Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  hlcms Structured version   Visualization version   GIF version

Theorem hlcms 22970
 Description: Every complex 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 22967 . 2 (𝑊 ∈ ℂHil → 𝑊 ∈ Ban)
2 bncms 22949 . 2 (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
31, 2syl 17 1 (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  CMetSpccms 22937  Bancbn 22938  ℂHilchl 22939 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-bn 22941  df-hl 22942 This theorem is referenced by:  pjthlem2  23017
 Copyright terms: Public domain W3C validator