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Theorem lnfnfi 29476
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 29319 . 2 (𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ)
31, 2ax-mp 5 1 𝑇: ℋ⟶ℂ
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  wf 6133  cc 10272  chba 28352  LinFnclf 28387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-hilex 28432
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-map 8144  df-lnfn 29283
This theorem is referenced by:  lnfn0i  29477  lnfnaddi  29478  lnfnmuli  29479  lnfnsubi  29481  nmbdfnlbi  29484  nmcfnexi  29486  nmcfnlbi  29487  lnfnconi  29490  nlelshi  29495  nlelchi  29496  riesz3i  29497  riesz4i  29498
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