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Theorem chssii 28935
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 28931 . 2 𝐻S
32shssii 28917 1 𝐻 ⊆ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  wss 3933  chba 28623   C cch 28633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-hilex 28703
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356  df-ov 7148  df-sh 28911  df-ch 28925
This theorem is referenced by:  cheli  28936  chelii  28937  hhsscms  28982  chocvali  29003  chm1i  29160  chsscon3i  29165  chsscon2i  29167  chjoi  29192  chj1i  29193  shjshsi  29196  sshhococi  29250  h1dei  29254  spansnpji  29282  spanunsni  29283  h1datomi  29285  spansnji  29350  pjfi  29408  riesz3i  29766  hmopidmpji  29856  pjoccoi  29882  pjinvari  29895  stcltr2i  29979  mdsymi  30115  mdcompli  30133  dmdcompli  30134
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