Theorem cheli 27473

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 27472	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3564	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 1977   ℋchil 27160   C_ℋ cch 27170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-hilex 27240

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-ov 6552  df-sh 27448  df-ch 27462

This theorem is referenced by:  pjhthlem1  27634  pjhthlem2  27635  h1de2ci  27799  spanunsni  27822  spansncvi  27895  3oalem1  27905  pjcompi  27915  pjocini  27941  pjjsi  27943  pjrni  27945  pjdsi  27955  pjds3i  27956  mayete3i  27971  riesz3i  28305  pjnmopi  28391  pjnormssi  28411  pjimai  28419  pjclem4a  28441  pjclem4  28442  pj3lem1  28449  pj3si  28450  strlem1  28493  strlem3  28496  strlem5  28498  hstrlem3  28504  hstrlem5  28506  sumdmdii  28658  sumdmdlem  28661  sumdmdlem2  28662