Theorem shss 28067

Description: A subspace is a subset of Hilbert space. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
shss	|- ( H e. SH -> H C_ ~H )

Proof of Theorem shss

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	simpld 475	1 |- ( H e. SH -> H C_ ~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:  shel  28068  shex  28069  shssii  28070  shsubcl  28077  chss  28086  shsspwh  28103  hhsssh  28126  shocel  28141  shocsh  28143  ocss  28144  shocss  28145  shocorth  28151  shococss  28153  shorth  28154  shoccl  28164  shsel  28173  shintcli  28188  spanid  28206  shjval  28210  shjcl  28215  shlej1  28219  shlub  28273  chscllem2  28497  chscllem4  28499