Theorem hlobn 31520

Description: Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
hlobn	⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)

Proof of Theorem hlobn

Step	Hyp	Ref	Expression
1		ishlo 31519	. 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
2	1	simplbi 502	1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  CPreHil_OLDccphlo 31444  CBanccbn 31494  CHil_OLDchlo 31517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-hlo 31518

This theorem is used by:  hlrel  31522  hlnv  31523  hlcmet  31526  htthlem  31549