Theorem cheli 28217

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 28216	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3632	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2030   ℋchil 27904   C_ℋ cch 27914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-hilex 27984

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fv 5934  df-ov 6693  df-sh 28192  df-ch 28206

This theorem is referenced by:  pjhthlem1  28378  pjhthlem2  28379  h1de2ci  28543  spanunsni  28566  spansncvi  28639  3oalem1  28649  pjcompi  28659  pjocini  28685  pjjsi  28687  pjrni  28689  pjdsi  28699  pjds3i  28700  mayete3i  28715  riesz3i  29049  pjnmopi  29135  pjnormssi  29155  pjimai  29163  pjclem4a  29185  pjclem4  29186  pj3lem1  29193  pj3si  29194  strlem1  29237  strlem3  29240  strlem5  29242  hstrlem3  29248  hstrlem5  29250  sumdmdii  29402  sumdmdlem  29405  sumdmdlem2  29406