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Theorem issh 26837
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 26628 . . . 4  |-  ~H  e.  _V
21elpw2 4581 . . 3  |-  ( H  e.  ~P ~H  <->  H  C_  ~H )
3 3anass 986 . . 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 701 . 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 2493 . . . 4  |-  ( h  =  H  ->  ( 0h  e.  h  <->  0h  e.  H ) )
6 id 23 . . . . . . 7  |-  ( h  =  H  ->  h  =  H )
76sqxpeqd 4872 . . . . . 6  |-  ( h  =  H  ->  (
h  X.  h )  =  ( H  X.  H ) )
87imaeq2d 5180 . . . . 5  |-  ( h  =  H  ->  (  +h  " ( h  X.  h ) )  =  (  +h  " ( H  X.  H ) ) )
98, 6sseq12d 3490 . . . 4  |-  ( h  =  H  ->  (
(  +h  " (
h  X.  h ) )  C_  h  <->  (  +h  " ( H  X.  H
) )  C_  H
) )
10 xpeq2 4861 . . . . . 6  |-  ( h  =  H  ->  ( CC  X.  h )  =  ( CC  X.  H
) )
1110imaeq2d 5180 . . . . 5  |-  ( h  =  H  ->  (  .h  " ( CC  X.  h ) )  =  (  .h  " ( CC  X.  H ) ) )
1211, 6sseq12d 3490 . . . 4  |-  ( h  =  H  ->  (
(  .h  " ( CC  X.  h ) ) 
C_  h  <->  (  .h  " ( CC  X.  H
) )  C_  H
) )
135, 9, 123anbi123d 1335 . . 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 26836 . . 3  |-  SH  =  { h  e.  ~P ~H  |  ( 0h  e.  h  /\  (  +h  " ( h  X.  h ) )  C_  h  /\  (  .h  "
( CC  X.  h
) )  C_  h
) }
1513, 14elrab2 3228 . 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 653 . 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 280 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    C_ wss 3433   ~Pcpw 3976    X. cxp 4844   "cima 4849   CCcc 9533   ~Hchil 26548    +h cva 26549    .h csm 26550   0hc0v 26553   SHcsh 26557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-hilex 26628
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4477  df-xp 4852  df-cnv 4854  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-sh 26836
This theorem is referenced by:  issh2  26838  shss  26839  sh0  26845
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