Theorem cheli 31251

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 31250	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3979	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2108   ℋchba 30938   C_ℋ cch 30948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-hilex 31018

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569  df-ov 7434  df-sh 31226  df-ch 31240

This theorem is referenced by:  pjhthlem1  31410  pjhthlem2  31411  h1de2ci  31575  spanunsni  31598  spansncvi  31671  3oalem1  31681  pjcompi  31691  pjocini  31717  pjjsi  31719  pjrni  31721  pjdsi  31731  pjds3i  31732  mayete3i  31747  riesz3i  32081  pjnmopi  32167  pjnormssi  32187  pjimai  32195  pjclem4a  32217  pjclem4  32218  pj3lem1  32225  pj3si  32226  strlem1  32269  strlem3  32272  strlem5  32274  hstrlem3  32280  hstrlem5  32282  sumdmdii  32434  sumdmdlem  32437  sumdmdlem2  32438