Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  chel Structured version   Visualization version   GIF version

Theorem chel 29057
 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 29056 . 2 (𝐻C𝐻 ⊆ ℋ)
21sselda 3917 1 ((𝐻C𝐴𝐻) → 𝐴 ∈ ℋ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111   ℋchba 28746   Cℋ cch 28756 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-hilex 28826 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3444  df-un 3888  df-in 3890  df-ss 3900  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fv 6340  df-ov 7148  df-sh 29034  df-ch 29048 This theorem is referenced by:  pjhtheu2  29243  pjspansn  29404  pjid  29522  atom1d  30180  sumdmdii  30242
 Copyright terms: Public domain W3C validator