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Theorem chsh 28928
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 28926 . 2 (𝐻C ↔ (𝐻S ∧ ( ⇝𝑣 “ (𝐻m ℕ)) ⊆ 𝐻))
21simplbi 498 1 (𝐻C𝐻S )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wss 3933  cima 5551  (class class class)co 7145  m cmap 8395  cn 11626  𝑣 chli 28631   S csh 28632   C cch 28633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356  df-ov 7148  df-ch 28925
This theorem is referenced by:  chsssh  28929  chshii  28931  ch0  28932  chss  28933  choccl  29010  chjval  29056  chjcl  29061  pjhth  29097  pjhtheu  29098  pjpreeq  29102  pjpjpre  29123  ch0le  29145  chle0  29147  chslej  29202  chjcom  29210  chub1  29211  chlub  29213  chlej1  29214  chlej2  29215  spansnsh  29265  fh1  29322  fh2  29323  chscllem1  29341  chscllem2  29342  chscllem3  29343  chscllem4  29344  chscl  29345  pjorthi  29373  pjoi0  29421  hstoc  29926  hstnmoc  29927  ch1dle  30056  atomli  30086  chirredlem3  30096  sumdmdii  30119
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