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Theorem sh0 28996
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 (𝐻S → 0𝐻)

Proof of Theorem sh0
StepHypRef Expression
1 issh 28988 . . 3 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
21simplbi 500 . 2 (𝐻S → (𝐻 ⊆ ℋ ∧ 0𝐻))
32simprd 498 1 (𝐻S → 0𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2113  wss 3939   × cxp 5556  cima 5561  cc 10538  chba 28699   + cva 28700   · csm 28701  0c0v 28704   S csh 28708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-hilex 28779
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-sh 28987
This theorem is referenced by:  ch0  29008  hhssabloilem  29041  hhssnv  29044  oc0  29070  ocin  29076  shscli  29097  shsel1  29101  shintcli  29109  shunssi  29148  omlsii  29183  sh0le  29220  imaelshi  29838  shatomistici  30141
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