Theorem cheli 31212

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 31211	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3925	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2111   ℋchba 30899   C_ℋ cch 30909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-hilex 30979

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fv 6489  df-ov 7349  df-sh 31187  df-ch 31201

This theorem is referenced by:  pjhthlem1  31371  pjhthlem2  31372  h1de2ci  31536  spanunsni  31559  spansncvi  31632  3oalem1  31642  pjcompi  31652  pjocini  31678  pjjsi  31680  pjrni  31682  pjdsi  31692  pjds3i  31693  mayete3i  31708  riesz3i  32042  pjnmopi  32128  pjnormssi  32148  pjimai  32156  pjclem4a  32178  pjclem4  32179  pj3lem1  32186  pj3si  32187  strlem1  32230  strlem3  32233  strlem5  32235  hstrlem3  32241  hstrlem5  32243  sumdmdii  32395  sumdmdlem  32398  sumdmdlem2  32399