Theorem cheli 31307

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 31306	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3929	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2113   ℋchba 30994   C_ℋ cch 31004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-hilex 31074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fv 6500  df-ov 7361  df-sh 31282  df-ch 31296

This theorem is referenced by:  pjhthlem1  31466  pjhthlem2  31467  h1de2ci  31631  spanunsni  31654  spansncvi  31727  3oalem1  31737  pjcompi  31747  pjocini  31773  pjjsi  31775  pjrni  31777  pjdsi  31787  pjds3i  31788  mayete3i  31803  riesz3i  32137  pjnmopi  32223  pjnormssi  32243  pjimai  32251  pjclem4a  32273  pjclem4  32274  pj3lem1  32281  pj3si  32282  strlem1  32325  strlem3  32328  strlem5  32330  hstrlem3  32336  hstrlem5  32338  sumdmdii  32490  sumdmdlem  32493  sumdmdlem2  32494