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Theorem issh 21787
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. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
issh  |-  ( H  e.  SH  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  (
(  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) )

Proof of Theorem issh
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 21579 . . . 4  |-  ~H  e.  _V
21elpw2 4175 . . 3  |-  ( H  e.  ~P ~H  <->  H  C_  ~H )
3 3anass 938 . . 3  |-  ( ( 0h  e.  H  /\  (  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H )  <->  ( 0h  e.  H  /\  (
(  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) )
42, 3anbi12i 678 . 2  |-  ( ( H  e.  ~P ~H  /\  ( 0h  e.  H  /\  (  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) )  <->  ( H  C_ 
~H  /\  ( 0h  e.  H  /\  (
(  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) ) )
5 eleq2 2344 . . . 4  |-  ( h  =  H  ->  ( 0h  e.  h  <->  0h  e.  H ) )
6 id 19 . . . . . . 7  |-  ( h  =  H  ->  h  =  H )
76, 6xpeq12d 4714 . . . . . 6  |-  ( h  =  H  ->  (
h  X.  h )  =  ( H  X.  H ) )
87imaeq2d 5012 . . . . 5  |-  ( h  =  H  ->  (  +h  " ( h  X.  h ) )  =  (  +h  " ( H  X.  H ) ) )
98, 6sseq12d 3207 . . . 4  |-  ( h  =  H  ->  (
(  +h  " (
h  X.  h ) )  C_  h  <->  (  +h  " ( H  X.  H
) )  C_  H
) )
10 xpeq2 4704 . . . . . 6  |-  ( h  =  H  ->  ( CC  X.  h )  =  ( CC  X.  H
) )
1110imaeq2d 5012 . . . . 5  |-  ( h  =  H  ->  (  .h  " ( CC  X.  h ) )  =  (  .h  " ( CC  X.  H ) ) )
1211, 6sseq12d 3207 . . . 4  |-  ( h  =  H  ->  (
(  .h  " ( CC  X.  h ) ) 
C_  h  <->  (  .h  " ( CC  X.  H
) )  C_  H
) )
135, 9, 123anbi123d 1252 . . 3  |-  ( h  =  H  ->  (
( 0h  e.  h  /\  (  +h  " (
h  X.  h ) )  C_  h  /\  (  .h  " ( CC  X.  h ) ) 
C_  h )  <->  ( 0h  e.  H  /\  (  +h  " ( H  X.  H ) )  C_  H  /\  (  .h  "
( CC  X.  H
) )  C_  H
) ) )
14 df-sh 21786 . . 3  |-  SH  =  { h  e.  ~P ~H  |  ( 0h  e.  h  /\  (  +h  " ( h  X.  h ) )  C_  h  /\  (  .h  "
( CC  X.  h
) )  C_  h
) }
1513, 14elrab2 2925 . 2  |-  ( H  e.  SH  <->  ( H  e.  ~P ~H  /\  ( 0h  e.  H  /\  (  +h  " ( H  X.  H ) )  C_  H  /\  (  .h  "
( CC  X.  H
) )  C_  H
) ) )
16 anass 630 . 2  |-  ( ( ( H  C_  ~H  /\ 
0h  e.  H )  /\  ( (  +h  " ( H  X.  H ) )  C_  H  /\  (  .h  "
( CC  X.  H
) )  C_  H
) )  <->  ( H  C_ 
~H  /\  ( 0h  e.  H  /\  (
(  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) ) )
174, 15, 163bitr4i 268 1  |-  ( H  e.  SH  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  (
(  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625    X. cxp 4687   "cima 4692   CCcc 8735   ~Hchil 21499    +h cva 21500    .h csm 21501   0hc0v 21504   SHcsh 21508
This theorem is referenced by:  issh2  21788  shss  21789  sh0  21795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-sh 21786
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