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Theorem shmulcl 25811
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 25802 . . . . 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 ) )
32simprd 463 . . 3  |-  ( H  e.  SH  ->  A. x  e.  CC  A. y  e.  H  ( x  .h  y )  e.  H
)
4 oveq1 6289 . . . . 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 6290 . . . . 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.  CC  /\  B  e.  H )  ->  ( A. x  e.  CC  A. y  e.  H  ( x  .h  y )  e.  H  ->  ( A  .h  B
)  e.  H ) )
93, 8syl5com 30 . 2  |-  ( H  e.  SH  ->  (
( A  e.  CC  /\  B  e.  H )  ->  ( A  .h  B )  e.  H
) )
1093impib 1194 1  |-  ( ( H  e.  SH  /\  A  e.  CC  /\  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 6282   CCcc 9486   ~Hchil 25512    +h cva 25513    .h csm 25514   0hc0v 25517   SHcsh 25521
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 25592  ax-hfvadd 25593  ax-hfvmul 25598
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-sh 25800
This theorem is referenced by:  shsubcl  25814  norm1exi  25844  hhssabloi  25854  hhssnv  25856  shsel3  25909  shscli  25911  shintcli  25923  pjhthlem1  25985  h1de2bi  26148  h1de2ctlem  26149  spansni  26151  spansnmul  26158  spansnss  26165  spanunsni  26173  h1datomi  26175  pjmulii  26271  mayete3i  26322  imaelshi  26653  strlem1  26845  cdj1i  27028  cdj3lem1  27029
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