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Theorem hlcph 25214
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 25212 . 2 (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
21simprbi 496 1 (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  ℂPreHilccph 25016  Bancbn 25183  ℂHilchl 25184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3947  df-hl 25187
This theorem is referenced by:  hlphl  25215  hlprlem  25217  cmslsschl  25227  chlcsschl  25228  pjthlem1  25287  pjthlem2  25288  cldcss  25291
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