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Theorem hlcms 25421
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 25418 . 2 (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
2 bncms 25399 . 2 (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
31, 2syl 17 1 (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  CMetSpccms 25387  Bancbn 25388  ℂHilchl 25389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6527  df-fv 6583  df-bn 25391  df-hl 25392
This theorem is referenced by:  pjthlem2  25493
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