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Theorem chssii 31250
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 31246 . 2 𝐻S
32shssii 31232 1 𝐻 ⊆ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  wss 3951  chba 30938   C cch 30948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-hilex 31018
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569  df-ov 7434  df-sh 31226  df-ch 31240
This theorem is referenced by:  cheli  31251  chelii  31252  hhsscms  31297  chocvali  31318  chm1i  31475  chsscon3i  31480  chsscon2i  31482  chjoi  31507  chj1i  31508  shjshsi  31511  sshhococi  31565  h1dei  31569  spansnpji  31597  spanunsni  31598  h1datomi  31600  spansnji  31665  pjfi  31723  riesz3i  32081  hmopidmpji  32171  pjoccoi  32197  pjinvari  32210  stcltr2i  32294  mdsymi  32430  mdcompli  32448  dmdcompli  32449
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