Theorem cheli 29008

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 29007	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3962	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2110   ℋchba 28695   C_ℋ cch 28705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-hilex 28775

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-xp 5560  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fv 6362  df-ov 7158  df-sh 28983  df-ch 28997

This theorem is referenced by:  pjhthlem1  29167  pjhthlem2  29168  h1de2ci  29332  spanunsni  29355  spansncvi  29428  3oalem1  29438  pjcompi  29448  pjocini  29474  pjjsi  29476  pjrni  29478  pjdsi  29488  pjds3i  29489  mayete3i  29504  riesz3i  29838  pjnmopi  29924  pjnormssi  29944  pjimai  29952  pjclem4a  29974  pjclem4  29975  pj3lem1  29982  pj3si  29983  strlem1  30026  strlem3  30029  strlem5  30031  hstrlem3  30037  hstrlem5  30039  sumdmdii  30191  sumdmdlem  30194  sumdmdlem2  30195