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Theorem ellkr 29279
Description: Membership in the kernel of a functional. (elnlfn 22508 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 29278 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( K `  G
)  =  ( `' G " {  .0.  } ) )
65eleq2d 2350 . 2  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
X  e.  ( `' G " {  .0.  } ) ) )
7 eqid 2283 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
8 lkrfval2.v . . . . 5  |-  V  =  ( Base `  W
)
91, 7, 8, 3lflf 29253 . . . 4  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  G : V --> ( Base `  D ) )
10 ffn 5389 . . . 4  |-  ( G : V --> ( Base `  D )  ->  G  Fn  V )
11 elpreima 5645 . . . 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 5539 . . . . 5  |-  ( G `
 X )  e. 
_V
1413elsnc 3663 . . . 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 1623    e. wcel 1684   {csn 3640   `'ccnv 4688   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255   Basecbs 13148  Scalarcsca 13211   0gc0g 13400  LFnlclfn 29247  LKerclk 29275
This theorem is referenced by:  lkrval2  29280  ellkr2  29281  lkrcl  29282  lkrf0  29283  lkrlss  29285  lkrsc  29287  eqlkr  29289  lkrlsp  29292  lkrlsp2  29293  lshpkr  29307  lkrin  29354  dochfln0  31667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-lfl 29248  df-lkr 29276
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