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Theorem chsh 21820
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 21818 . 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 1696    C_ wss 3165   "cima 4708  (class class class)co 5874    ^m cmap 6788   NNcn 9762    ~~>v chli 21523   SHcsh 21524   CHcch 21525
This theorem is referenced by:  chsssh  21821  chshii  21823  ch0  21824  chss  21825  choccl  21901  chjval  21947  chjcl  21952  pjhth  21988  pjhtheu  21989  pjpreeq  21993  pjpjpre  22014  ch0le  22036  chle0  22038  chslej  22093  chjcom  22101  chub1  22102  chlub  22104  chlej1  22105  chlej2  22106  spansnsh  22156  fh1  22213  fh2  22214  chscllem1  22232  chscllem2  22233  chscllem3  22234  chscllem4  22235  chscl  22236  pjorthi  22264  pjoi0  22312  hstoc  22818  hstnmoc  22819  ch1dle  22948  atomli  22978  chirredlem3  22988  sumdmdii  23011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877  df-ch 21817
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