Theorem cheli 27279

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 27278	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3563	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 1976   ℋchil 26966   C_ℋ cch 26976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-hilex 27046

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-xp 5034  df-cnv 5036  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fv 5798  df-ov 6530  df-sh 27254  df-ch 27268

This theorem is referenced by:  pjhthlem1  27440  pjhthlem2  27441  h1de2ci  27605  spanunsni  27628  spansncvi  27701  3oalem1  27711  pjcompi  27721  pjocini  27747  pjjsi  27749  pjrni  27751  pjdsi  27761  pjds3i  27762  mayete3i  27777  riesz3i  28111  pjnmopi  28197  pjnormssi  28217  pjimai  28225  pjclem4a  28247  pjclem4  28248  pj3lem1  28255  pj3si  28256  strlem1  28299  strlem3  28302  strlem5  28304  hstrlem3  28310  hstrlem5  28312  sumdmdii  28464  sumdmdlem  28467  sumdmdlem2  28468