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Theorem hlph 28593
Description: Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
hlph (𝑈 ∈ CHilOLD𝑈 ∈ CPreHilOLD)

Proof of Theorem hlph
StepHypRef Expression
1 ishlo 28591 . 2 (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
21simprbi 497 1 (𝑈 ∈ CHilOLD𝑈 ∈ CPreHilOLD)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  CPreHilOLDccphlo 28516  CBanccbn 28566  CHilOLDchlo 28589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940  df-hlo 28590
This theorem is referenced by:  hlpar2  28600  hlpar  28601  hlipdir  28616  hlipass  28617  htthlem  28621
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