Theorem sheli 9083

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

Hypotheses

Ref	Expression
shssi.1	|- H e. SH
sheli.1	|- A e. H

Assertion

Ref	Expression
sheli	|- A e. H~

Proof of Theorem sheli

Step	Hyp	Ref	Expression
1		shssi.1	. . 3 |- H e. SH
2	1	shssi 9081	. 2 |- H (_ H~
3		sheli.1	. 2 |- A e. H
4	2, 3	sselii 2066	1 |- A e. H~

Syntax hints:   e. wcel 958  H~chil 8788  SHcsh 8797

This theorem is referenced by:  omlsilem 9244

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-sh 9076

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