Theorem sh2 9091

Description: Subspace H of a Hilbert space.

Assertion

Ref	Expression
sh2	|- (H (_ H~ -> (H e. SH <-> (0h e. H /\ (A.x e. H A.y e. H (x +h y) e. H /\ A.x e. CC A.y e. H (x .h y) e. H))))

Distinct variable group:   x,y,H

Proof of Theorem sh2

Step	Hyp	Ref	Expression
1		anass 439	. . 3 |- (((H (_ H~ /\ 0h e. H) /\ (A.x e. H A.y e. H (x +h y) e. H /\ A.x e. CC A.y e. H (x .h y) e. H)) <-> (H (_ H~ /\ (0h e. H /\ (A.x e. H A.y e. H (x +h y) e. H /\ A.x e. CC A.y e. H (x .h y) e. H))))
2	1	baib 685	. 2 |- (H (_ H~ -> (((H (_ H~ /\ 0h e. H) /\ (A.x e. H A.y e. H (x +h y) e. H /\ A.x e. CC A.y e. H (x .h y) e. H)) <-> (0h e. H /\ (A.x e. H A.y e. H (x +h y) e. H /\ A.x e. CC A.y e. H (x .h y) e. H))))
3		sh 9078	. 2 |- (H e. SH <-> ((H (_ H~ /\ 0h e. H) /\ (A.x e. H A.y e. H (x +h y) e. H /\ A.x e. CC A.y e. H (x .h y) e. H)))
4	2, 3	syl5bb 532	1 |- (H (_ H~ -> (H e. SH <-> (0h e. H /\ (A.x e. H A.y e. H (x +h y) e. H /\ A.x e. CC A.y e. H (x .h y) e. H))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 958  A.wral 1645   (_ wss 2047  (class class class)co 3963  CCcc 5232  H~chil 8788   +h cva 8789   .h csm 8790  0hc0v 8791  SHcsh 8797

This theorem is referenced by:  nlelsh 9993  hmopidmch 10079

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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