Theorem chssi 9096

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 9092	. 2 |- H e. SH
3	2	shssi 9076	1 |- H (_ H~

Syntax hints:   e. wcel 960   (_ wss 2050  H~chil 8783  CHcch 8793

This theorem is referenced by:  chel 9097  cheli 9098  hhsscms 9145  chocval 9166  choccl 9180  projlem26 9206  projlem29 9209  shlub 9341  chm1 9374  chsscon3 9379  chj1 9407  shjshs 9410  sshhococ 9464  h1det 9468  spansnpj 9496  spanunsn 9497  h1datom 9499  osumlem4 9576  osumlem8 9580  osum 9581  spansnj 9586  pjf 9644  riesz3 9990  pjocco 10101  pjinvar 10114  stcltr2 10197  mdsym 10333  mdcompl 10351  dmdcompl 10352

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-in 2054  df-ss 2056  df-sh 9071  df-ch 9087

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