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Theorem chsh 22765
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 22763 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  "
( H  ^m  NN ) )  C_  H
) )
21simplbi 448 1  |-  ( H  e.  CH  ->  H  e.  SH )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1728    C_ wss 3309   "cima 4916  (class class class)co 6117    ^m cmap 7054   NNcn 10038    ~~>v chli 22468   SHcsh 22469   CHcch 22470
This theorem is referenced by:  chsssh  22766  chshii  22768  ch0  22769  chss  22770  choccl  22846  chjval  22892  chjcl  22897  pjhth  22933  pjhtheu  22934  pjpreeq  22938  pjpjpre  22959  ch0le  22981  chle0  22983  chslej  23038  chjcom  23046  chub1  23047  chlub  23049  chlej1  23050  chlej2  23051  spansnsh  23101  fh1  23158  fh2  23159  chscllem1  23177  chscllem2  23178  chscllem3  23179  chscllem4  23180  chscl  23181  pjorthi  23209  pjoi0  23257  hstoc  23763  hstnmoc  23764  ch1dle  23893  atomli  23923  chirredlem3  23933  sumdmdii  23956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-xp 4919  df-cnv 4921  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fv 5497  df-ov 6120  df-ch 22762
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