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Theorem sh0 26533
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 26525 . . 3  |-  ( H  e.  SH  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  (
(  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) )
21simplbi 458 . 2  |-  ( H  e.  SH  ->  ( H  C_  ~H  /\  0h  e.  H ) )
32simprd 461 1  |-  ( H  e.  SH  ->  0h  e.  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1842    C_ wss 3413    X. cxp 4820   "cima 4825   CCcc 9519   ~Hchil 26236    +h cva 26237    .h csm 26238   0hc0v 26241   SHcsh 26245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-hilex 26316
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-sh 26524
This theorem is referenced by:  ch0  26546  hhssabloi  26578  hhssnv  26580  oc0  26608  ocin  26614  shscli  26635  shsel1  26639  shintcli  26647  shunssi  26686  omlsii  26721  sh0le  26758  imaelshi  27376  shatomistici  27679
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