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Theorem issh 22702
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 22494 . . . 4  |-  ~H  e.  _V
21elpw2 4356 . . 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 2496 . . . 4  |-  ( h  =  H  ->  ( 0h  e.  h  <->  0h  e.  H ) )
6 id 20 . . . . . . 7  |-  ( h  =  H  ->  h  =  H )
76, 6xpeq12d 4895 . . . . . 6  |-  ( h  =  H  ->  (
h  X.  h )  =  ( H  X.  H ) )
87imaeq2d 5195 . . . . 5  |-  ( h  =  H  ->  (  +h  " ( h  X.  h ) )  =  (  +h  " ( H  X.  H ) ) )
98, 6sseq12d 3369 . . . 4  |-  ( h  =  H  ->  (
(  +h  " (
h  X.  h ) )  C_  h  <->  (  +h  " ( H  X.  H
) )  C_  H
) )
10 xpeq2 4885 . . . . . 6  |-  ( h  =  H  ->  ( CC  X.  h )  =  ( CC  X.  H
) )
1110imaeq2d 5195 . . . . 5  |-  ( h  =  H  ->  (  .h  " ( CC  X.  h ) )  =  (  .h  " ( CC  X.  H ) ) )
1211, 6sseq12d 3369 . . . 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 22701 . . 3  |-  SH  =  { h  e.  ~P ~H  |  ( 0h  e.  h  /\  (  +h  " ( h  X.  h ) )  C_  h  /\  (  .h  "
( CC  X.  h
) )  C_  h
) }
1513, 14elrab2 3086 . 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 1652    e. wcel 1725    C_ wss 3312   ~Pcpw 3791    X. cxp 4868   "cima 4873   CCcc 8980   ~Hchil 22414    +h cva 22415    .h csm 22416   0hc0v 22419   SHcsh 22423
This theorem is referenced by:  issh2  22703  shss  22704  sh0  22710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-hilex 22494
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-sh 22701
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