Theorem chelii 31168

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 31166	. 2 ⊢ 𝐻 ⊆ ℋ
3		cheli.1	. 2 ⊢ 𝐴 ∈ 𝐻
4	2, 3	sselii 3945	1 ⊢ 𝐴 ∈ ℋ

Syntax hints:   ∈ wcel 2109   ℋchba 30854   C_ℋ cch 30864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-hilex 30934

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-xp 5646  df-cnv 5648  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fv 6521  df-ov 7392  df-sh 31142  df-ch 31156

This theorem is referenced by: (None)