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Theorem chsh 28209
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 28207 . 2 (𝐻C ↔ (𝐻S ∧ ( ⇝𝑣 “ (𝐻𝑚 ℕ)) ⊆ 𝐻))
21simplbi 475 1 (𝐻C𝐻S )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  wss 3607  cima 5146  (class class class)co 6690  𝑚 cmap 7899  cn 11058  𝑣 chli 27912   S csh 27913   C cch 27914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fv 5934  df-ov 6693  df-ch 28206
This theorem is referenced by:  chsssh  28210  chshii  28212  ch0  28213  chss  28214  choccl  28293  chjval  28339  chjcl  28344  pjhth  28380  pjhtheu  28381  pjpreeq  28385  pjpjpre  28406  ch0le  28428  chle0  28430  chslej  28485  chjcom  28493  chub1  28494  chlub  28496  chlej1  28497  chlej2  28498  spansnsh  28548  fh1  28605  fh2  28606  chscllem1  28624  chscllem2  28625  chscllem3  28626  chscllem4  28627  chscl  28628  pjorthi  28656  pjoi0  28704  hstoc  29209  hstnmoc  29210  ch1dle  29339  atomli  29369  chirredlem3  29379  sumdmdii  29402
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