Theorem hlobn 30980

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 30979	. 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
2	1	simplbi 497	1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan)

Syntax hints:   → wi 4   ∈ wcel 2115  CPreHil_OLDccphlo 30904  CBanccbn 30954  CHil_OLDchlo 30977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-ext 2708

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1546  df-ex 1783  df-sb 2070  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3430  df-in 3893  df-hlo 30978

This theorem is referenced by:  hlrel  30982  hlnv  30983  hlcmet  30986  htthlem  31009