HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  chssii Structured version   Visualization version   GIF version

Theorem chssii 30484
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 30480 . 2 𝐻S
32shssii 30466 1 𝐻 ⊆ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  wss 3949  chba 30172   C cch 30182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-hilex 30252
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fv 6552  df-ov 7412  df-sh 30460  df-ch 30474
This theorem is referenced by:  cheli  30485  chelii  30486  hhsscms  30531  chocvali  30552  chm1i  30709  chsscon3i  30714  chsscon2i  30716  chjoi  30741  chj1i  30742  shjshsi  30745  sshhococi  30799  h1dei  30803  spansnpji  30831  spanunsni  30832  h1datomi  30834  spansnji  30899  pjfi  30957  riesz3i  31315  hmopidmpji  31405  pjoccoi  31431  pjinvari  31444  stcltr2i  31528  mdsymi  31664  mdcompli  31682  dmdcompli  31683
  Copyright terms: Public domain W3C validator