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Theorem sh0 31235
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 31227 . . 3 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
21simplbi 497 . 2 (𝐻S → (𝐻 ⊆ ℋ ∧ 0𝐻))
32simprd 495 1 (𝐻S → 0𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wss 3951   × cxp 5683  cima 5688  cc 11153  chba 30938   + cva 30939   · csm 30940  0c0v 30943   S csh 30947
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-hilex 31018
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-sh 31226
This theorem is referenced by:  ch0  31247  hhssabloilem  31280  hhssnv  31283  oc0  31309  ocin  31315  shscli  31336  shsel1  31340  shintcli  31348  shunssi  31387  omlsii  31422  sh0le  31459  imaelshi  32077  shatomistici  32380
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