Theorem hlbn 25479

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 25478	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2	1	simplbi 501	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)

Syntax hints:   → wi 4   ∈ wcel 2145  ℂPreHilccph 25282  Bancbn 25449  ℂHilchl 25450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-in 3914  df-hl 25453

This theorem is referenced by:  hlcms  25482  hlprlem  25483  cmslsschl  25493  chlcsschl  25494