Theorem cheli 31320

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 31319	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3931	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2114   ℋchba 31007   C_ℋ cch 31017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-hilex 31087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fv 6508  df-ov 7371  df-sh 31295  df-ch 31309

This theorem is referenced by:  pjhthlem1  31479  pjhthlem2  31480  h1de2ci  31644  spanunsni  31667  spansncvi  31740  3oalem1  31750  pjcompi  31760  pjocini  31786  pjjsi  31788  pjrni  31790  pjdsi  31800  pjds3i  31801  mayete3i  31816  riesz3i  32150  pjnmopi  32236  pjnormssi  32256  pjimai  32264  pjclem4a  32286  pjclem4  32287  pj3lem1  32294  pj3si  32295  strlem1  32338  strlem3  32341  strlem5  32343  hstrlem3  32349  hstrlem5  32351  sumdmdii  32503  sumdmdlem  32506  sumdmdlem2  32507