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Theorem sh0 31244
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 31236 . . 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 2105  wss 3962   × cxp 5686  cima 5691  cc 11150  chba 30947   + cva 30948   · csm 30949  0c0v 30952   S csh 30956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-hilex 31027
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-sh 31235
This theorem is referenced by:  ch0  31256  hhssabloilem  31289  hhssnv  31292  oc0  31318  ocin  31324  shscli  31345  shsel1  31349  shintcli  31357  shunssi  31396  omlsii  31431  sh0le  31468  imaelshi  32086  shatomistici  32389
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