Theorem hlbn 24871

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 24870	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2	1	simplbi 498	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)

Syntax hints:   → wi 4   ∈ wcel 2106  ℂPreHilccph 24674  Bancbn 24841  ℂHilchl 24842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-hl 24845

This theorem is referenced by:  hlcms  24874  hlprlem  24875  cmslsschl  24885  chlcsschl  24886