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Theorem hlobn 30917
Description: Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
hlobn (𝑈 ∈ CHilOLD𝑈 ∈ CBan)

Proof of Theorem hlobn
StepHypRef Expression
1 ishlo 30916 . 2 (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
21simplbi 497 1 (𝑈 ∈ CHilOLD𝑈 ∈ CBan)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  CPreHilOLDccphlo 30841  CBanccbn 30891  CHilOLDchlo 30914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-in 3970  df-hlo 30915
This theorem is referenced by:  hlrel  30919  hlnv  30920  hlcmet  30923  htthlem  30946
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