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Theorem chss 28070
 Description: A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)
Assertion
Ref Expression
chss (𝐻C𝐻 ⊆ ℋ)

Proof of Theorem chss
StepHypRef Expression
1 chsh 28065 . 2 (𝐻C𝐻S )
2 shss 28051 . 2 (𝐻S𝐻 ⊆ ℋ)
31, 2syl 17 1 (𝐻C𝐻 ⊆ ℋ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1989   ⊆ wss 3572   ℋchil 27760   Sℋ csh 27769   Cℋ cch 27770 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-hilex 27840 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-xp 5118  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fv 5894  df-ov 6650  df-sh 28048  df-ch 28062 This theorem is referenced by:  chel  28071  pjhcl  28244  dfch2  28250  shlub  28257  chsscon2  28345  chscllem2  28481  pjvec  28539  pjocvec  28540  pjhf  28551  elpjrn  29033
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