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Theorem chsh 21804
Description: A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
chsh  |-  ( H  e.  CH  ->  H  e.  SH )

Proof of Theorem chsh
StepHypRef Expression
1 isch 21802 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  "
( H  ^m  NN ) )  C_  H
) )
21simplbi 446 1  |-  ( H  e.  CH  ->  H  e.  SH )
Colors of variables: wff set class
Syntax hints:    -> wi 4    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:  chsssh  21805  chshii  21807  ch0  21808  chss  21809  choccl  21885  chjval  21931  chjcl  21936  pjhth  21972  pjhtheu  21973  pjpreeq  21977  pjpjpre  21998  ch0le  22020  chle0  22022  chslej  22077  chjcom  22085  chub1  22086  chlub  22088  chlej1  22089  chlej2  22090  spansnsh  22140  fh1  22197  fh2  22198  chscllem1  22216  chscllem2  22217  chscllem3  22218  chscllem4  22219  chscl  22220  pjorthi  22248  pjoi0  22296  hstoc  22802  hstnmoc  22803  ch1dle  22932  atomli  22962  chirredlem3  22972  sumdmdii  22995
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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