Theorem cheli 31213

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 31212	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3954	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2108   ℋchba 30900   C_ℋ cch 30910

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-hilex 30980

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fv 6539  df-ov 7408  df-sh 31188  df-ch 31202

This theorem is referenced by:  pjhthlem1  31372  pjhthlem2  31373  h1de2ci  31537  spanunsni  31560  spansncvi  31633  3oalem1  31643  pjcompi  31653  pjocini  31679  pjjsi  31681  pjrni  31683  pjdsi  31693  pjds3i  31694  mayete3i  31709  riesz3i  32043  pjnmopi  32129  pjnormssi  32149  pjimai  32157  pjclem4a  32179  pjclem4  32180  pj3lem1  32187  pj3si  32188  strlem1  32231  strlem3  32234  strlem5  32236  hstrlem3  32242  hstrlem5  32244  sumdmdii  32396  sumdmdlem  32399  sumdmdlem2  32400