Theorem chelt 9100

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 9099	. . 3 |- (H e. CH -> H (_ H~)
2	1	sseld 2067	. 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 958  H~chil 8788  CHcch 8798

This theorem is referenced by:  pjspansnt 9500  pjidt 9640  atom1d 10280  sumdmdi 10342

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  df-ch 9092

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