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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

Proof of Theorem sh0
StepHypRef Expression
1 issh 25801 . . 3
21simplbi 460 . 2
32simprd 463 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wcel 1767   wss 3476   cxp 4997  cima 5002  cc 9486  chil 25512   cva 25513   csm 25514  c0v 25517  csh 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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