Theorem sh0 28073

Description: The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sh0

Step	Hyp	Ref	Expression
1		issh 28065	. . 3 |- ( H e. SH <-> ( ( H C_ ~H /\ 0h e. H ) /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) )
2	1	simplbi 476	. 2 |- ( H e. SH -> ( H C_ ~H /\ 0h e. H ) )
3	2	simprd 479	1 |- ( H e. SH -> 0h e. H )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   C_ wss 3574   X. cxp 5112   "cima 5117   CCcc 9934   ~Hchil 27776   +h cva 27777   .h csm 27778   0hc0v 27781   SHcsh 27785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-sh 28064

This theorem is referenced by:  ch0  28085  hhssabloilem  28118  hhssnv  28121  oc0  28149  ocin  28155  shscli  28176  shsel1  28180  shintcli  28188  shunssi  28227  omlsii  28262  sh0le  28299  imaelshi  28917  shatomistici  29220