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Theorem chsh 24632
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 24630 . 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 1756    C_ wss 3333   "cima 4848  (class class class)co 6096    ^m cmap 7219   NNcn 10327    ~~>v chli 24334   SHcsh 24335   CHcch 24336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-xp 4851  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fv 5431  df-ov 6099  df-ch 24629
This theorem is referenced by:  chsssh  24633  chshii  24635  ch0  24636  chss  24637  choccl  24714  chjval  24760  chjcl  24765  pjhth  24801  pjhtheu  24802  pjpreeq  24806  pjpjpre  24827  ch0le  24849  chle0  24851  chslej  24906  chjcom  24914  chub1  24915  chlub  24917  chlej1  24918  chlej2  24919  spansnsh  24969  fh1  25026  fh2  25027  chscllem1  25045  chscllem2  25046  chscllem3  25047  chscllem4  25048  chscl  25049  pjorthi  25077  pjoi0  25125  hstoc  25631  hstnmoc  25632  ch1dle  25761  atomli  25791  chirredlem3  25801  sumdmdii  25824
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