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Theorem hlbn 24115
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 24114 . 2 (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
21simplbi 501 1 (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  ℂPreHilccph 23918  Bancbn 24085  ℂHilchl 24086
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 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-in 3850  df-hl 24089
This theorem is referenced by:  hlcms  24118  hlprlem  24119  cmslsschl  24129  chlcsschl  24130
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