Theorem cheli 31655

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 31654	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3934	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ℋchba 31342   C_ℋ cch 31352

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-hilex 31422

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fv 6548  df-ov 7422  df-sh 31630  df-ch 31644

This theorem is used by:  pjhthlem1  31814  pjhthlem2  31815  h1de2ci  31979  spanunsni  32002  spansncvi  32075  3oalem1  32085  pjcompi  32095  pjocini  32121  pjjsi  32123  pjrni  32125  pjdsi  32135  pjds3i  32136  mayete3i  32151  riesz3i  32485  pjnmopi  32571  pjnormssi  32591  pjimai  32599  pjclem4a  32621  pjclem4  32622  pj3lem1  32629  pj3si  32630  strlem1  32673  strlem3  32676  strlem5  32678  hstrlem3  32684  hstrlem5  32686  sumdmdii  32838  sumdmdlem  32841  sumdmdlem2  32842