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Theorem hlbn 25215
Description: Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.)
Assertion
Ref Expression
hlbn (π‘Š ∈ β„‚Hil β†’ π‘Š ∈ Ban)

Proof of Theorem hlbn
StepHypRef Expression
1 ishl 25214 . 2 (π‘Š ∈ β„‚Hil ↔ (π‘Š ∈ Ban ∧ π‘Š ∈ β„‚PreHil))
21simplbi 497 1 (π‘Š ∈ β„‚Hil β†’ π‘Š ∈ Ban)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2098  β„‚PreHilccph 25018  Bancbn 25185  β„‚Hilchl 25186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-hl 25189
This theorem is referenced by:  hlcms  25218  hlprlem  25219  cmslsschl  25229  chlcsschl  25230
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