Theorem chsssh 5129

Description: Closed subspaces are subspaces in a Hilbert space.

Assertion

Ref	Expression
chsssh	⊢ Cℋ ⊆ Sℋ

Proof of Theorem chsssh

Step	Hyp	Ref	Expression
1		df-ch 5127	. 2 ⊢ Cℋ = {h∣(h ∈ Sℋ ∧ ∀f∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h))}
2		ssab 1555	. 2 ⊢ {h∣(h ∈ Sℋ ∧ ∀f∀x((f:ℕ–→h ∧ f ⇝v x) → x ∈ h))} ⊆ Sℋ
3	1, 2	eqsstr 1530	1 ⊢ Cℋ ⊆ Sℋ

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   ∈ wel 803  {cab 1090   ∈ wcel 1092   ⊆ wss 1487   class class class wbr 2054  –→wf 2418  ℕcn 4093   ⇝_v chli 4966   S_ℋ csh 4967   C_ℋ cch 4968

This theorem is referenced by:  chex 5130  chsh 5131  chsspwh 5154  chintcl 5296

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-ch 5127

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