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Theorem issh 22559
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 22351 . . . 4  |-  ~H  e.  _V
21elpw2 4306 . . 3  |-  ( H  e.  ~P ~H  <->  H  C_  ~H )
3 3anass 940 . . 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 679 . 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 2449 . . . 4  |-  ( h  =  H  ->  ( 0h  e.  h  <->  0h  e.  H ) )
6 id 20 . . . . . . 7  |-  ( h  =  H  ->  h  =  H )
76, 6xpeq12d 4844 . . . . . 6  |-  ( h  =  H  ->  (
h  X.  h )  =  ( H  X.  H ) )
87imaeq2d 5144 . . . . 5  |-  ( h  =  H  ->  (  +h  " ( h  X.  h ) )  =  (  +h  " ( H  X.  H ) ) )
98, 6sseq12d 3321 . . . 4  |-  ( h  =  H  ->  (
(  +h  " (
h  X.  h ) )  C_  h  <->  (  +h  " ( H  X.  H
) )  C_  H
) )
10 xpeq2 4834 . . . . . 6  |-  ( h  =  H  ->  ( CC  X.  h )  =  ( CC  X.  H
) )
1110imaeq2d 5144 . . . . 5  |-  ( h  =  H  ->  (  .h  " ( CC  X.  h ) )  =  (  .h  " ( CC  X.  H ) ) )
1211, 6sseq12d 3321 . . . 4  |-  ( h  =  H  ->  (
(  .h  " ( CC  X.  h ) ) 
C_  h  <->  (  .h  " ( CC  X.  H
) )  C_  H
) )
135, 9, 123anbi123d 1254 . . 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 22558 . . 3  |-  SH  =  { h  e.  ~P ~H  |  ( 0h  e.  h  /\  (  +h  " ( h  X.  h ) )  C_  h  /\  (  .h  "
( CC  X.  h
) )  C_  h
) }
1513, 14elrab2 3038 . 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 631 . 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 269 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3264   ~Pcpw 3743    X. cxp 4817   "cima 4822   CCcc 8922   ~Hchil 22271    +h cva 22272    .h csm 22273   0hc0v 22276   SHcsh 22280
This theorem is referenced by:  issh2  22560  shss  22561  sh0  22567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-hilex 22351
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-cnv 4827  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-sh 22558
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