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Theorem chshii 24565
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 24562 . 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 1761   SHcsh 24265   CHcch 24266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-xp 4842  df-cnv 4844  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fv 5423  df-ov 6093  df-ch 24559
This theorem is referenced by:  chssii  24569  helsh  24583  h0elsh  24594  hhsscms  24615  hhssbn  24616  hhsshl  24617  chocunii  24639  shsleji  24708  shjshcli  24714  pjhthlem1  24729  pjhthlem2  24730  omlsii  24741  ococi  24743  pjoc1i  24769  chne0i  24791  chocini  24792  chjcli  24795  chsleji  24796  chseli  24797  chunssji  24805  chjcomi  24806  chub1i  24807  chlubi  24809  chlej1i  24811  chlej2i  24812  h1de2bi  24892  h1de2ctlem  24893  spansnpji  24916  spanunsni  24917  h1datomi  24919  pjoml2i  24923  qlaxr3i  24974  osumi  24980  osumcor2i  24982  spansnji  24984  spansnm0i  24988  nonbooli  24989  spansncvi  24990  5oai  24999  3oalem2  25001  3oalem5  25004  3oalem6  25005  pjaddii  25013  pjmulii  25015  pjss2i  25018  pjssmii  25019  pj0i  25031  pjocini  25036  pjjsi  25038  pjpythi  25060  mayete3i  25066  mayete3iOLD  25067  pjnmopi  25487  pjimai  25515  pjclem4  25538  pj3si  25546  sto1i  25575  stlei  25579  strlem1  25589  hatomici  25698  hatomistici  25701  atomli  25721  chirredlem3  25731  sumdmdii  25754  sumdmdlem  25757  sumdmdlem2  25758
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