Theorem chssi 9101

Description: A closed subspace of a Hilbert space is a subset of Hilbert space.

Hypothesis

Ref	Expression
chssi.1	|- H e. CH

Assertion

Ref	Expression
chssi	|- H (_ H~

Proof of Theorem chssi

Step	Hyp	Ref	Expression
1		chssi.1	. . 3 |- H e. CH
2	1	chshi 9097	. 2 |- H e. SH
3	2	shssi 9081	1 |- H (_ H~

Syntax hints:   e. wcel 958   (_ wss 2047  H~chil 8788  CHcch 8798

This theorem is referenced by:  chel 9102  cheli 9103  hhsscms 9150  chocval 9171  choccl 9185  projlem26 9211  projlem29 9214  shlub 9346  chm1 9379  chsscon3 9384  chj1 9412  shjshs 9415  sshhococ 9469  h1det 9473  spansnpj 9501  spanunsn 9502  h1datom 9504  osumlem4 9581  osumlem8 9585  osum 9586  spansnj 9591  pjf 9649  riesz3 9995  pjocco 10106  pjinvar 10119  stcltr2 10202  mdsym 10338  mdcompl 10356  dmdcompl 10357

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-sh 9076  df-ch 9092

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