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Theorem sh0 21811
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 21803 . . 3  |-  ( H  e.  SH  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  (
(  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) )
21simplbi 446 . 2  |-  ( H  e.  SH  ->  ( H  C_  ~H  /\  0h  e.  H ) )
32simprd 449 1  |-  ( H  e.  SH  ->  0h  e.  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696    C_ wss 3165    X. cxp 4703   "cima 4708   CCcc 8751   ~Hchil 21515    +h cva 21516    .h csm 21517   0hc0v 21520   SHcsh 21524
This theorem is referenced by:  ch0  21824  hhssabloi  21855  hhssnv  21857  oc0  21885  ocin  21891  shscli  21912  shsel1  21916  shintcli  21924  shunssi  21963  omlsii  21998  sh0le  22035  imaelshi  22654  shatomistici  22957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-sh 21802
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