Theorem cheli 31161

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 31160	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3942	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2109   ℋchba 30848   C_ℋ cch 30858

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 5251  ax-hilex 30928

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fv 6519  df-ov 7390  df-sh 31136  df-ch 31150

This theorem is referenced by:  pjhthlem1  31320  pjhthlem2  31321  h1de2ci  31485  spanunsni  31508  spansncvi  31581  3oalem1  31591  pjcompi  31601  pjocini  31627  pjjsi  31629  pjrni  31631  pjdsi  31641  pjds3i  31642  mayete3i  31657  riesz3i  31991  pjnmopi  32077  pjnormssi  32097  pjimai  32105  pjclem4a  32127  pjclem4  32128  pj3lem1  32135  pj3si  32136  strlem1  32179  strlem3  32182  strlem5  32184  hstrlem3  32190  hstrlem5  32192  sumdmdii  32344  sumdmdlem  32347  sumdmdlem2  32348