Theorem cheli 30485

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 30484	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3979	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2107   ℋchba 30172   C_ℋ cch 30182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-hilex 30252

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fv 6552  df-ov 7412  df-sh 30460  df-ch 30474

This theorem is referenced by:  pjhthlem1  30644  pjhthlem2  30645  h1de2ci  30809  spanunsni  30832  spansncvi  30905  3oalem1  30915  pjcompi  30925  pjocini  30951  pjjsi  30953  pjrni  30955  pjdsi  30965  pjds3i  30966  mayete3i  30981  riesz3i  31315  pjnmopi  31401  pjnormssi  31421  pjimai  31429  pjclem4a  31451  pjclem4  31452  pj3lem1  31459  pj3si  31460  strlem1  31503  strlem3  31506  strlem5  31508  hstrlem3  31514  hstrlem5  31516  sumdmdii  31668  sumdmdlem  31671  sumdmdlem2  31672