Theorem shel 9082

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

Hypothesis

Ref	Expression
shssi.1	|- H e. SH

Assertion

Ref	Expression
shel	|- (A e. H -> A e. H~)

Proof of Theorem shel

Step	Hyp	Ref	Expression
1		shssi.1	. . 3 |- H e. SH
2	1	shssi 9081	. 2 |- H (_ H~
3	2	sseli 2065	1 |- (A e. H -> A e. H~)

Syntax hints:   -> wi 3   e. wcel 958  H~chil 8788  SHcsh 8797

This theorem is referenced by:  norm1ex 9122  hhssabl 9132  hhssnv 9134  chocuni 9172  omlsi 9245  shscl 9281  shunss 9337  shmods 9362  5oalem1 9599  5oalem2 9600  5oalem3 9601  5oalem5 9603  nlelch 9994  pjima 10104  shatomic 10285  shatomistic 10288  cdjreu 10359  cdj1 10360  cdj3lem1 10361  cdj3lem2b 10364  cdj3lem3 10365  cdj3lem3b 10367  cdj3 10368

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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