Theorem cheli 30740

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 30739	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3978	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2106   ℋchba 30427   C_ℋ cch 30437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-hilex 30507

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fv 6551  df-ov 7414  df-sh 30715  df-ch 30729

This theorem is referenced by:  pjhthlem1  30899  pjhthlem2  30900  h1de2ci  31064  spanunsni  31087  spansncvi  31160  3oalem1  31170  pjcompi  31180  pjocini  31206  pjjsi  31208  pjrni  31210  pjdsi  31220  pjds3i  31221  mayete3i  31236  riesz3i  31570  pjnmopi  31656  pjnormssi  31676  pjimai  31684  pjclem4a  31706  pjclem4  31707  pj3lem1  31714  pj3si  31715  strlem1  31758  strlem3  31761  strlem5  31763  hstrlem3  31769  hstrlem5  31771  sumdmdii  31923  sumdmdlem  31926  sumdmdlem2  31927