Theorem hlbn 23960

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 23959	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2	1	simplbi 500	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)

Syntax hints:   → wi 4   ∈ wcel 2110  ℂPreHilccph 23764  Bancbn 23930  ℂHilchl 23931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-in 3942  df-hl 23934

This theorem is referenced by:  hlcms  23963  hlprlem  23964  cmslsschl  23974  chlcsschl  23975