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Theorem chssii 31260
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 31256 . 2 𝐻S
32shssii 31242 1 𝐻 ⊆ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wss 3963  chba 30948   C cch 30958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571  df-ov 7434  df-sh 31236  df-ch 31250
This theorem is referenced by:  cheli  31261  chelii  31262  hhsscms  31307  chocvali  31328  chm1i  31485  chsscon3i  31490  chsscon2i  31492  chjoi  31517  chj1i  31518  shjshsi  31521  sshhococi  31575  h1dei  31579  spansnpji  31607  spanunsni  31608  h1datomi  31610  spansnji  31675  pjfi  31733  riesz3i  32091  hmopidmpji  32181  pjoccoi  32207  pjinvari  32220  stcltr2i  32304  mdsymi  32440  mdcompli  32458  dmdcompli  32459
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