Theorem chelii 29595

Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
chssi.1	⊢ 𝐻 ∈ Cℋ
cheli.1	⊢ 𝐴 ∈ 𝐻

Assertion

Ref	Expression
chelii	⊢ 𝐴 ∈ ℋ

Proof of Theorem chelii

Step	Hyp	Ref	Expression
1		chssi.1	. . 3 ⊢ 𝐻 ∈ Cℋ
2	1	chssii 29593	. 2 ⊢ 𝐻 ⊆ ℋ
3		cheli.1	. 2 ⊢ 𝐴 ∈ 𝐻
4	2, 3	sselii 3918	1 ⊢ 𝐴 ∈ ℋ

Syntax hints:   ∈ wcel 2106   ℋchba 29281   C_ℋ cch 29291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-hilex 29361

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fv 6441  df-ov 7278  df-sh 29569  df-ch 29583

This theorem is referenced by: (None)