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Theorem homulcl 26469
Description: The scalar product of a Hilbert space operator is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
homulcl  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )

Proof of Theorem homulcl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ffvelrn 6029 . . . . 5  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
2 hvmulcl 25721 . . . . 5  |-  ( ( A  e.  CC  /\  ( T `  x )  e.  ~H )  -> 
( A  .h  ( T `  x )
)  e.  ~H )
31, 2sylan2 474 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  x  e.  ~H )
)  ->  ( A  .h  ( T `  x
) )  e.  ~H )
43anassrs 648 . . 3  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( T `  x ) )  e.  ~H )
5 eqid 2467 . . 3  |-  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( A  .h  ( T `  x ) ) )
64, 5fmptd 6055 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( x  e.  ~H  |->  ( A  .h  ( T `  x )
) ) : ~H --> ~H )
7 hommval 26446 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
)  =  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) )
87feq1d 5722 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( ( A  .op  T ) : ~H --> ~H  <->  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) : ~H --> ~H ) )
96, 8mpbird 232 1  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767    |-> cmpt 4510   -->wf 5589   ` cfv 5593  (class class class)co 6294   CCcc 9500   ~Hchil 25627    .h csm 25629    .op chot 25647
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-hilex 25707  ax-hfvmul 25713
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-map 7432  df-homul 26441
This theorem is referenced by:  honegsubi  26506  homulid2  26510  homco1  26511  homulass  26512  hoadddi  26513  hoadddir  26514  hosubneg  26517  hosubdi  26518  honegsubdi  26520  honegsubdi2  26521  hosub4  26523  hosubsub4  26528  hosubeq0i  26536  nmopnegi  26675  homco2  26687  lnopmi  26710  hmopm  26731  nmophmi  26741  adjmul  26802  opsqrlem1  26850  opsqrlem6  26855
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