " {  .0.  } ) )  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) )
1311, 12fvmptg 5929 . . 3  |-  ( ( G  e.  F  /\  ( `' G " {  .0.  } )  e.  _V )  ->  ( ( f  e.  F  |->  ( `' f
" {  .0.  }
) ) `  G
)  =  ( `' G " {  .0.  } ) )
149, 13mpdan 666 . 2  |-  ( G  e.  F  ->  (
( f  e.  F  |->  ( `' f " {  .0.  } ) ) `
 G )  =  ( `' G " {  .0.  } ) )
156, 14sylan9eq 2515 1  |-  ( ( W  e.  X  /\  G  e.  F )  ->  ( K `  G
)  =  ( `' G " {  .0.  } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   {csn 4016    |-> cmpt 4497   `'ccnv 4987   "cima 4991   ` cfv 5570  Scalarcsca 14790   0gc0g 14932  LFnlclfn 35198  LKerclk 35226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-lkr 35227
This theorem is referenced by:  ellkr  35230  lkr0f  35235
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