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Theorem sh0 22710
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 22702 . . 3  |-  ( H  e.  SH  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  (
(  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) )
21simplbi 447 . 2  |-  ( H  e.  SH  ->  ( H  C_  ~H  /\  0h  e.  H ) )
32simprd 450 1  |-  ( H  e.  SH  ->  0h  e.  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    C_ wss 3312    X. cxp 4868   "cima 4873   CCcc 8980   ~Hchil 22414    +h cva 22415    .h csm 22416   0hc0v 22419   SHcsh 22423
This theorem is referenced by:  ch0  22723  hhssabloi  22754  hhssnv  22756  oc0  22784  ocin  22790  shscli  22811  shsel1  22815  shintcli  22823  shunssi  22862  omlsii  22897  sh0le  22934  imaelshi  23553  shatomistici  23856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-hilex 22494
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-sh 22701
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