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Theorem hlcms 23652
 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 23649 . 2 (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
2 bncms 23630 . 2 (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
31, 2syl 17 1 (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2081  CMetSpccms 23618  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-3an 1082  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-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-iota 6189  df-fv 6233  df-bn 23622  df-hl 23623 This theorem is referenced by:  pjthlem2  23724
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