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Theorem ch0 25822
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  |-  ( H  e.  CH  ->  0h  e.  H )

Proof of Theorem ch0
StepHypRef Expression
1 chsh 25818 . 2  |-  ( H  e.  CH  ->  H  e.  SH )
2 sh0 25809 . 2  |-  ( H  e.  SH  ->  0h  e.  H )
31, 2syl 16 1  |-  ( H  e.  CH  ->  0h  e.  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   0hc0v 25517   SHcsh 25521   CHcch 25522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-hilex 25592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fv 5594  df-ov 6285  df-sh 25800  df-ch 25815
This theorem is referenced by:  omlsii  25997  nonbooli  26245  strlem1  26845
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