Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  chsh Structured version   Visualization version   GIF version

Theorem chsh 28653
 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 28651 . 2 (𝐻C ↔ (𝐻S ∧ ( ⇝𝑣 “ (𝐻𝑚 ℕ)) ⊆ 𝐻))
21simplbi 493 1 (𝐻C𝐻S )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2106   ⊆ wss 3791   “ cima 5358  (class class class)co 6922   ↑𝑚 cmap 8140  ℕcn 11374   ⇝𝑣 chli 28356   Sℋ csh 28357   Cℋ cch 28358 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-xp 5361  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fv 6143  df-ov 6925  df-ch 28650 This theorem is referenced by:  chsssh  28654  chshii  28656  ch0  28657  chss  28658  choccl  28737  chjval  28783  chjcl  28788  pjhth  28824  pjhtheu  28825  pjpreeq  28829  pjpjpre  28850  ch0le  28872  chle0  28874  chslej  28929  chjcom  28937  chub1  28938  chlub  28940  chlej1  28941  chlej2  28942  spansnsh  28992  fh1  29049  fh2  29050  chscllem1  29068  chscllem2  29069  chscllem3  29070  chscllem4  29071  chscl  29072  pjorthi  29100  pjoi0  29148  hstoc  29653  hstnmoc  29654  ch1dle  29783  atomli  29813  chirredlem3  29823  sumdmdii  29846
 Copyright terms: Public domain W3C validator