Theorem hlbn 25288

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

Step	Hyp	Ref	Expression
1		ishl 25287	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2	1	simplbi 497	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)

Syntax hints:   → wi 4   ∈ wcel 2111  ℂPreHilccph 25091  Bancbn 25258  ℂHilchl 25259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-in 3909  df-hl 25262

This theorem is referenced by:  hlcms  25291  hlprlem  25292  cmslsschl  25302  chlcsschl  25303