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Theorem lkrcl 29828
Description: A member of the kernel of a functional is a vector. (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
lkrcl.v  |-  V  =  ( Base `  W
)
lkrcl.f  |-  F  =  (LFnl `  W )
lkrcl.k  |-  K  =  (LKer `  W )
Assertion
Ref Expression
lkrcl  |-  ( ( W  e.  Y  /\  G  e.  F  /\  X  e.  ( K `  G ) )  ->  X  e.  V )

Proof of Theorem lkrcl
StepHypRef Expression
1 lkrcl.v . . . 4  |-  V  =  ( Base `  W
)
2 eqid 2436 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2436 . . . 4  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
4 lkrcl.f . . . 4  |-  F  =  (LFnl `  W )
5 lkrcl.k . . . 4  |-  K  =  (LKer `  W )
61, 2, 3, 4, 5ellkr 29825 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
( X  e.  V  /\  ( G `  X
)  =  ( 0g
`  (Scalar `  W )
) ) ) )
76simprbda 607 . 2  |-  ( ( ( W  e.  Y  /\  G  e.  F
)  /\  X  e.  ( K `  G ) )  ->  X  e.  V )
873impa 1148 1  |-  ( ( W  e.  Y  /\  G  e.  F  /\  X  e.  ( K `  G ) )  ->  X  e.  V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5447   Basecbs 13462  Scalarcsca 13525   0gc0g 13716  LFnlclfn 29793  LKerclk 29821
This theorem is referenced by:  lkrlss  29831  lkrin  29900
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 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
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 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-map 7013  df-lfl 29794  df-lkr 29822
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