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Theorem hlph 27129
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 27127 . 2 (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
21simprbi 479 1 (𝑈 ∈ CHilOLD𝑈 ∈ CPreHilOLD)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  CPreHilOLDccphlo 27051  CBanccbn 27102  CHilOLDchlo 27125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-hlo 27126
This theorem is referenced by:  hlpar2  27136  hlpar  27137  hlipdir  27152  hlipass  27153  htthlem  27158
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