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Theorem hlph 29539
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 29537 . 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 29462  CBanccbn 29512  CHilOLDchlo 29535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-in 3905  df-hlo 29536
This theorem is referenced by:  hlpar2  29546  hlpar  29547  hlipdir  29562  hlipass  29563  htthlem  29567
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