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Theorem shaddcl 22568
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 22560 . . . . 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 451 . . . 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 446 . . 3  |-  ( H  e.  SH  ->  A. x  e.  H  A. y  e.  H  ( x  +h  y )  e.  H
)
4 oveq1 6028 . . . . 5  |-  ( x  =  A  ->  (
x  +h  y )  =  ( A  +h  y ) )
54eleq1d 2454 . . . 4  |-  ( x  =  A  ->  (
( x  +h  y
)  e.  H  <->  ( A  +h  y )  e.  H
) )
6 oveq2 6029 . . . . 5  |-  ( y  =  B  ->  ( A  +h  y )  =  ( A  +h  B
) )
76eleq1d 2454 . . . 4  |-  ( y  =  B  ->  (
( A  +h  y
)  e.  H  <->  ( A  +h  B )  e.  H
) )
85, 7rspc2v 3002 . . 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 28 . 2  |-  ( H  e.  SH  ->  (
( A  e.  H  /\  B  e.  H
)  ->  ( A  +h  B )  e.  H
) )
1093impib 1151 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 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650    C_ wss 3264  (class class class)co 6021   CCcc 8922   ~Hchil 22271    +h cva 22272    .h csm 22273   0hc0v 22276   SHcsh 22280
This theorem is referenced by:  shsubcl  22572  hhssabloi  22611  hhssnv  22613  shscli  22668  shintcli  22680  shsleji  22721  shsidmi  22735  pjhthlem1  22742  spanuni  22895  spanunsni  22930  sumspansn  23000  pjaddii  23026  imaelshi  23410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-hilex 22351  ax-hfvadd 22352  ax-hfvmul 22357
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-ov 6024  df-sh 22558
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