HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  ch0 Structured version   Visualization version   GIF version

Theorem ch0 31248
Description: The zero vector belongs to any closed subspace of a Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)
Assertion
Ref Expression
ch0 (𝐻C → 0𝐻)

Proof of Theorem ch0
StepHypRef Expression
1 chsh 31244 . 2 (𝐻C𝐻S )
2 sh0 31236 . 2 (𝐻S → 0𝐻)
31, 2syl 17 1 (𝐻C → 0𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  0c0v 30944   S csh 30948   C cch 30949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-hilex 31019
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fv 6568  df-ov 7435  df-sh 31227  df-ch 31241
This theorem is referenced by:  omlsii  31423  nonbooli  31671  strlem1  32270
  Copyright terms: Public domain W3C validator