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Theorem issh 26942
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 26733 . . . 4  |-  ~H  e.  _V
21elpw2 4565 . . 3  |-  ( H  e.  ~P ~H  <->  H  C_  ~H )
3 3anass 1011 . . 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 711 . 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 2538 . . . 4  |-  ( h  =  H  ->  ( 0h  e.  h  <->  0h  e.  H ) )
6 id 22 . . . . . . 7  |-  ( h  =  H  ->  h  =  H )
76sqxpeqd 4865 . . . . . 6  |-  ( h  =  H  ->  (
h  X.  h )  =  ( H  X.  H ) )
87imaeq2d 5174 . . . . 5  |-  ( h  =  H  ->  (  +h  " ( h  X.  h ) )  =  (  +h  " ( H  X.  H ) ) )
98, 6sseq12d 3447 . . . 4  |-  ( h  =  H  ->  (
(  +h  " (
h  X.  h ) )  C_  h  <->  (  +h  " ( H  X.  H
) )  C_  H
) )
10 xpeq2 4854 . . . . . 6  |-  ( h  =  H  ->  ( CC  X.  h )  =  ( CC  X.  H
) )
1110imaeq2d 5174 . . . . 5  |-  ( h  =  H  ->  (  .h  " ( CC  X.  h ) )  =  (  .h  " ( CC  X.  H ) ) )
1211, 6sseq12d 3447 . . . 4  |-  ( h  =  H  ->  (
(  .h  " ( CC  X.  h ) ) 
C_  h  <->  (  .h  " ( CC  X.  H
) )  C_  H
) )
135, 9, 123anbi123d 1365 . . 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 26941 . . 3  |-  SH  =  { h  e.  ~P ~H  |  ( 0h  e.  h  /\  (  +h  " ( h  X.  h ) )  C_  h  /\  (  .h  "
( CC  X.  h
) )  C_  h
) }
1513, 14elrab2 3186 . 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 661 . 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 285 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 setvar class
Syntax hints:    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    C_ wss 3390   ~Pcpw 3942    X. cxp 4837   "cima 4842   CCcc 9555   ~Hchil 26653    +h cva 26654    .h csm 26655   0hc0v 26658   SHcsh 26662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-hilex 26733
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-sh 26941
This theorem is referenced by:  issh2  26943  shss  26944  sh0  26950
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