Theorem cheli 31134

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 31133	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3939	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2109   ℋchba 30821   C_ℋ cch 30831

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 30901

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 31109  df-ch 31123

This theorem is referenced by:  pjhthlem1  31293  pjhthlem2  31294  h1de2ci  31458  spanunsni  31481  spansncvi  31554  3oalem1  31564  pjcompi  31574  pjocini  31600  pjjsi  31602  pjrni  31604  pjdsi  31614  pjds3i  31615  mayete3i  31630  riesz3i  31964  pjnmopi  32050  pjnormssi  32070  pjimai  32078  pjclem4a  32100  pjclem4  32101  pj3lem1  32108  pj3si  32109  strlem1  32152  strlem3  32155  strlem5  32157  hstrlem3  32163  hstrlem5  32165  sumdmdii  32317  sumdmdlem  32320  sumdmdlem2  32321