Theorem chsssh 9049

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 9047	. 2 |- CH = {h | (h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h))}
2		ssab2 2127	. 2 |- {h | (h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h))} (_ SH
3	1, 2	eqsstr 2088	1 |- CH (_ SH

Syntax hints:   -> wi 3   /\ wa 223  A.wal 953   e. wcel 957  {cab 1462   (_ wss 2044   class class class wbr 2615  -->wf 3174  NNcn 5279   ~~>v chli 8751  SHcsh 8752  CHcch 8753

This theorem is referenced by:  chex 9050  chsh 9051  chsspwh 9074  chintcl 9250  shatomistic 10244

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-in 2048  df-ss 2050  df-ch 9047

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