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Theorem sh0 25809
Description: The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
sh0  |-  ( H  e.  SH  ->  0h  e.  H )

Proof of Theorem sh0
StepHypRef Expression
1 issh 25801 . . 3  |-  ( H  e.  SH  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  (
(  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) )
21simplbi 460 . 2  |-  ( H  e.  SH  ->  ( H  C_  ~H  /\  0h  e.  H ) )
32simprd 463 1  |-  ( H  e.  SH  ->  0h  e.  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767    C_ wss 3476    X. cxp 4997   "cima 5002   CCcc 9486   ~Hchil 25512    +h cva 25513    .h csm 25514   0hc0v 25517   SHcsh 25521
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  ax-sep 4568  ax-hilex 25592
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-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  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-sh 25800
This theorem is referenced by:  ch0  25822  hhssabloi  25854  hhssnv  25856  oc0  25884  ocin  25890  shscli  25911  shsel1  25915  shintcli  25923  shunssi  25962  omlsii  25997  sh0le  26034  imaelshi  26653  shatomistici  26956
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