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Theorem hlobn 30141
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 30140 . 2 (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
21simplbi 499 1 (𝑈 ∈ CHilOLD𝑈 ∈ CBan)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  CPreHilOLDccphlo 30065  CBanccbn 30115  CHilOLDchlo 30138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-hlo 30139
This theorem is referenced by:  hlrel  30143  hlnv  30144  hlcmet  30147  htthlem  30170
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