Theorem cheli 28089

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 28088	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3599	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 1990   ℋchil 27776   C_ℋ cch 27786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-ov 6653  df-sh 28064  df-ch 28078

This theorem is referenced by:  pjhthlem1  28250  pjhthlem2  28251  h1de2ci  28415  spanunsni  28438  spansncvi  28511  3oalem1  28521  pjcompi  28531  pjocini  28557  pjjsi  28559  pjrni  28561  pjdsi  28571  pjds3i  28572  mayete3i  28587  riesz3i  28921  pjnmopi  29007  pjnormssi  29027  pjimai  29035  pjclem4a  29057  pjclem4  29058  pj3lem1  29065  pj3si  29066  strlem1  29109  strlem3  29112  strlem5  29114  hstrlem3  29120  hstrlem5  29122  sumdmdii  29274  sumdmdlem  29277  sumdmdlem2  29278