Theorem cheli 31261

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 31260	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3991	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2106   ℋchba 30948   C_ℋ cch 30958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-hilex 31028

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571  df-ov 7434  df-sh 31236  df-ch 31250

This theorem is referenced by:  pjhthlem1  31420  pjhthlem2  31421  h1de2ci  31585  spanunsni  31608  spansncvi  31681  3oalem1  31691  pjcompi  31701  pjocini  31727  pjjsi  31729  pjrni  31731  pjdsi  31741  pjds3i  31742  mayete3i  31757  riesz3i  32091  pjnmopi  32177  pjnormssi  32197  pjimai  32205  pjclem4a  32227  pjclem4  32228  pj3lem1  32235  pj3si  32236  strlem1  32279  strlem3  32282  strlem5  32284  hstrlem3  32290  hstrlem5  32292  sumdmdii  32444  sumdmdlem  32447  sumdmdlem2  32448