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Theorem hlcms 24131
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 24128 . 2 (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
2 bncms 24109 . 2 (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
31, 2syl 17 1 (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  CMetSpccms 24097  Bancbn 24098  ℂHilchl 24099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-v 3402  df-un 3858  df-in 3860  df-ss 3870  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-iota 6308  df-fv 6358  df-bn 24101  df-hl 24102
This theorem is referenced by:  pjthlem2  24203
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