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Theorem lflf 36358
Description: A linear functional is a function from vectors to scalars. (lnfnfi 29828 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 ((𝑊𝑋𝐺𝐹) → 𝐺:𝑉𝐾)

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 2801 . . 3 (+g𝑊) = (+g𝑊)
3 lflf.d . . 3 𝐷 = (Scalar‘𝑊)
4 eqid 2801 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
5 lflf.k . . 3 𝐾 = (Base‘𝐷)
6 eqid 2801 . . 3 (+g𝐷) = (+g𝐷)
7 eqid 2801 . . 3 (.r𝐷) = (.r𝐷)
8 lflf.f . . 3 𝐹 = (LFnl‘𝑊)
91, 2, 3, 4, 5, 6, 7, 8islfl 36355 . 2 (𝑊𝑋 → (𝐺𝐹 ↔ (𝐺:𝑉𝐾 ∧ ∀𝑟𝐾𝑥𝑉𝑦𝑉 (𝐺‘((𝑟( ·𝑠𝑊)𝑥)(+g𝑊)𝑦)) = ((𝑟(.r𝐷)(𝐺𝑥))(+g𝐷)(𝐺𝑦)))))
109simprbda 502 1 ((𝑊𝑋𝐺𝐹) → 𝐺:𝑉𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  wf 6324  cfv 6328  (class class class)co 7139  Basecbs 16479  +gcplusg 16561  .rcmulr 16562  Scalarcsca 16564   ·𝑠 cvsca 16565  LFnlclfn 36352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-lfl 36353
This theorem is referenced by:  lflcl  36359  lfl1  36365  lfladdcl  36366  lfladdcom  36367  lfladdass  36368  lfladd0l  36369  lflnegl  36371  lflvscl  36372  lflvsdi1  36373  lflvsdi2  36374  lflvsass  36376  lfl0sc  36377  lfl1sc  36379  ellkr  36384  lkr0f  36389  lkrsc  36392  eqlkr2  36395  eqlkr3  36396  ldualvaddval  36426  ldualvsval  36433
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