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Theorem hlcph 22968
 Description: Every complex Hilbert space is a complex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
hlcph (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)

Proof of Theorem hlcph
StepHypRef Expression
1 ishl 22966 . 2 (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
21simprbi 479 1 (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  ℂPreHilccph 22774  Bancbn 22938  ℂHilchl 22939 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-hl 22942 This theorem is referenced by:  hlphl  22969  hlprlem  22971  pjthlem1  23016  pjthlem2  23017  cldcss  23020
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