Theorem hlph 8524

Description: Every complex Hilbert space is an inner product space (also called a pre-Hilbert space).

Assertion

Ref	Expression
hlph	|- (U e. CHil -> U e. CPreHil)

Proof of Theorem hlph

Step	Hyp	Ref	Expression
1		ishl 8522	. 2 |- (U e. CHil <-> (U e. CBan /\ U e. CPreHil))
2	1	pm3.27bi 326	1 |- (U e. CHil -> U e. CPreHil)

Syntax hints:   -> wi 3   e. wcel 955  CPreHilcphl 8402  CBancbn 8453  CHilchl 8520

This theorem is referenced by:  hlipdir 8544  hlipass 8545  htthlem5 8554  htthlem6 8555

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-in 2041  df-hl 8521

Copyright terms: Public domain