Theorem cheli 31041

Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Hypothesis

Ref	Expression
chssi.1	⊢ 𝐻 ∈ Cℋ

Assertion

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

Proof of Theorem cheli

Step	Hyp	Ref	Expression
1		chssi.1	. . 3 ⊢ 𝐻 ∈ Cℋ
2	1	chssii 31040	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3976	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2099   ℋchba 30728   C_ℋ cch 30738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-hilex 30808

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fv 6556  df-ov 7423  df-sh 31016  df-ch 31030

This theorem is referenced by:  pjhthlem1  31200  pjhthlem2  31201  h1de2ci  31365  spanunsni  31388  spansncvi  31461  3oalem1  31471  pjcompi  31481  pjocini  31507  pjjsi  31509  pjrni  31511  pjdsi  31521  pjds3i  31522  mayete3i  31537  riesz3i  31871  pjnmopi  31957  pjnormssi  31977  pjimai  31985  pjclem4a  32007  pjclem4  32008  pj3lem1  32015  pj3si  32016  strlem1  32059  strlem3  32062  strlem5  32064  hstrlem3  32070  hstrlem5  32072  sumdmdii  32224  sumdmdlem  32227  sumdmdlem2  32228