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Theorem shmulcl 22712
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 22703 . . . . 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 6080 . . . . 5  |-  ( x  =  A  ->  (
x  .h  y )  =  ( A  .h  y ) )
54eleq1d 2501 . . . 4  |-  ( x  =  A  ->  (
( x  .h  y
)  e.  H  <->  ( A  .h  y )  e.  H
) )
6 oveq2 6081 . . . . 5  |-  ( y  =  B  ->  ( A  .h  y )  =  ( A  .h  B ) )
76eleq1d 2501 . . . 4  |-  ( y  =  B  ->  (
( A  .h  y
)  e.  H  <->  ( A  .h  B )  e.  H
) )
85, 7rspc2v 3050 . . 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 1652    e. wcel 1725   A.wral 2697    C_ wss 3312  (class class class)co 6073   CCcc 8980   ~Hchil 22414    +h cva 22415    .h csm 22416   0hc0v 22419   SHcsh 22423
This theorem is referenced by:  shsubcl  22715  norm1exi  22744  hhssabloi  22754  hhssnv  22756  shsel3  22809  shscli  22811  shintcli  22823  pjhthlem1  22885  h1de2bi  23048  h1de2ctlem  23049  spansni  23051  spansnmul  23058  spansnss  23065  spanunsni  23073  h1datomi  23075  pjmulii  23171  mayete3i  23222  imaelshi  23553  strlem1  23745  cdj1i  23928  cdj3lem1  23929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-hilex 22494  ax-hfvadd 22495  ax-hfvmul 22500
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-sh 22701
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