HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  shaddcl Unicode version

Theorem shaddcl 21812
Description: Closure of vector addition in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
Assertion
Ref Expression
shaddcl  |-  ( ( H  e.  SH  /\  A  e.  H  /\  B  e.  H )  ->  ( A  +h  B
)  e.  H )

Proof of Theorem shaddcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issh2 21804 . . . . 5  |-  ( H  e.  SH  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  ( A. x  e.  H  A. y  e.  H  ( x  +h  y
)  e.  H  /\  A. x  e.  CC  A. y  e.  H  (
x  .h  y )  e.  H ) ) )
21simprbi 450 . . . 4  |-  ( H  e.  SH  ->  ( A. x  e.  H  A. y  e.  H  ( x  +h  y
)  e.  H  /\  A. x  e.  CC  A. y  e.  H  (
x  .h  y )  e.  H ) )
32simpld 445 . . 3  |-  ( H  e.  SH  ->  A. x  e.  H  A. y  e.  H  ( x  +h  y )  e.  H
)
4 oveq1 5881 . . . . 5  |-  ( x  =  A  ->  (
x  +h  y )  =  ( A  +h  y ) )
54eleq1d 2362 . . . 4  |-  ( x  =  A  ->  (
( x  +h  y
)  e.  H  <->  ( A  +h  y )  e.  H
) )
6 oveq2 5882 . . . . 5  |-  ( y  =  B  ->  ( A  +h  y )  =  ( A  +h  B
) )
76eleq1d 2362 . . . 4  |-  ( y  =  B  ->  (
( A  +h  y
)  e.  H  <->  ( A  +h  B )  e.  H
) )
85, 7rspc2v 2903 . . 3  |-  ( ( A  e.  H  /\  B  e.  H )  ->  ( A. x  e.  H  A. y  e.  H  ( x  +h  y )  e.  H  ->  ( A  +h  B
)  e.  H ) )
93, 8syl5com 26 . 2  |-  ( H  e.  SH  ->  (
( A  e.  H  /\  B  e.  H
)  ->  ( A  +h  B )  e.  H
) )
1093impib 1149 1  |-  ( ( H  e.  SH  /\  A  e.  H  /\  B  e.  H )  ->  ( A  +h  B
)  e.  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165  (class class class)co 5874   CCcc 8751   ~Hchil 21515    +h cva 21516    .h csm 21517   0hc0v 21520   SHcsh 21524
This theorem is referenced by:  shsubcl  21816  hhssabloi  21855  hhssnv  21857  shscli  21912  shintcli  21924  shsleji  21965  shsidmi  21979  pjhthlem1  21986  spanuni  22139  spanunsni  22174  sumspansn  22244  pjaddii  22270  imaelshi  22654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hilex 21595  ax-hfvadd 21596  ax-hfvmul 21601
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-sh 21802
  Copyright terms: Public domain W3C validator