Theorem cheli 31176

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 31175	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3931	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2109   ℋchba 30863   C_ℋ cch 30873

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-hilex 30943

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-xp 5625  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fv 6490  df-ov 7352  df-sh 31151  df-ch 31165

This theorem is referenced by:  pjhthlem1  31335  pjhthlem2  31336  h1de2ci  31500  spanunsni  31523  spansncvi  31596  3oalem1  31606  pjcompi  31616  pjocini  31642  pjjsi  31644  pjrni  31646  pjdsi  31656  pjds3i  31657  mayete3i  31672  riesz3i  32006  pjnmopi  32092  pjnormssi  32112  pjimai  32120  pjclem4a  32142  pjclem4  32143  pj3lem1  32150  pj3si  32151  strlem1  32194  strlem3  32197  strlem5  32199  hstrlem3  32205  hstrlem5  32207  sumdmdii  32359  sumdmdlem  32362  sumdmdlem2  32363