Theorem cheli 31303

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 31302	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3917	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2114   ℋchba 30990   C_ℋ cch 31000

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 2708  ax-sep 5231  ax-hilex 31070

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 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fv 6506  df-ov 7370  df-sh 31278  df-ch 31292

This theorem is referenced by:  pjhthlem1  31462  pjhthlem2  31463  h1de2ci  31627  spanunsni  31650  spansncvi  31723  3oalem1  31733  pjcompi  31743  pjocini  31769  pjjsi  31771  pjrni  31773  pjdsi  31783  pjds3i  31784  mayete3i  31799  riesz3i  32133  pjnmopi  32219  pjnormssi  32239  pjimai  32247  pjclem4a  32269  pjclem4  32270  pj3lem1  32277  pj3si  32278  strlem1  32321  strlem3  32324  strlem5  32326  hstrlem3  32332  hstrlem5  32334  sumdmdii  32486  sumdmdlem  32489  sumdmdlem2  32490