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

Theorem shmulcl 22568
Description: Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)
Assertion
Ref Expression
shmulcl  |-  ( ( H  e.  SH  /\  A  e.  CC  /\  B  e.  H )  ->  ( A  .h  B )  e.  H )

Proof of Theorem shmulcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issh2 22559 . . . . 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 ) )
32simprd 450 . . 3  |-  ( H  e.  SH  ->  A. x  e.  CC  A. y  e.  H  ( x  .h  y )  e.  H
)
4 oveq1 6027 . . . . 5  |-  ( x  =  A  ->  (
x  .h  y )  =  ( A  .h  y ) )
54eleq1d 2453 . . . 4  |-  ( x  =  A  ->  (
( x  .h  y
)  e.  H  <->  ( A  .h  y )  e.  H
) )
6 oveq2 6028 . . . . 5  |-  ( y  =  B  ->  ( A  .h  y )  =  ( A  .h  B ) )
76eleq1d 2453 . . . 4  |-  ( y  =  B  ->  (
( A  .h  y
)  e.  H  <->  ( A  .h  B )  e.  H
) )
85, 7rspc2v 3001 . . 3  |-  ( ( A  e.  CC  /\  B  e.  H )  ->  ( A. x  e.  CC  A. y  e.  H  ( x  .h  y )  e.  H  ->  ( A  .h  B
)  e.  H ) )
93, 8syl5com 28 . 2  |-  ( H  e.  SH  ->  (
( A  e.  CC  /\  B  e.  H )  ->  ( A  .h  B )  e.  H
) )
1093impib 1151 1  |-  ( ( H  e.  SH  /\  A  e.  CC  /\  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 2649    C_ wss 3263  (class class class)co 6020   CCcc 8921   ~Hchil 22270    +h cva 22271    .h csm 22272   0hc0v 22275   SHcsh 22279
This theorem is referenced by:  shsubcl  22571  norm1exi  22600  hhssabloi  22610  hhssnv  22612  shsel3  22665  shscli  22667  shintcli  22679  pjhthlem1  22741  h1de2bi  22904  h1de2ctlem  22905  spansni  22907  spansnmul  22914  spansnss  22921  spanunsni  22929  h1datomi  22931  pjmulii  23027  mayete3i  23078  imaelshi  23409  strlem1  23601  cdj1i  23784  cdj3lem1  23785
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 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-hilex 22350  ax-hfvadd 22351  ax-hfvmul 22356
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-ov 6023  df-sh 22557
  Copyright terms: Public domain W3C validator