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Theorem chssii 31164
Description: A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
chssi.1 𝐻C
Assertion
Ref Expression
chssii 𝐻 ⊆ ℋ

Proof of Theorem chssii
StepHypRef Expression
1 chssi.1 . . 3 𝐻C
21chshii 31160 . 2 𝐻S
32shssii 31146 1 𝐻 ⊆ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 2099  wss 3947  chba 30852   C cch 30862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-hilex 30932
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fv 6562  df-ov 7427  df-sh 31140  df-ch 31154
This theorem is referenced by:  cheli  31165  chelii  31166  hhsscms  31211  chocvali  31232  chm1i  31389  chsscon3i  31394  chsscon2i  31396  chjoi  31421  chj1i  31422  shjshsi  31425  sshhococi  31479  h1dei  31483  spansnpji  31511  spanunsni  31512  h1datomi  31514  spansnji  31579  pjfi  31637  riesz3i  31995  hmopidmpji  32085  pjoccoi  32111  pjinvari  32124  stcltr2i  32208  mdsymi  32344  mdcompli  32362  dmdcompli  32363
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