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

Theorem shaddcl 25838
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 25830 . . . . 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 464 . . . 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 459 . . 3  |-  ( H  e.  SH  ->  A. x  e.  H  A. y  e.  H  ( x  +h  y )  e.  H
)
4 oveq1 6291 . . . . 5  |-  ( x  =  A  ->  (
x  +h  y )  =  ( A  +h  y ) )
54eleq1d 2536 . . . 4  |-  ( x  =  A  ->  (
( x  +h  y
)  e.  H  <->  ( A  +h  y )  e.  H
) )
6 oveq2 6292 . . . . 5  |-  ( y  =  B  ->  ( A  +h  y )  =  ( A  +h  B
) )
76eleq1d 2536 . . . 4  |-  ( y  =  B  ->  (
( A  +h  y
)  e.  H  <->  ( A  +h  B )  e.  H
) )
85, 7rspc2v 3223 . . 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 30 . 2  |-  ( H  e.  SH  ->  (
( A  e.  H  /\  B  e.  H
)  ->  ( A  +h  B )  e.  H
) )
1093impib 1194 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476  (class class class)co 6284   CCcc 9490   ~Hchil 25540    +h cva 25541    .h csm 25542   0hc0v 25545   SHcsh 25549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-hilex 25620  ax-hfvadd 25621  ax-hfvmul 25626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-sh 25828
This theorem is referenced by:  shsubcl  25842  hhssabloi  25882  hhssnv  25884  shscli  25939  shintcli  25951  shsleji  25992  shsidmi  26006  pjhthlem1  26013  spanuni  26166  spanunsni  26201  sumspansn  26271  pjaddii  26297  imaelshi  26681
  Copyright terms: Public domain W3C validator