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Theorem chss 28637
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 28632 . 2 (𝐻C𝐻S )
2 shss 28618 . 2 (𝐻S𝐻 ⊆ ℋ)
31, 2syl 17 1 (𝐻C𝐻 ⊆ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2164  wss 3798  chba 28327   S csh 28336   C cch 28337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-hilex 28407
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-xp 5352  df-cnv 5354  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fv 6135  df-ov 6913  df-sh 28615  df-ch 28629
This theorem is referenced by:  chel  28638  pjhcl  28811  dfch2  28817  shlub  28824  chsscon2  28912  chscllem2  29048  pjvec  29106  pjocvec  29107  pjhf  29118  elpjrn  29600
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