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Theorem lflf 35204
Description: A linear functional is a function from vectors to scalars. (lnfnfi 27161 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 2454 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
3 lflf.d . . 3  |-  D  =  (Scalar `  W )
4 eqid 2454 . . 3  |-  ( .s
`  W )  =  ( .s `  W
)
5 lflf.k . . 3  |-  K  =  ( Base `  D
)
6 eqid 2454 . . 3  |-  ( +g  `  D )  =  ( +g  `  D )
7 eqid 2454 . . 3  |-  ( .r
`  D )  =  ( .r `  D
)
8 lflf.f . . 3  |-  F  =  (LFnl `  W )
91, 2, 3, 4, 5, 6, 7, 8islfl 35201 . 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 621 1  |-  ( ( W  e.  X  /\  G  e.  F )  ->  G : V --> K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787   .rcmulr 14788  Scalarcsca 14790   .scvsca 14791  LFnlclfn 35198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-lfl 35199
This theorem is referenced by:  lflcl  35205  lfl1  35211  lfladdcl  35212  lfladdcom  35213  lfladdass  35214  lfladd0l  35215  lflnegl  35217  lflvscl  35218  lflvsdi1  35219  lflvsdi2  35220  lflvsass  35222  lfl0sc  35223  lfl1sc  35225  ellkr  35230  lkr0f  35235  lkrsc  35238  eqlkr2  35241  eqlkr3  35242  ldualvaddval  35272  ldualvsval  35279
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