Theorem chss 9099

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

Assertion

Ref	Expression
chss	|- (H e. CH -> H (_ H~)

Proof of Theorem chss

Step	Hyp	Ref	Expression
1		chsh 9096	. 2 |- (H e. CH -> H e. SH)
2		shss 9079	. 2 |- (H e. SH -> H (_ H~)
3	1, 2	syl 10	1 |- (H e. CH -> H (_ H~)

Syntax hints:   -> wi 3   e. wcel 958   (_ wss 2047  H~chil 8788  SHcsh 8797  CHcch 8798

This theorem is referenced by:  chelt 9100  pjhclt 9243  dfch2 9249  pjvect 9641  pjocvect 9642  pjft 9653

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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