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Theorem lkrf0 29891
Description: The value of a functional at a member of its kernel is zero. (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
lkrf0.d  |-  D  =  (Scalar `  W )
lkrf0.o  |-  .0.  =  ( 0g `  D )
lkrf0.f  |-  F  =  (LFnl `  W )
lkrf0.k  |-  K  =  (LKer `  W )
Assertion
Ref Expression
lkrf0  |-  ( ( W  e.  Y  /\  G  e.  F  /\  X  e.  ( K `  G ) )  -> 
( G `  X
)  =  .0.  )

Proof of Theorem lkrf0
StepHypRef Expression
1 eqid 2436 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 lkrf0.d . . . 4  |-  D  =  (Scalar `  W )
3 lkrf0.o . . . 4  |-  .0.  =  ( 0g `  D )
4 lkrf0.f . . . 4  |-  F  =  (LFnl `  W )
5 lkrf0.k . . . 4  |-  K  =  (LKer `  W )
61, 2, 3, 4, 5ellkr 29887 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
( X  e.  (
Base `  W )  /\  ( G `  X
)  =  .0.  )
) )
76simplbda 608 . 2  |-  ( ( ( W  e.  Y  /\  G  e.  F
)  /\  X  e.  ( K `  G ) )  ->  ( G `  X )  =  .0.  )
873impa 1148 1  |-  ( ( W  e.  Y  /\  G  e.  F  /\  X  e.  ( K `  G ) )  -> 
( G `  X
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454   Basecbs 13469  Scalarcsca 13532   0gc0g 13723  LFnlclfn 29855  LKerclk 29883
This theorem is referenced by:  lkrlss  29893  lkrshp  29903  lkrin  29962  lcfrlem12N  32352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-lfl 29856  df-lkr 29884
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