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Theorem shel 31231
Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
Assertion
Ref Expression
shel ((𝐻S𝐴𝐻) → 𝐴 ∈ ℋ)

Proof of Theorem shel
StepHypRef Expression
1 shss 31230 . 2 (𝐻S𝐻 ⊆ ℋ)
21sselda 3982 1 ((𝐻S𝐴𝐻) → 𝐴 ∈ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2107  chba 30939   S csh 30948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-hilex 31019
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-sh 31227
This theorem is referenced by:  shuni  31320  shsel3  31335  shscom  31339  shsel1  31341  elspancl  31357  pjpjpre  31439  spansnss  31591  sh1dle  32371
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