Theorem chel 5137

Description: A member of a closed subspace of a Hilbert space is a vector.

Hypothesis

Ref	Expression
chssi.1	⊢ H ∈ Cℋ

Assertion

Ref	Expression
chel	⊢ (A ∈ H → A ∈ ℋ )

Proof of Theorem chel

Step	Hyp	Ref	Expression
1		chssi.1	. . 3 ⊢ H ∈ Cℋ
2	1	chssi 5136	. 2 ⊢ H ⊆ ℋ
3	2	sseli 1504	1 ⊢ (A ∈ H → A ∈ ℋ )

Syntax hints:   → wi 2   ∈ wcel 1092   ℋ chil 4958   C_ℋ cch 4968

This theorem is referenced by:  chocuni 5179  projlem8 5200  projlem10 5202  projlem12 5204  projlem13 5205  projlem15 5207  projlem26 5218  projlem28 5220  pjpj0 5259  h1de2ct 5461  spanunsn 5482  osumlem1 5530  spansncv 5542  3oalem1 5552  pjocin 5583  pjin 5584  pjjs 5585  pjrn 5587  pjv 5589  pjnormss 5638  pjclem4a 5652  pjclem4 5653  pj3lem1 5658  pj3s 5659  strlem1 5691  strlem3 5694  strlem4 5695  strlem5 5696  sumdmdi 5785  sumdmdlem 5786  sumdmd 5787

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-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-hilex 4983

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-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-sh 5114  df-ch 5127

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