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Theorem lnfnf 29819
Description: A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
lnfnf (𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ)

Proof of Theorem lnfnf
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ellnfn 29818 . 2 (𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
21simplbi 501 1 (𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2113  wral 3053  wf 6336  cfv 6340  (class class class)co 7171  cc 10614   + caddc 10619   · cmul 10621  chba 28854   + cva 28855   · csm 28856  LinFnclf 28889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7480  ax-cnex 10672  ax-hilex 28934
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3683  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-ov 7174  df-oprab 7175  df-mpo 7176  df-map 8440  df-lnfn 29783
This theorem is referenced by:  nmfn0  29922  lnfnfi  29976  rnbra  30042
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