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Theorem chelii 27474
Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
chssi.1 𝐻C
cheli.1 𝐴𝐻
Assertion
Ref Expression
chelii 𝐴 ∈ ℋ

Proof of Theorem chelii
StepHypRef Expression
1 chssi.1 . . 3 𝐻C
21chssii 27472 . 2 𝐻 ⊆ ℋ
3 cheli.1 . 2 𝐴𝐻
42, 3sselii 3565 1 𝐴 ∈ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 1977  chil 27160   C cch 27170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-hilex 27240
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-ov 6552  df-sh 27448  df-ch 27462
This theorem is referenced by: (None)
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