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Theorem shmulclOLD 22571
Description: Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
shmulclOLD  |-  ( H  e.  SH  ->  (
( A  e.  CC  /\  B  e.  H )  ->  ( A  .h  B )  e.  H
) )

Proof of Theorem shmulclOLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issh2 22561 . . . 4  |-  ( 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 . . 3  |-  ( 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 . 2  |-  ( H  e.  SH  ->  A. x  e.  CC  A. y  e.  H  ( x  .h  y )  e.  H
)
4 oveq1 6029 . . . 4  |-  ( x  =  A  ->  (
x  .h  y )  =  ( A  .h  y ) )
54eleq1d 2455 . . 3  |-  ( x  =  A  ->  (
( x  .h  y
)  e.  H  <->  ( A  .h  y )  e.  H
) )
6 oveq2 6030 . . . 4  |-  ( y  =  B  ->  ( A  .h  y )  =  ( A  .h  B ) )
76eleq1d 2455 . . 3  |-  ( y  =  B  ->  (
( A  .h  y
)  e.  H  <->  ( A  .h  B )  e.  H
) )
85, 7rspc2v 3003 . 2  |-  ( ( 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 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    = wceq 1649    e. wcel 1717   A.wral 2651    C_ wss 3265  (class class class)co 6022   CCcc 8923   ~Hchil 22272    +h cva 22273    .h csm 22274   0hc0v 22277   SHcsh 22281
This theorem is referenced by:  mayete3iOLD  23081
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 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-hilex 22352  ax-hfvadd 22353  ax-hfvmul 22358
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-sh 22559
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