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Theorem chsh 28421
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 28419 . 2 (𝐻C ↔ (𝐻S ∧ ( ⇝𝑣 “ (𝐻𝑚 ℕ)) ⊆ 𝐻))
21simplbi 485 1 (𝐻C𝐻S )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  wss 3723  cima 5252  (class class class)co 6793  𝑚 cmap 8009  cn 11222  𝑣 chli 28124   S csh 28125   C cch 28126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fv 6039  df-ov 6796  df-ch 28418
This theorem is referenced by:  chsssh  28422  chshii  28424  ch0  28425  chss  28426  choccl  28505  chjval  28551  chjcl  28556  pjhth  28592  pjhtheu  28593  pjpreeq  28597  pjpjpre  28618  ch0le  28640  chle0  28642  chslej  28697  chjcom  28705  chub1  28706  chlub  28708  chlej1  28709  chlej2  28710  spansnsh  28760  fh1  28817  fh2  28818  chscllem1  28836  chscllem2  28837  chscllem3  28838  chscllem4  28839  chscl  28840  pjorthi  28868  pjoi0  28916  hstoc  29421  hstnmoc  29422  ch1dle  29551  atomli  29581  chirredlem3  29591  sumdmdii  29614
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