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Theorem isch 22718
Description: Closed subspace  H of a Hilbert space. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
isch  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  "
( H  ^m  NN ) )  C_  H
) )

Proof of Theorem isch
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 oveq1 6081 . . . 4  |-  ( h  =  H  ->  (
h  ^m  NN )  =  ( H  ^m  NN ) )
21imaeq2d 5196 . . 3  |-  ( h  =  H  ->  (  ~~>v  " ( h  ^m  NN ) )  =  ( 
~~>v  " ( H  ^m  NN ) ) )
3 id 20 . . 3  |-  ( h  =  H  ->  h  =  H )
42, 3sseq12d 3370 . 2  |-  ( h  =  H  ->  (
(  ~~>v  " ( h  ^m  NN ) )  C_  h  <->  ( 
~~>v  " ( H  ^m  NN ) )  C_  H
) )
5 df-ch 22717 . 2  |-  CH  =  { h  e.  SH  |  (  ~~>v  " (
h  ^m  NN )
)  C_  h }
64, 5elrab2 3087 1  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  "
( H  ^m  NN ) )  C_  H
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3313   "cima 4874  (class class class)co 6074    ^m cmap 7011   NNcn 9993    ~~>v chli 22423   SHcsh 22424   CHcch 22425
This theorem is referenced by:  isch2  22719  chsh  22720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-xp 4877  df-cnv 4879  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fv 5455  df-ov 6077  df-ch 22717
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