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Theorem ch0 26560
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 26556 . 2  |-  ( H  e.  CH  ->  H  e.  SH )
2 sh0 26547 . 2  |-  ( H  e.  SH  ->  0h  e.  H )
31, 2syl 17 1  |-  ( H  e.  CH  ->  0h  e.  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842   0hc0v 26255   SHcsh 26259   CHcch 26260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-hilex 26330
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-xp 4829  df-cnv 4831  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fv 5577  df-ov 6281  df-sh 26538  df-ch 26553
This theorem is referenced by:  omlsii  26735  nonbooli  26983  strlem1  27582
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