Theorem sh 5116

Description: Subspace H of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. Definition of [Beran] p. 95.

Assertion

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

Distinct variable group(s):   x,y,H

Proof of Theorem sh

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 |- (H e. SH -> H e. V)
2		ax-hilex 4983	. . . 4 |- H~ e. V
3	2	ssex 1700	. . 3 |- (H (_ H~ -> H e. V)
4	3	ad2antll 320	. 2 |- (((H (_ H~ /\ 0v e. H) /\ (A.x e. H A.y e. H (x +v y) e. H /\ A.x e. CC A.y e. H (x .s y) e. H)) -> H e. V)
5		sseq1 1521	. . . . 5 |- (h = H -> (h (_ H~ <-> H (_ H~))
6		eleq2 1150	. . . . 5 |- (h = H -> (0v e. h <-> 0v e. H))
7	5, 6	anbi12d 476	. . . 4 |- (h = H -> ((h (_ H~ /\ 0v e. h) <-> (H (_ H~ /\ 0v e. H)))
8		eleq2 1150	. . . . . . 7 |- (h = H -> ((x +v y) e. h <-> (x +v y) e. H))
9	8	raleqd 1327	. . . . . 6 |- (h = H -> (A.y e. h (x +v y) e. h <-> A.y e. H (x +v y) e. H))
10	9	raleqd 1327	. . . . 5 |- (h = H -> (A.x e. h A.y e. h (x +v y) e. h <-> A.x e. H A.y e. H (x +v y) e. H))
11		eleq2 1150	. . . . . . 7 |- (h = H -> ((x .s y) e. h <-> (x .s y) e. H))
12	11	raleqd 1327	. . . . . 6 |- (h = H -> (A.y e. h (x .s y) e. h <-> A.y e. H (x .s y) e. H))
13	12	biraldv 1219	. . . . 5 |- (h = H -> (A.x e. CC A.y e. h (x .s y) e. h <-> A.x e. CC A.y e. H (x .s y) e. H))
14	10, 13	anbi12d 476	. . . 4 |- (h = H -> ((A.x e. h A.y e. h (x +v y) e. h /\ A.x e. CC A.y e. h (x .s y) e. h) <-> (A.x e. H A.y e. H (x +v y) e. H /\ A.x e. CC A.y e. H (x .s y) e. H)))
15	7, 14	anbi12d 476	. . 3 |- (h = H -> (((h (_ H~ /\ 0v e. h) /\ (A.x e. h A.y e. h (x +v y) e. h /\ A.x e. CC A.y e. h (x .s y) e. h)) <-> ((H (_ H~ /\ 0v e. H) /\ (A.x e. H A.y e. H (x +v y) e. H /\ A.x e. CC A.y e. H (x .s y) e. H))))
16		df-sh 5114	. . 3 |- SH = {h | ((h (_ H~ /\ 0v e. h) /\ (A.x e. h A.y e. h (x +v y) e. h /\ A.x e. CC A.y e. h (x .s y) e. h))}
17	15, 16	elab2g 1418	. 2 |- (H e. V -> (H e. SH <-> ((H (_ H~ /\ 0v e. H) /\ (A.x e. H A.y e. H (x +v y) e. H /\ A.x e. CC A.y e. H (x .s y) e. H))))
18	1, 4, 17	pm5.21nii 504	1 |- (H e. SH <-> ((H (_ H~ /\ 0v e. H) /\ (A.x e. H A.y e. H (x +v y) e. H /\ A.x e. CC A.y e. H (x .s y) e. H)))

Syntax hints:   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201  Vcvv 1348   (_ wss 1487  (class class class)co 3001  CCcc 4026  H~chil 4958   +v cva 4959   .s csm 4960  0vc0v 4961  SHcsh 4967

This theorem is referenced by:  shss 5117  sh0 5122  shaddclt 5123  shmulclt 5124  sh2 5126  helch 5151  hsn0elch 5155  ocsh 5164  shscl 5282  shintcl 5294

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-sh 5114

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