Theorem sh0 5122

Description: The zero vector belongs to any subspace of a Hilbert space.

Assertion

Ref	Expression
sh0	|- (H e. SH -> 0v e. H)

Proof of Theorem sh0

Step	Hyp	Ref	Expression
1		sh 5116	. . 3 |- (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)))
2	1	pm3.26bd 259	. 2 |- (H e. SH -> (H (_ H~ /\ 0v e. H))
3	2	pm3.27d 262	1 |- (H e. SH -> 0v e. H)

Syntax hints:   -> wi 2   /\ wa 196   e. wcel 1092  A.wral 1201   (_ 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:  ch0 5133  oc0 5171  ocin 5177  omlsi 5250  shscl 5282  shsel1t 5286  shintcl 5294  shunss 5338  sh0let 5365

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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