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Theorem hlbn 25285
Description: Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
Assertion
Ref Expression
hlbn (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)

Proof of Theorem hlbn
StepHypRef Expression
1 ishl 25284 . 2 (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
21simplbi 497 1 (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  ℂPreHilccph 25088  Bancbn 25255  ℂHilchl 25256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-in 3952  df-hl 25259
This theorem is referenced by:  hlcms  25288  hlprlem  25289  cmslsschl  25299  chlcsschl  25300
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