HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  lnfnfi Structured version   Visualization version   GIF version

Theorem lnfnfi 31864
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: β„‹βŸΆβ„‚

Proof of Theorem lnfnfi
StepHypRef Expression
1 lnfnl.1 . 2 𝑇 ∈ LinFn
2 lnfnf 31707 . 2 (𝑇 ∈ LinFn β†’ 𝑇: β„‹βŸΆβ„‚)
31, 2ax-mp 5 1 𝑇: β„‹βŸΆβ„‚
Colors of variables: wff setvar class
Syntax hints:   ∈ wcel 2099  βŸΆwf 6544  β„‚cc 11137   β„‹chba 30742  LinFnclf 30777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-hilex 30822
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8847  df-lnfn 31671
This theorem is referenced by:  lnfn0i  31865  lnfnaddi  31866  lnfnmuli  31867  lnfnsubi  31869  nmbdfnlbi  31872  nmcfnexi  31874  nmcfnlbi  31875  lnfnconi  31878  nlelshi  31883  nlelchi  31884  riesz3i  31885  riesz4i  31886
  Copyright terms: Public domain W3C validator