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Theorem chel 31163
Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
Assertion
Ref Expression
chel ((𝐻C𝐴𝐻) → 𝐴 ∈ ℋ)

Proof of Theorem chel
StepHypRef Expression
1 chss 31162 . 2 (𝐻C𝐻 ⊆ ℋ)
21sselda 3979 1 ((𝐻C𝐴𝐻) → 𝐴 ∈ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2099  chba 30852   C cch 30862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-hilex 30932
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fv 6562  df-ov 7427  df-sh 31140  df-ch 31154
This theorem is referenced by:  pjhtheu2  31349  pjspansn  31510  pjid  31628  atom1d  32286  sumdmdii  32348
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