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Theorem ch0 31257
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 31253 . 2 (𝐻C𝐻S )
2 sh0 31245 . 2 (𝐻S → 0𝐻)
31, 2syl 17 1 (𝐻C → 0𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  0c0v 30953   S csh 30957   C cch 30958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571  df-ov 7434  df-sh 31236  df-ch 31250
This theorem is referenced by:  omlsii  31432  nonbooli  31680  strlem1  32279
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