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Theorem hlobn 30907
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 30906 . 2 (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
21simplbi 497 1 (𝑈 ∈ CHilOLD𝑈 ∈ CBan)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  CPreHilOLDccphlo 30831  CBanccbn 30881  CHilOLDchlo 30904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958  df-hlo 30905
This theorem is referenced by:  hlrel  30909  hlnv  30910  hlcmet  30913  htthlem  30936
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