Theorem cheli 31891

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 31890	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3909	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ℋchba 31578   C_ℋ cch 31588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-hilex 31658

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-xp 5664  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fv 6574  df-ov 7454  df-sh 31866  df-ch 31880

This theorem is used by:  pjhthlem1  32050  pjhthlem2  32051  h1de2ci  32215  spanunsni  32238  spansncvi  32311  3oalem1  32321  pjcompi  32331  pjocini  32357  pjjsi  32359  pjrni  32361  pjdsi  32371  pjds3i  32372  mayete3i  32387  riesz3i  32721  pjnmopi  32807  pjnormssi  32827  pjimai  32835  pjclem4a  32857  pjclem4  32858  pj3lem1  32865  pj3si  32866  strlem1  32909  strlem3  32912  strlem5  32914  hstrlem3  32920  hstrlem5  32922  sumdmdii  33074  sumdmdlem  33077  sumdmdlem2  33078