Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lkrval Unicode version

Theorem lkrval 29900
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 29899 . . 3  |-  ( W  e.  X  ->  K  =  ( f  e.  F  |->  ( `' f
" {  .0.  }
) ) )
65fveq1d 5543 . 2  |-  ( W  e.  X  ->  ( K `  G )  =  ( ( f  e.  F  |->  ( `' f " {  .0.  } ) ) `  G
) )
7 cnvexg 5224 . . . 4  |-  ( G  e.  F  ->  `' G  e.  _V )
8 imaexg 5042 . . . 4  |-  ( `' G  e.  _V  ->  ( `' G " {  .0.  } )  e.  _V )
97, 8syl 15 . . 3  |-  ( G  e.  F  ->  ( `' G " {  .0.  } )  e.  _V )
10 cnveq 4871 . . . . 5  |-  ( f  =  G  ->  `' f  =  `' G
)
1110imaeq1d 5027 . . . 4  |-  ( f  =  G  ->  ( `' f " {  .0.  } )  =  ( `' G " {  .0.  } ) )
12 eqid 2296 . . . 4  |-  ( f  e.  F  |->  ( `' f " {  .0.  } ) )  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) )
1311, 12fvmptg 5616 . . 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 2348 1  |-  ( ( W  e.  X  /\  G  e.  F )  ->  ( K `  G
)  =  ( `' G " {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    e. cmpt 4093   `'ccnv 4704   "cima 4708   ` cfv 5271  Scalarcsca 13227   0gc0g 13416  LFnlclfn 29869  LKerclk 29897
This theorem is referenced by:  ellkr  29901  lkr0f  29906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-lkr 29898
  Copyright terms: Public domain W3C validator