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Theorem ellkr 35227
Description: Membership in the kernel of a functional. (elnlfn 26963 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 35226 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( K `  G
)  =  ( `' G " {  .0.  } ) )
65eleq2d 2452 . 2  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
X  e.  ( `' G " {  .0.  } ) ) )
7 eqid 2382 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
8 lkrfval2.v . . . . 5  |-  V  =  ( Base `  W
)
91, 7, 8, 3lflf 35201 . . . 4  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  G : V --> ( Base `  D ) )
10 ffn 5639 . . . 4  |-  ( G : V --> ( Base `  D )  ->  G  Fn  V )
11 elpreima 5909 . . . 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 5784 . . . . 5  |-  ( G `
 X )  e. 
_V
1413elsnc 3968 . . . 4  |-  ( ( G `  X )  e.  {  .0.  }  <->  ( G `  X )  =  .0.  )
1514anbi2i 692 . . 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 367    = wceq 1399    e. wcel 1826   {csn 3944   `'ccnv 4912   "cima 4916    Fn wfn 5491   -->wf 5492   ` cfv 5496   Basecbs 14634  Scalarcsca 14705   0gc0g 14847  LFnlclfn 35195  LKerclk 35223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-map 7340  df-lfl 35196  df-lkr 35224
This theorem is referenced by:  lkrval2  35228  ellkr2  35229  lkrcl  35230  lkrf0  35231  lkrlss  35233  lkrsc  35235  eqlkr  35237  lkrlsp  35240  lkrlsp2  35241  lshpkr  35255  lkrin  35302  dochfln0  37617
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