Theorem hlbn 8577

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 8576	. 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 958  CPreHilcphl 8456  CBancbn 8507  CHilchl 8574

This theorem is referenced by:  hlrel 8579  hlnv 8580  hlcms 8583  htthlem11 8615

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-hl 8575

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