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Theorem lflf 33014
Description: A linear functional is a function from vectors to scalars. (lnfnfi 25580 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 2451 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
3 lflf.d . . 3  |-  D  =  (Scalar `  W )
4 eqid 2451 . . 3  |-  ( .s
`  W )  =  ( .s `  W
)
5 lflf.k . . 3  |-  K  =  ( Base `  D
)
6 eqid 2451 . . 3  |-  ( +g  `  D )  =  ( +g  `  D )
7 eqid 2451 . . 3  |-  ( .r
`  D )  =  ( .r `  D
)
8 lflf.f . . 3  |-  F  =  (LFnl `  W )
91, 2, 3, 4, 5, 6, 7, 8islfl 33011 . 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 623 1  |-  ( ( W  e.  X  /\  G  e.  F )  ->  G : V --> K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   -->wf 5512   ` cfv 5516  (class class class)co 6190   Basecbs 14276   +g cplusg 14340   .rcmulr 14341  Scalarcsca 14343   .scvsca 14344  LFnlclfn 33008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-map 7316  df-lfl 33009
This theorem is referenced by:  lflcl  33015  lfl1  33021  lfladdcl  33022  lfladdcom  33023  lfladdass  33024  lfladd0l  33025  lflnegl  33027  lflvscl  33028  lflvsdi1  33029  lflvsdi2  33030  lflvsass  33032  lfl0sc  33033  lfl1sc  33035  ellkr  33040  lkr0f  33045  lkrsc  33048  eqlkr2  33051  eqlkr3  33052  ldualvaddval  33082  ldualvsval  33089
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