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Theorem chelii 28395
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 28393 . 2 𝐻 ⊆ ℋ
3 cheli.1 . 2 𝐴𝐻
42, 3sselii 3737 1 𝐴 ∈ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 2135  chil 28081   C cch 28091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-hilex 28161
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-rex 3052  df-rab 3055  df-v 3338  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-xp 5268  df-cnv 5270  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fv 6053  df-ov 6812  df-sh 28369  df-ch 28383
This theorem is referenced by: (None)
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