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

Theorem chelii 28697
 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 28695 . 2 𝐻 ⊆ ℋ
3 cheli.1 . 2 𝐴𝐻
42, 3sselii 3892 1 𝐴 ∈ ℋ
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2083   ℋchba 28383   Cℋ cch 28393 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771  ax-sep 5101  ax-hilex 28463 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-xp 5456  df-cnv 5458  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fv 6240  df-ov 7026  df-sh 28671  df-ch 28685 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator