Theorem hlbn 23968

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 23967	. 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
2	1	simplbi 500	1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)

Syntax hints:   → wi 4   ∈ wcel 2114  ℂPreHilccph 23772  Bancbn 23938  ℂHilchl 23939

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-in 3945  df-hl 23942

This theorem is referenced by:  hlcms  23971  hlprlem  23972  cmslsschl  23982  chlcsschl  23983