Theorem chelii 31319

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 31317	. 2 ⊢ 𝐻 ⊆ ℋ
3		cheli.1	. 2 ⊢ 𝐴 ∈ 𝐻
4	2, 3	sselii 3919	1 ⊢ 𝐴 ∈ ℋ

Syntax hints:   ∈ wcel 2114   ℋchba 31005   C_ℋ cch 31015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-hilex 31085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fv 6500  df-ov 7363  df-sh 31293  df-ch 31307

This theorem is referenced by: (None)