Theorem cheli 31256

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 31255	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3927	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2113   ℋchba 30943   C_ℋ cch 30953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-hilex 31023

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-xp 5628  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fv 6498  df-ov 7359  df-sh 31231  df-ch 31245

This theorem is referenced by:  pjhthlem1  31415  pjhthlem2  31416  h1de2ci  31580  spanunsni  31603  spansncvi  31676  3oalem1  31686  pjcompi  31696  pjocini  31722  pjjsi  31724  pjrni  31726  pjdsi  31736  pjds3i  31737  mayete3i  31752  riesz3i  32086  pjnmopi  32172  pjnormssi  32192  pjimai  32200  pjclem4a  32222  pjclem4  32223  pj3lem1  32230  pj3si  32231  strlem1  32274  strlem3  32277  strlem5  32279  hstrlem3  32285  hstrlem5  32287  sumdmdii  32439  sumdmdlem  32442  sumdmdlem2  32443