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Theorem ellkr 33040
Description: Membership in the kernel of a functional. (elnlfn 25467 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 33039 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( K `  G
)  =  ( `' G " {  .0.  } ) )
65eleq2d 2521 . 2  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
X  e.  ( `' G " {  .0.  } ) ) )
7 eqid 2451 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
8 lkrfval2.v . . . . 5  |-  V  =  ( Base `  W
)
91, 7, 8, 3lflf 33014 . . . 4  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  G : V --> ( Base `  D ) )
10 ffn 5657 . . . 4  |-  ( G : V --> ( Base `  D )  ->  G  Fn  V )
11 elpreima 5922 . . . 4  |-  ( G  Fn  V  ->  ( X  e.  ( `' G " {  .0.  }
)  <->  ( X  e.  V  /\  ( G `
 X )  e. 
{  .0.  } ) ) )
129, 10, 113syl 20 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( `' G " {  .0.  } )  <->  ( X  e.  V  /\  ( G `
 X )  e. 
{  .0.  } ) ) )
13 fvex 5799 . . . . 5  |-  ( G `
 X )  e. 
_V
1413elsnc 3999 . . . 4  |-  ( ( G `  X )  e.  {  .0.  }  <->  ( G `  X )  =  .0.  )
1514anbi2i 694 . . 3  |-  ( ( X  e.  V  /\  ( G `  X )  e.  {  .0.  }
)  <->  ( X  e.  V  /\  ( G `
 X )  =  .0.  ) )
1612, 15syl6bb 261 . 2  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( `' G " {  .0.  } )  <->  ( X  e.  V  /\  ( G `
 X )  =  .0.  ) ) )
176, 16bitrd 253 1  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
( X  e.  V  /\  ( G `  X
)  =  .0.  )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {csn 3975   `'ccnv 4937   "cima 4941    Fn wfn 5511   -->wf 5512   ` cfv 5516   Basecbs 14276  Scalarcsca 14343   0gc0g 14480  LFnlclfn 33008  LKerclk 33036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-map 7316  df-lfl 33009  df-lkr 33037
This theorem is referenced by:  lkrval2  33041  ellkr2  33042  lkrcl  33043  lkrf0  33044  lkrlss  33046  lkrsc  33048  eqlkr  33050  lkrlsp  33053  lkrlsp2  33054  lshpkr  33068  lkrin  33115  dochfln0  35428
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