Theorem shelt 9019

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

Assertion

Ref	Expression
shelt	|- ((H e. SH /\ A e. H) -> A e. H~)

Proof of Theorem shelt

Step	Hyp	Ref	Expression
1		shss 9018	. . 3 |- (H e. SH -> H (_ H~)
2	1	sseld 2063	. 2 |- (H e. SH -> (A e. H -> A e. H~))
3	2	imp 350	1 |- ((H e. SH /\ A e. H) -> A e. H~)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 956  H~chil 8727  SHcsh 8736

This theorem is referenced by:  shsel3t 9217  shscomt 9221  shsel1t 9223  elspanclt 9243  spansnsst 9434  sh1dle 10215

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-hilex 8808

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808  df-in 2047  df-ss 2049  df-sh 9015

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