Theorem cheli 31435

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 31434	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3932	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2142   ℋchba 31122   C_ℋ cch 31132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-hilex 31202

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5653  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fv 6529  df-ov 7399  df-sh 31410  df-ch 31424

This theorem is referenced by:  pjhthlem1  31594  pjhthlem2  31595  h1de2ci  31759  spanunsni  31782  spansncvi  31855  3oalem1  31865  pjcompi  31875  pjocini  31901  pjjsi  31903  pjrni  31905  pjdsi  31915  pjds3i  31916  mayete3i  31931  riesz3i  32265  pjnmopi  32351  pjnormssi  32371  pjimai  32379  pjclem4a  32401  pjclem4  32402  pj3lem1  32409  pj3si  32410  strlem1  32453  strlem3  32456  strlem5  32458  hstrlem3  32464  hstrlem5  32466  sumdmdii  32618  sumdmdlem  32621  sumdmdlem2  32622