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Theorem lkrcl 29258
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 2380 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2380 . . . 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 29255 . . 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 1649    e. wcel 1717   ` cfv 5387   Basecbs 13389  Scalarcsca 13452   0gc0g 13643  LFnlclfn 29223  LKerclk 29251
This theorem is referenced by:  lkrlss  29261  lkrin  29330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-map 6949  df-lfl 29224  df-lkr 29252
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