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Theorem ellkr2 35232
Description: Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.)
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 )
ellkr2.w  |-  ( ph  ->  W  e.  Y )
ellkr2.g  |-  ( ph  ->  G  e.  F )
ellkr2.x  |-  ( ph  ->  X  e.  V )
Assertion
Ref Expression
ellkr2  |-  ( ph  ->  ( X  e.  ( K `  G )  <-> 
( G `  X
)  =  .0.  )
)

Proof of Theorem ellkr2
StepHypRef Expression
1 ellkr2.w . . 3  |-  ( ph  ->  W  e.  Y )
2 ellkr2.g . . 3  |-  ( ph  ->  G  e.  F )
3 lkrfval2.v . . . 4  |-  V  =  ( Base `  W
)
4 lkrfval2.d . . . 4  |-  D  =  (Scalar `  W )
5 lkrfval2.o . . . 4  |-  .0.  =  ( 0g `  D )
6 lkrfval2.f . . . 4  |-  F  =  (LFnl `  W )
7 lkrfval2.k . . . 4  |-  K  =  (LKer `  W )
83, 4, 5, 6, 7ellkr 35230 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
( X  e.  V  /\  ( G `  X
)  =  .0.  )
) )
91, 2, 8syl2anc 659 . 2  |-  ( ph  ->  ( X  e.  ( K `  G )  <-> 
( X  e.  V  /\  ( G `  X
)  =  .0.  )
) )
10 ellkr2.x . . 3  |-  ( ph  ->  X  e.  V )
1110biantrurd 506 . 2  |-  ( ph  ->  ( ( G `  X )  =  .0.  <->  ( X  e.  V  /\  ( G `  X )  =  .0.  ) ) )
129, 11bitr4d 256 1  |-  ( ph  ->  ( X  e.  ( K `  G )  <-> 
( G `  X
)  =  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   ` cfv 5570   Basecbs 14719  Scalarcsca 14790   0gc0g 14932  LFnlclfn 35198  LKerclk 35226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-lfl 35199  df-lkr 35227
This theorem is referenced by:  lclkrlem2f  37655  lclkrlem2n  37663  lcfrlem3  37687  lcfrlem25  37710  hdmapellkr  38060  hdmapip0  38061  hdmapinvlem1  38064
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