Theorem sh0 9079

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

Assertion

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

Proof of Theorem sh0

Step	Hyp	Ref	Expression
1		sh 9073	. . 3 |- (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)))
2	1	pm3.26bi 322	. 2 |- (H e. SH -> (H (_ H~ /\ 0h e. H))
3	2	pm3.27d 325	1 |- (H e. SH -> 0h e. H)

Syntax hints:   -> wi 3   /\ 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:  ch0 9093  hhssabl 9127  hhssnv 9129  oc0 9158  ocin 9164  omlsi 9240  shscl 9276  shsel1t 9280  shintcl 9288  shunss 9332  sh0let 9359  shatomistic 10283

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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