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Theorem sh0 27457
 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 27449 . . 3 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
21simplbi 475 . 2 (𝐻S → (𝐻 ⊆ ℋ ∧ 0𝐻))
32simprd 478 1 (𝐻S → 0𝐻)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ⊆ wss 3540   × cxp 5036   “ cima 5041  ℂcc 9813   ℋchil 27160   +ℎ cva 27161   ·ℎ csm 27162  0ℎc0v 27165   Sℋ csh 27169 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-hilex 27240 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-sh 27448 This theorem is referenced by:  ch0  27469  hhssabloilem  27502  hhssnv  27505  oc0  27533  ocin  27539  shscli  27560  shsel1  27564  shintcli  27572  shunssi  27611  omlsii  27646  sh0le  27683  imaelshi  28301  shatomistici  28604
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