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Theorem chsh 24562
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 24560 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  "
( H  ^m  NN ) )  C_  H
) )
21simplbi 457 1  |-  ( H  e.  CH  ->  H  e.  SH )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1761    C_ wss 3325   "cima 4839  (class class class)co 6090    ^m cmap 7210   NNcn 10318    ~~>v chli 24264   SHcsh 24265   CHcch 24266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-xp 4842  df-cnv 4844  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fv 5423  df-ov 6093  df-ch 24559
This theorem is referenced by:  chsssh  24563  chshii  24565  ch0  24566  chss  24567  choccl  24644  chjval  24690  chjcl  24695  pjhth  24731  pjhtheu  24732  pjpreeq  24736  pjpjpre  24757  ch0le  24779  chle0  24781  chslej  24836  chjcom  24844  chub1  24845  chlub  24847  chlej1  24848  chlej2  24849  spansnsh  24899  fh1  24956  fh2  24957  chscllem1  24975  chscllem2  24976  chscllem3  24977  chscllem4  24978  chscl  24979  pjorthi  25007  pjoi0  25055  hstoc  25561  hstnmoc  25562  ch1dle  25691  atomli  25721  chirredlem3  25731  sumdmdii  25754
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