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Theorem ch0 28999
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 28995 . 2 (𝐻C𝐻S )
2 sh0 28987 . 2 (𝐻S → 0𝐻)
31, 2syl 17 1 (𝐻C → 0𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  0c0v 28695   S csh 28699   C cch 28700
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-hilex 28770
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-xp 5555  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fv 6357  df-ov 7153  df-sh 28978  df-ch 28992
This theorem is referenced by:  omlsii  29174  nonbooli  29422  strlem1  30021
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