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Theorem lkrcl 29282
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 2283 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2283 . . . 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 29279 . . 3  |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
( X  e.  V  /\  ( G `  X
)  =  ( 0g
`  (Scalar `  W )
) ) ) )
76simprbda 606 . 2  |-  ( ( ( W  e.  Y  /\  G  e.  F
)  /\  X  e.  ( K `  G ) )  ->  X  e.  V )
873impa 1146 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255   Basecbs 13148  Scalarcsca 13211   0gc0g 13400  LFnlclfn 29247  LKerclk 29275
This theorem is referenced by:  lkrlss  29285  lkrin  29354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-lfl 29248  df-lkr 29276
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