Theorem sh2 9086

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 441	. . 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 687	. 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 9073	. 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 534	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 960  A.wral 1648   (_ wss 2050  (class class class)co 3969  CCcc 5244  H~chil 8783   +h cva 8784   .h csm 8785  0hc0v 8786  SHcsh 8792

This theorem is referenced by:  nlelsh 9988  hmopidmch 10074

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

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