Theorem chsssh 9015

Description: Closed subspaces are subspaces in a Hilbert space.

Assertion

Ref	Expression
chsssh	|- CH (_ SH

Proof of Theorem chsssh

Step	Hyp	Ref	Expression
1		df-ch 9013	. 2 |- CH = {h | (h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h))}
2		ssab2 2120	. 2 |- {h | (h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h))} (_ SH
3	1, 2	eqsstr 2081	1 |- CH (_ SH

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   e. wcel 955  {cab 1456   (_ wss 2037   class class class wbr 2609  -->wf 3168  NNcn 5268   ~~>v chli 8735  SHcsh 8736  CHcch 8737

This theorem is referenced by:  chex 9016  chsh 9017  chsspwh 9040  chintcl 9210  shatomistic 10196

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-in 2041  df-ss 2043  df-ch 9013

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