Theorem cheli 31211

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 31210	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3939	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2109   ℋchba 30898   C_ℋ cch 30908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-hilex 30978

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fv 6507  df-ov 7372  df-sh 31186  df-ch 31200

This theorem is referenced by:  pjhthlem1  31370  pjhthlem2  31371  h1de2ci  31535  spanunsni  31558  spansncvi  31631  3oalem1  31641  pjcompi  31651  pjocini  31677  pjjsi  31679  pjrni  31681  pjdsi  31691  pjds3i  31692  mayete3i  31707  riesz3i  32041  pjnmopi  32127  pjnormssi  32147  pjimai  32155  pjclem4a  32177  pjclem4  32178  pj3lem1  32185  pj3si  32186  strlem1  32229  strlem3  32232  strlem5  32234  hstrlem3  32240  hstrlem5  32242  sumdmdii  32394  sumdmdlem  32397  sumdmdlem2  32398