Theorem cheli 9098

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

Hypotheses

Ref	Expression
chssi.1	|- H e. CH
cheli.1	|- A e. H

Assertion

Ref	Expression
cheli	|- A e. H~

Proof of Theorem cheli

Step	Hyp	Ref	Expression
1		chssi.1	. . 3 |- H e. CH
2	1	chssi 9096	. 2 |- H (_ H~
3		cheli.1	. 2 |- A e. H
4	2, 3	sselii 2069	1 |- A e. H~

Syntax hints:   e. wcel 960  H~chil 8783  CHcch 8793

This theorem is referenced by:  projlem14 9194  projlem18 9198  projlem19 9199  pjthlem12 9225  pjthlem13 9226  pjthlem14 9227

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-in 2054  df-ss 2056  df-sh 9071  df-ch 9087

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