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Theorem chsh 22719
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 22717 . 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 4873  (class class class)co 6073    ^m cmap 7010   NNcn 9992    ~~>v chli 22422   SHcsh 22423   CHcch 22424
This theorem is referenced by:  chsssh  22720  chshii  22722  ch0  22723  chss  22724  choccl  22800  chjval  22846  chjcl  22851  pjhth  22887  pjhtheu  22888  pjpreeq  22892  pjpjpre  22913  ch0le  22935  chle0  22937  chslej  22992  chjcom  23000  chub1  23001  chlub  23003  chlej1  23004  chlej2  23005  spansnsh  23055  fh1  23112  fh2  23113  chscllem1  23131  chscllem2  23132  chscllem3  23133  chscllem4  23134  chscl  23135  pjorthi  23163  pjoi0  23211  hstoc  23717  hstnmoc  23718  ch1dle  23847  atomli  23877  chirredlem3  23887  sumdmdii  23910
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 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fv 5454  df-ov 6076  df-ch 22716
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