Theorem chsssh 31517

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	⊢ Cℋ ⊆ Sℋ

Proof of Theorem chsssh

Step	Hyp	Ref	Expression
1		chsh 31516	. 2 ⊢ (𝑥 ∈ Cℋ → 𝑥 ∈ Sℋ )
2	1	ssriv 3949	1 ⊢ Cℋ ⊆ Sℋ

Syntax hints:   ⊆ wss 3913   S_ℋ csh 31220   C_ℋ cch 31221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fv 6545  df-ov 7414  df-ch 31513

This theorem is referenced by:  chex  31518  chsspwh  31539  chintcli  31623  shatomistici  32653