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Theorem lnfnfi 29209
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 29052 . 2 (𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ)
31, 2ax-mp 5 1 𝑇: ℋ⟶ℂ
Colors of variables: wff setvar class
Syntax hints:  wcel 2139  wf 6045  cc 10126  chil 28085  LinFnclf 28120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-hilex 28165
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-lnfn 29016
This theorem is referenced by:  lnfn0i  29210  lnfnaddi  29211  lnfnmuli  29212  lnfnsubi  29214  nmbdfnlbi  29217  nmcfnexi  29219  nmcfnlbi  29220  lnfnconi  29223  nlelshi  29228  nlelchi  29229  riesz3i  29230  riesz4i  29231
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