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Theorem chsssh 31053
Description: Closed subspaces are subspaces in a Hilbert space. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
chsssh CS

Proof of Theorem chsssh
StepHypRef Expression
1 chsh 31052 . 2 (𝑥C𝑥S )
21ssriv 3984 1 CS
Colors of variables: wff setvar class
Syntax hints:  wss 3947   S csh 30756   C cch 30757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-xp 5686  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fv 6559  df-ov 7427  df-ch 31049
This theorem is referenced by:  chex  31054  chsspwh  31075  chintcli  31159  shatomistici  32189
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