Theorem cheli 31834

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 31833	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3927	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ℋchba 31521   C_ℋ cch 31531

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-hilex 31601

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fv 6546  df-ov 7423  df-sh 31809  df-ch 31823

This theorem is used by:  pjhthlem1  31993  pjhthlem2  31994  h1de2ci  32158  spanunsni  32181  spansncvi  32254  3oalem1  32264  pjcompi  32274  pjocini  32300  pjjsi  32302  pjrni  32304  pjdsi  32314  pjds3i  32315  mayete3i  32330  riesz3i  32664  pjnmopi  32750  pjnormssi  32770  pjimai  32778  pjclem4a  32800  pjclem4  32801  pj3lem1  32808  pj3si  32809  strlem1  32852  strlem3  32855  strlem5  32857  hstrlem3  32863  hstrlem5  32865  sumdmdii  33017  sumdmdlem  33020  sumdmdlem2  33021