Theorem chelt 9095

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

Assertion

Ref	Expression
chelt	|- ((H e. CH /\ A e. H) -> A e. H~)

Proof of Theorem chelt

Step	Hyp	Ref	Expression
1		chss 9094	. . 3 |- (H e. CH -> H (_ H~)
2	1	sseld 2070	. 2 |- (H e. CH -> (A e. H -> A e. H~))
3	2	imp 350	1 |- ((H e. CH /\ A e. H) -> A e. H~)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 960  H~chil 8783  CHcch 8793

This theorem is referenced by:  pjspansnt 9495  pjidt 9635  atom1d 10275  sumdmdi 10337

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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