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Theorem isch 22566
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 6020 . . . 4  |-  ( h  =  H  ->  (
h  ^m  NN )  =  ( H  ^m  NN ) )
21imaeq2d 5136 . . 3  |-  ( h  =  H  ->  (  ~~>v  " ( h  ^m  NN ) )  =  ( 
~~>v  " ( H  ^m  NN ) ) )
3 id 20 . . 3  |-  ( h  =  H  ->  h  =  H )
42, 3sseq12d 3313 . 2  |-  ( h  =  H  ->  (
(  ~~>v  " ( h  ^m  NN ) )  C_  h  <->  ( 
~~>v  " ( H  ^m  NN ) )  C_  H
) )
5 df-ch 22565 . 2  |-  CH  =  { h  e.  SH  |  (  ~~>v  " (
h  ^m  NN )
)  C_  h }
64, 5elrab2 3030 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 1649    e. wcel 1717    C_ wss 3256   "cima 4814  (class class class)co 6013    ^m cmap 6947   NNcn 9925    ~~>v chli 22271   SHcsh 22272   CHcch 22273
This theorem is referenced by:  isch2  22567  chsh  22568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-xp 4817  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fv 5395  df-ov 6016  df-ch 22565
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