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Theorem shmulcl 24765
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 24756 . . . . 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 6200 . . . . 5  |-  ( x  =  A  ->  (
x  .h  y )  =  ( A  .h  y ) )
54eleq1d 2520 . . . 4  |-  ( x  =  A  ->  (
( x  .h  y
)  e.  H  <->  ( A  .h  y )  e.  H
) )
6 oveq2 6201 . . . . 5  |-  ( y  =  B  ->  ( A  .h  y )  =  ( A  .h  B ) )
76eleq1d 2520 . . . 4  |-  ( y  =  B  ->  (
( A  .h  y
)  e.  H  <->  ( A  .h  B )  e.  H
) )
85, 7rspc2v 3179 . . 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 1186 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 965    = wceq 1370    e. wcel 1758   A.wral 2795    C_ wss 3429  (class class class)co 6193   CCcc 9384   ~Hchil 24466    +h cva 24467    .h csm 24468   0hc0v 24471   SHcsh 24475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-hilex 24546  ax-hfvadd 24547  ax-hfvmul 24552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-sh 24754
This theorem is referenced by:  shsubcl  24768  norm1exi  24798  hhssabloi  24808  hhssnv  24810  shsel3  24863  shscli  24865  shintcli  24877  pjhthlem1  24939  h1de2bi  25102  h1de2ctlem  25103  spansni  25105  spansnmul  25112  spansnss  25119  spanunsni  25127  h1datomi  25129  pjmulii  25225  mayete3i  25276  imaelshi  25607  strlem1  25799  cdj1i  25982  cdj3lem1  25983
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