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Theorem sh0 28598
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 28590 . . 3 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
21simplbi 492 . 2 (𝐻S → (𝐻 ⊆ ℋ ∧ 0𝐻))
32simprd 490 1 (𝐻S → 0𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wcel 2157  wss 3769   × cxp 5310  cima 5315  cc 10222  chba 28301   + cva 28302   · csm 28303  0c0v 28306   S csh 28310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-hilex 28381
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-sh 28589
This theorem is referenced by:  ch0  28610  hhssabloilem  28643  hhssnv  28646  oc0  28674  ocin  28680  shscli  28701  shsel1  28705  shintcli  28713  shunssi  28752  omlsii  28787  sh0le  28824  imaelshi  29442  shatomistici  29745
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