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Theorem chshii 25821
Description: A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
chshi.1  |-  H  e. 
CH
Assertion
Ref Expression
chshii  |-  H  e.  SH

Proof of Theorem chshii
StepHypRef Expression
1 chshi.1 . 2  |-  H  e. 
CH
2 chsh 25818 . 2  |-  ( H  e.  CH  ->  H  e.  SH )
31, 2ax-mp 5 1  |-  H  e.  SH
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767   SHcsh 25521   CHcch 25522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fv 5594  df-ov 6285  df-ch 25815
This theorem is referenced by:  chssii  25825  helsh  25839  h0elsh  25850  hhsscms  25871  hhssbn  25872  hhsshl  25873  chocunii  25895  shsleji  25964  shjshcli  25970  pjhthlem1  25985  pjhthlem2  25986  omlsii  25997  ococi  25999  pjoc1i  26025  chne0i  26047  chocini  26048  chjcli  26051  chsleji  26052  chseli  26053  chunssji  26061  chjcomi  26062  chub1i  26063  chlubi  26065  chlej1i  26067  chlej2i  26068  h1de2bi  26148  h1de2ctlem  26149  spansnpji  26172  spanunsni  26173  h1datomi  26175  pjoml2i  26179  qlaxr3i  26230  osumi  26236  osumcor2i  26238  spansnji  26240  spansnm0i  26244  nonbooli  26245  spansncvi  26246  5oai  26255  3oalem2  26257  3oalem5  26260  3oalem6  26261  pjaddii  26269  pjmulii  26271  pjss2i  26274  pjssmii  26275  pj0i  26287  pjocini  26292  pjjsi  26294  pjpythi  26316  mayete3i  26322  mayete3iOLD  26323  pjnmopi  26743  pjimai  26771  pjclem4  26794  pj3si  26802  sto1i  26831  stlei  26835  strlem1  26845  hatomici  26954  hatomistici  26957  atomli  26977  chirredlem3  26987  sumdmdii  27010  sumdmdlem  27013  sumdmdlem2  27014
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