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Theorem lnfnfi 23536
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 23379 . 2  |-  ( T  e.  LinFn  ->  T : ~H
--> CC )
31, 2ax-mp 8 1  |-  T : ~H
--> CC
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   -->wf 5442   CCcc 8980   ~Hchil 22414   LinFnclf 22449
This theorem is referenced by:  lnfn0i  23537  lnfnaddi  23538  lnfnmuli  23539  lnfnsubi  23541  nmbdfnlbi  23544  nmcfnexi  23546  nmcfnlbi  23547  lnfnconi  23550  nlelshi  23555  nlelchi  23556  riesz3i  23557  riesz4i  23558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-hilex 22494
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-lnfn 23343
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