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Theorem isch 21802
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 5865 . . . 4  |-  ( h  =  H  ->  (
h  ^m  NN )  =  ( H  ^m  NN ) )
21imaeq2d 5012 . . 3  |-  ( h  =  H  ->  (  ~~>v  " ( h  ^m  NN ) )  =  ( 
~~>v  " ( H  ^m  NN ) ) )
3 id 19 . . 3  |-  ( h  =  H  ->  h  =  H )
42, 3sseq12d 3207 . 2  |-  ( h  =  H  ->  (
(  ~~>v  " ( h  ^m  NN ) )  C_  h  <->  ( 
~~>v  " ( H  ^m  NN ) )  C_  H
) )
5 df-ch 21801 . 2  |-  CH  =  { h  e.  SH  |  (  ~~>v  " (
h  ^m  NN )
)  C_  h }
64, 5elrab2 2925 1  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  "
( H  ^m  NN ) )  C_  H
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   "cima 4692  (class class class)co 5858    ^m cmap 6772   NNcn 9746    ~~>v chli 21507   SHcsh 21508   CHcch 21509
This theorem is referenced by:  isch2  21803  chsh  21804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861  df-ch 21801
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