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Theorem ch0 22732
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 22728 . 2  |-  ( H  e.  CH  ->  H  e.  SH )
2 sh0 22719 . 2  |-  ( H  e.  SH  ->  0h  e.  H )
31, 2syl 16 1  |-  ( H  e.  CH  ->  0h  e.  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   0hc0v 22428   SHcsh 22432   CHcch 22433
This theorem is referenced by:  omlsii  22906  nonbooli  23154  strlem1  23754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-hilex 22503
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-xp 4885  df-cnv 4887  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fv 5463  df-ov 6085  df-sh 22710  df-ch 22725
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