Theorem hlbn 8588

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

Assertion

Ref	Expression
hlbn	|- (U e. CHil -> U e. CBan)

Proof of Theorem hlbn

Step	Hyp	Ref	Expression
1		ishl 8587	. 2 |- (U e. CHil <-> (U e. CBan /\ U e. CPreHil))
2	1	pm3.26bi 322	1 |- (U e. CHil -> U e. CBan)

Syntax hints:   -> wi 3   e. wcel 960  CPreHilcphl 8467  CBancbn 8518  CHilchl 8585

This theorem is referenced by:  hlrel 8590  hlnv 8591  hlcms 8594  htthlem11 8626

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-hl 8586

Copyright terms: Public domain