Theorem cheli 31264

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 31263	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 4004	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2108   ℋchba 30951   C_ℋ cch 30961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-hilex 31031

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fv 6581  df-ov 7451  df-sh 31239  df-ch 31253

This theorem is referenced by:  pjhthlem1  31423  pjhthlem2  31424  h1de2ci  31588  spanunsni  31611  spansncvi  31684  3oalem1  31694  pjcompi  31704  pjocini  31730  pjjsi  31732  pjrni  31734  pjdsi  31744  pjds3i  31745  mayete3i  31760  riesz3i  32094  pjnmopi  32180  pjnormssi  32200  pjimai  32208  pjclem4a  32230  pjclem4  32231  pj3lem1  32238  pj3si  32239  strlem1  32282  strlem3  32285  strlem5  32287  hstrlem3  32293  hstrlem5  32295  sumdmdii  32447  sumdmdlem  32450  sumdmdlem2  32451