MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hlbn Structured version   Visualization version   GIF version

Theorem hlbn 23881
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 23880 . 2 (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
21simplbi 498 1 (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  ℂPreHilccph 23685  Bancbn 23851  ℂHilchl 23852
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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-v 3501  df-in 3946  df-hl 23855
This theorem is referenced by:  hlcms  23884  hlprlem  23885  cmslsschl  23895  chlcsschl  23896
  Copyright terms: Public domain W3C validator