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Theorem lflf 29253
Description: A linear functional is a function from vectors to scalars. (lnfnfi 22621 analog.) (Contributed by NM, 15-Apr-2014.)
Hypotheses
Ref Expression
lflf.d  |-  D  =  (Scalar `  W )
lflf.k  |-  K  =  ( Base `  D
)
lflf.v  |-  V  =  ( Base `  W
)
lflf.f  |-  F  =  (LFnl `  W )
Assertion
Ref Expression
lflf  |-  ( ( W  e.  X  /\  G  e.  F )  ->  G : V --> K )

Proof of Theorem lflf
Dummy variables  x  r  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lflf.v . . 3  |-  V  =  ( Base `  W
)
2 eqid 2283 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
3 lflf.d . . 3  |-  D  =  (Scalar `  W )
4 eqid 2283 . . 3  |-  ( .s
`  W )  =  ( .s `  W
)
5 lflf.k . . 3  |-  K  =  ( Base `  D
)
6 eqid 2283 . . 3  |-  ( +g  `  D )  =  ( +g  `  D )
7 eqid 2283 . . 3  |-  ( .r
`  D )  =  ( .r `  D
)
8 lflf.f . . 3  |-  F  =  (LFnl `  W )
91, 2, 3, 4, 5, 6, 7, 8islfl 29250 . 2  |-  ( W  e.  X  ->  ( G  e.  F  <->  ( G : V --> K  /\  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( G `  ( (
r ( .s `  W ) x ) ( +g  `  W
) y ) )  =  ( ( r ( .r `  D
) ( G `  x ) ) ( +g  `  D ) ( G `  y
) ) ) ) )
109simprbda 606 1  |-  ( ( W  e.  X  /\  G  e.  F )  ->  G : V --> K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212  LFnlclfn 29247
This theorem is referenced by:  lflcl  29254  lfl1  29260  lfladdcl  29261  lfladdcom  29262  lfladdass  29263  lfladd0l  29264  lflnegl  29266  lflvscl  29267  lflvsdi1  29268  lflvsdi2  29269  lflvsass  29271  lfl0sc  29272  lfl1sc  29274  ellkr  29279  lkr0f  29284  lkrsc  29287  eqlkr2  29290  eqlkr3  29291  ldualvaddval  29321  ldualvsval  29328
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-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-rab 2552  df-v 2790  df-sbc 2992  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-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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-lfl 29248
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