Theorem chelii 31329

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 31327	. 2 ⊢ 𝐻 ⊆ ℋ
3		cheli.1	. 2 ⊢ 𝐴 ∈ 𝐻
4	2, 3	sselii 3919	1 ⊢ 𝐴 ∈ ℋ

Syntax hints:   ∈ wcel 2119   ℋchba 31015   C_ℋ cch 31025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-hilex 31095

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fv 6500  df-ov 7366  df-sh 31303  df-ch 31317

This theorem is referenced by: (None)