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Theorem chel 27264
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 27263 . 2 (𝐻C𝐻 ⊆ ℋ)
21sselda 3567 1 ((𝐻C𝐴𝐻) → 𝐴 ∈ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1976  chil 26953   C cch 26963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-hilex 27033
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-xp 5033  df-cnv 5035  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fv 5797  df-ov 6529  df-sh 27241  df-ch 27255
This theorem is referenced by:  pjhtheu2  27452  pjspansn  27613  pjid  27731  atom1d  28389  sumdmdii  28451
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