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

Theorem shaddcl 26705
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 26697 . . . . 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 465 . . . 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 460 . . 3  |-  ( H  e.  SH  ->  A. x  e.  H  A. y  e.  H  ( x  +h  y )  e.  H
)
4 oveq1 6312 . . . . 5  |-  ( x  =  A  ->  (
x  +h  y )  =  ( A  +h  y ) )
54eleq1d 2498 . . . 4  |-  ( x  =  A  ->  (
( x  +h  y
)  e.  H  <->  ( A  +h  y )  e.  H
) )
6 oveq2 6313 . . . . 5  |-  ( y  =  B  ->  ( A  +h  y )  =  ( A  +h  B
) )
76eleq1d 2498 . . . 4  |-  ( y  =  B  ->  (
( A  +h  y
)  e.  H  <->  ( A  +h  B )  e.  H
) )
85, 7rspc2v 3197 . . 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 31 . 2  |-  ( H  e.  SH  ->  (
( A  e.  H  /\  B  e.  H
)  ->  ( A  +h  B )  e.  H
) )
1093impib 1203 1  |-  ( ( H  e.  SH  /\  A  e.  H  /\  B  e.  H )  ->  ( A  +h  B
)  e.  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782    C_ wss 3442  (class class class)co 6305   CCcc 9536   ~Hchil 26407    +h cva 26408    .h csm 26409   0hc0v 26412   SHcsh 26416
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 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-hilex 26487  ax-hfvadd 26488  ax-hfvmul 26493
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-sh 26695
This theorem is referenced by:  shsubcl  26708  hhssabloi  26748  hhssnv  26750  shscli  26805  shintcli  26817  shsleji  26858  shsidmi  26872  pjhthlem1  26879  spanuni  27032  spanunsni  27067  sumspansn  27137  pjaddii  27163  imaelshi  27546
  Copyright terms: Public domain W3C validator