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Theorem chsh 22715
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 22713 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  "
( H  ^m  NN ) )  C_  H
) )
21simplbi 447 1  |-  ( H  e.  CH  ->  H  e.  SH )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    C_ wss 3312   "cima 4872  (class class class)co 6072    ^m cmap 7009   NNcn 9989    ~~>v chli 22418   SHcsh 22419   CHcch 22420
This theorem is referenced by:  chsssh  22716  chshii  22718  ch0  22719  chss  22720  choccl  22796  chjval  22842  chjcl  22847  pjhth  22883  pjhtheu  22884  pjpreeq  22888  pjpjpre  22909  ch0le  22931  chle0  22933  chslej  22988  chjcom  22996  chub1  22997  chlub  22999  chlej1  23000  chlej2  23001  spansnsh  23051  fh1  23108  fh2  23109  chscllem1  23127  chscllem2  23128  chscllem3  23129  chscllem4  23130  chscl  23131  pjorthi  23159  pjoi0  23207  hstoc  23713  hstnmoc  23714  ch1dle  23843  atomli  23873  chirredlem3  23883  sumdmdii  23906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4875  df-cnv 4877  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fv 5453  df-ov 6075  df-ch 22712
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