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Theorem hlcms 23896
Description: Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015.)
Assertion
Ref Expression
hlcms (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)

Proof of Theorem hlcms
StepHypRef Expression
1 hlbn 23893 . 2 (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
2 bncms 23874 . 2 (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
31, 2syl 17 1 (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  CMetSpccms 23862  Bancbn 23863  ℂHilchl 23864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-bn 23866  df-hl 23867
This theorem is referenced by:  pjthlem2  23968
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