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Theorem lflf 29861
Description: A linear functional is a function from vectors to scalars. (lnfnfi 23544 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 2436 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
3 lflf.d . . 3  |-  D  =  (Scalar `  W )
4 eqid 2436 . . 3  |-  ( .s
`  W )  =  ( .s `  W
)
5 lflf.k . . 3  |-  K  =  ( Base `  D
)
6 eqid 2436 . . 3  |-  ( +g  `  D )  =  ( +g  `  D )
7 eqid 2436 . . 3  |-  ( .r
`  D )  =  ( .r `  D
)
8 lflf.f . . 3  |-  F  =  (LFnl `  W )
91, 2, 3, 4, 5, 6, 7, 8islfl 29858 . 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 607 1  |-  ( ( W  e.  X  /\  G  e.  F )  ->  G : V --> K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533  LFnlclfn 29855
This theorem is referenced by:  lflcl  29862  lfl1  29868  lfladdcl  29869  lfladdcom  29870  lfladdass  29871  lfladd0l  29872  lflnegl  29874  lflvscl  29875  lflvsdi1  29876  lflvsdi2  29877  lflvsass  29879  lfl0sc  29880  lfl1sc  29882  ellkr  29887  lkr0f  29892  lkrsc  29895  eqlkr2  29898  eqlkr3  29899  ldualvaddval  29929  ldualvsval  29936
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-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-rab 2714  df-v 2958  df-sbc 3162  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-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-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-lfl 29856
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