Theorem sh2 9030

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 684	. 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 9017	. 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 531	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 956  A.wral 1642   (_ wss 2043  (class class class)co 3954  CCcc 5212  H~chil 8727   +h cva 8728   .h csm 8729  0hc0v 8730  SHcsh 8736

This theorem is referenced by:  nlelsh 9931  hmopidmch 10017

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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