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Theorem chssii 27934
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 27930 . 2 𝐻S
32shssii 27916 1 𝐻 ⊆ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  wss 3555  chil 27622   C cch 27632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-hilex 27702
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fv 5855  df-ov 6607  df-sh 27910  df-ch 27924
This theorem is referenced by:  cheli  27935  chelii  27936  hhsscms  27982  chocvali  28004  chm1i  28161  chsscon3i  28166  chsscon2i  28168  chjoi  28193  chj1i  28194  shjshsi  28197  sshhococi  28251  h1dei  28255  spansnpji  28283  spanunsni  28284  h1datomi  28286  spansnji  28351  pjfi  28409  riesz3i  28767  hmopidmpji  28857  pjoccoi  28883  pjinvari  28896  stcltr2i  28980  mdsymi  29116  mdcompli  29134  dmdcompli  29135
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