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Theorem hlcph 23650
 Description: Every subcomplex Hilbert space is a subcomplex 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 23648 . 2 (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
21simprbi 497 1 (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2081  ℂPreHilccph 23453  Bancbn 23619  ℂHilchl 23620 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-in 3866  df-hl 23623 This theorem is referenced by:  hlphl  23651  hlprlem  23653  cmslsschl  23663  chlcsschl  23664  pjthlem1  23723  pjthlem2  23724  cldcss  23727
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