Theorem hlobn 27744

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 e. CHilOLD -> U e. CBan )

Proof of Theorem hlobn

Step	Hyp	Ref	Expression
1		ishlo 27743	. 2 |- ( U e. CHilOLD <-> ( U e. CBan /\ U e. CPreHil OLD ) )
2	1	simplbi 476	1 |- ( U e. CHilOLD -> U e. CBan )

Syntax hints:   -> wi 4   e. wcel 1990   CPreHil OLDccphlo 27667   CBanccbn 27718   CHilOLDchlo 27741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-hlo 27742

This theorem is referenced by:  hlrel  27746  hlnv  27747  hlcmet  27750  htthlem  27774