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Theorem shelii 31234
Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
shssi.1 𝐻S
sheli.1 𝐴𝐻
Assertion
Ref Expression
shelii 𝐴 ∈ ℋ

Proof of Theorem shelii
StepHypRef Expression
1 shssi.1 . . 3 𝐻S
21shssii 31232 . 2 𝐻 ⊆ ℋ
3 sheli.1 . 2 𝐴𝐻
42, 3sselii 3980 1 𝐴 ∈ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  chba 30938   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:  omlsilem  31421
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