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Theorem chsh 25818
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 25816 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  "
( H  ^m  NN ) )  C_  H
) )
21simplbi 460 1  |-  ( H  e.  CH  ->  H  e.  SH )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767    C_ wss 3476   "cima 5002  (class class class)co 6282    ^m cmap 7417   NNcn 10532    ~~>v chli 25520   SHcsh 25521   CHcch 25522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fv 5594  df-ov 6285  df-ch 25815
This theorem is referenced by:  chsssh  25819  chshii  25821  ch0  25822  chss  25823  choccl  25900  chjval  25946  chjcl  25951  pjhth  25987  pjhtheu  25988  pjpreeq  25992  pjpjpre  26013  ch0le  26035  chle0  26037  chslej  26092  chjcom  26100  chub1  26101  chlub  26103  chlej1  26104  chlej2  26105  spansnsh  26155  fh1  26212  fh2  26213  chscllem1  26231  chscllem2  26232  chscllem3  26233  chscllem4  26234  chscl  26235  pjorthi  26263  pjoi0  26311  hstoc  26817  hstnmoc  26818  ch1dle  26947  atomli  26977  chirredlem3  26987  sumdmdii  27010
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