Theorem chsssh 31650

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 31649	. 2 ⊢ (𝑥 ∈ Cℋ → 𝑥 ∈ Sℋ )
2	1	ssriv 3942	1 ⊢ Cℋ ⊆ Sℋ

This proof depends on syntax axioms:   ⊆ wss 3906   S_ℋ csh 31353   C_ℋ cch 31354

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fv 6548  df-ov 7422  df-ch 31646

This theorem is used by:  chex  31651  chsspwh  31672  chintcli  31756  shatomistici  32786