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Theorem lnfnfi 31576
Description: A linear Hilbert space functional is a functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnfnl.1 𝑇 ∈ LinFn
Assertion
Ref Expression
lnfnfi 𝑇: ℋ⟶ℂ

Proof of Theorem lnfnfi
StepHypRef Expression
1 lnfnl.1 . 2 𝑇 ∈ LinFn
2 lnfnf 31419 . 2 (𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ)
31, 2ax-mp 5 1 𝑇: ℋ⟶ℂ
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  wf 6539  cc 11114  chba 30454  LinFnclf 30489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-hilex 30534
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8828  df-lnfn 31383
This theorem is referenced by:  lnfn0i  31577  lnfnaddi  31578  lnfnmuli  31579  lnfnsubi  31581  nmbdfnlbi  31584  nmcfnexi  31586  nmcfnlbi  31587  lnfnconi  31590  nlelshi  31595  nlelchi  31596  riesz3i  31597  riesz4i  31598
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