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Theorem chsh 29004
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 (𝐻C𝐻S )

Proof of Theorem chsh
StepHypRef Expression
1 isch 29002 . 2 (𝐻C ↔ (𝐻S ∧ ( ⇝𝑣 “ (𝐻m ℕ)) ⊆ 𝐻))
21simplbi 500 1 (𝐻C𝐻S )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2113  wss 3939  cima 5561  (class class class)co 7159  m cmap 8409  cn 11641  𝑣 chli 28707   S csh 28708   C cch 28709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-xp 5564  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fv 6366  df-ov 7162  df-ch 29001
This theorem is referenced by:  chsssh  29005  chshii  29007  ch0  29008  chss  29009  choccl  29086  chjval  29132  chjcl  29137  pjhth  29173  pjhtheu  29174  pjpreeq  29178  pjpjpre  29199  ch0le  29221  chle0  29223  chslej  29278  chjcom  29286  chub1  29287  chlub  29289  chlej1  29290  chlej2  29291  spansnsh  29341  fh1  29398  fh2  29399  chscllem1  29417  chscllem2  29418  chscllem3  29419  chscllem4  29420  chscl  29421  pjorthi  29449  pjoi0  29497  hstoc  30002  hstnmoc  30003  ch1dle  30132  atomli  30162  chirredlem3  30172  sumdmdii  30195
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