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Theorem ellkr 29888
Description: Membership in the kernel of a functional. (elnlfn 23432 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 29887 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( K `  G
)  =  ( `' G " {  .0.  } ) )
65eleq2d 2504 . 2  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
X  e.  ( `' G " {  .0.  } ) ) )
7 eqid 2437 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
8 lkrfval2.v . . . . 5  |-  V  =  ( Base `  W
)
91, 7, 8, 3lflf 29862 . . . 4  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  G : V --> ( Base `  D ) )
10 ffn 5592 . . . 4  |-  ( G : V --> ( Base `  D )  ->  G  Fn  V )
11 elpreima 5851 . . . 4  |-  ( G  Fn  V  ->  ( X  e.  ( `' G " {  .0.  }
)  <->  ( X  e.  V  /\  ( G `
 X )  e. 
{  .0.  } ) ) )
129, 10, 113syl 19 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( `' G " {  .0.  } )  <->  ( X  e.  V  /\  ( G `
 X )  e. 
{  .0.  } ) ) )
13 fvex 5743 . . . . 5  |-  ( G `
 X )  e. 
_V
1413elsnc 3838 . . . 4  |-  ( ( G `  X )  e.  {  .0.  }  <->  ( G `  X )  =  .0.  )
1514anbi2i 677 . . 3  |-  ( ( X  e.  V  /\  ( G `  X )  e.  {  .0.  }
)  <->  ( X  e.  V  /\  ( G `
 X )  =  .0.  ) )
1612, 15syl6bb 254 . 2  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( `' G " {  .0.  } )  <->  ( X  e.  V  /\  ( G `
 X )  =  .0.  ) ) )
176, 16bitrd 246 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   {csn 3815   `'ccnv 4878   "cima 4882    Fn wfn 5450   -->wf 5451   ` cfv 5455   Basecbs 13470  Scalarcsca 13533   0gc0g 13724  LFnlclfn 29856  LKerclk 29884
This theorem is referenced by:  lkrval2  29889  ellkr2  29890  lkrcl  29891  lkrf0  29892  lkrlss  29894  lkrsc  29896  eqlkr  29898  lkrlsp  29901  lkrlsp2  29902  lshpkr  29916  lkrin  29963  dochfln0  32276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-map 7021  df-lfl 29857  df-lkr 29885
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