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Theorem ellkr 29901
Description: Membership in the kernel of a functional. (elnlfn 22524 analog.) (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
lkrfval2.v  |-  V  =  ( Base `  W
)
lkrfval2.d  |-  D  =  (Scalar `  W )
lkrfval2.o  |-  .0.  =  ( 0g `  D )
lkrfval2.f  |-  F  =  (LFnl `  W )
lkrfval2.k  |-  K  =  (LKer `  W )
Assertion
Ref Expression
ellkr  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
( X  e.  V  /\  ( G `  X
)  =  .0.  )
) )

Proof of Theorem ellkr
StepHypRef Expression
1 lkrfval2.d . . . 4  |-  D  =  (Scalar `  W )
2 lkrfval2.o . . . 4  |-  .0.  =  ( 0g `  D )
3 lkrfval2.f . . . 4  |-  F  =  (LFnl `  W )
4 lkrfval2.k . . . 4  |-  K  =  (LKer `  W )
51, 2, 3, 4lkrval 29900 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( K `  G
)  =  ( `' G " {  .0.  } ) )
65eleq2d 2363 . 2  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
X  e.  ( `' G " {  .0.  } ) ) )
7 eqid 2296 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
8 lkrfval2.v . . . . 5  |-  V  =  ( Base `  W
)
91, 7, 8, 3lflf 29875 . . . 4  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  G : V --> ( Base `  D ) )
10 ffn 5405 . . . 4  |-  ( G : V --> ( Base `  D )  ->  G  Fn  V )
11 elpreima 5661 . . . 4  |-  ( G  Fn  V  ->  ( X  e.  ( `' G " {  .0.  }
)  <->  ( X  e.  V  /\  ( G `
 X )  e. 
{  .0.  } ) ) )
129, 10, 113syl 18 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( `' G " {  .0.  } )  <->  ( X  e.  V  /\  ( G `
 X )  e. 
{  .0.  } ) ) )
13 fvex 5555 . . . . 5  |-  ( G `
 X )  e. 
_V
1413elsnc 3676 . . . 4  |-  ( ( G `  X )  e.  {  .0.  }  <->  ( G `  X )  =  .0.  )
1514anbi2i 675 . . 3  |-  ( ( X  e.  V  /\  ( G `  X )  e.  {  .0.  }
)  <->  ( X  e.  V  /\  ( G `
 X )  =  .0.  ) )
1612, 15syl6bb 252 . 2  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( `' G " {  .0.  } )  <->  ( X  e.  V  /\  ( G `
 X )  =  .0.  ) ) )
176, 16bitrd 244 1  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
( X  e.  V  /\  ( G `  X
)  =  .0.  )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653   `'ccnv 4704   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271   Basecbs 13164  Scalarcsca 13227   0gc0g 13416  LFnlclfn 29869  LKerclk 29897
This theorem is referenced by:  lkrval2  29902  ellkr2  29903  lkrcl  29904  lkrf0  29905  lkrlss  29907  lkrsc  29909  eqlkr  29911  lkrlsp  29914  lkrlsp2  29915  lshpkr  29929  lkrin  29976  dochfln0  32289
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-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-lfl 29870  df-lkr 29898
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