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Theorem lnfnfi 25624
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  e. 
LinFn
Assertion
Ref Expression
lnfnfi  |-  T : ~H
--> CC

Proof of Theorem lnfnfi
StepHypRef Expression
1 lnfnl.1 . 2  |-  T  e. 
LinFn
2 lnfnf 25467 . 2  |-  ( T  e.  LinFn  ->  T : ~H
--> CC )
31, 2ax-mp 5 1  |-  T : ~H
--> CC
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758   -->wf 5525   CCcc 9395   ~Hchil 24500   LinFnclf 24535
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-hilex 24580
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329  df-lnfn 25431
This theorem is referenced by:  lnfn0i  25625  lnfnaddi  25626  lnfnmuli  25627  lnfnsubi  25629  nmbdfnlbi  25632  nmcfnexi  25634  nmcfnlbi  25635  lnfnconi  25638  nlelshi  25643  nlelchi  25644  riesz3i  25645  riesz4i  25646
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