Theorem cheli 5138

Description: A member of a closed subspace of a Hilbert space is a vector.

Hypotheses

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

Assertion

Ref	Expression
cheli	⊢ A ∈ ℋ

Proof of Theorem cheli

Step	Hyp	Ref	Expression
1		chssi.1	. . 3 ⊢ H ∈ Cℋ
2	1	chssi 5136	. 2 ⊢ H ⊆ ℋ
3		cheli.1	. 2 ⊢ A ∈ H
4	2, 3	sselii 1505	1 ⊢ A ∈ ℋ

Syntax hints:   ∈ wcel 1092   ℋ chil 4958   C_ℋ cch 4968

This theorem is referenced by:  projlem14 5206  projlem18 5210  projlem19 5211  pjthlem12 5236  pjthlem13 5237  pjthlem14 5238

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-sh 5114  df-ch 5127

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