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Theorem homulcl 22773
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 5770 . . . . 5  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
2 hvmulcl 22027 . . . . 5  |-  ( ( A  e.  CC  /\  ( T `  x )  e.  ~H )  -> 
( A  .h  ( T `  x )
)  e.  ~H )
31, 2sylan2 460 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  x  e.  ~H )
)  ->  ( A  .h  ( T `  x
) )  e.  ~H )
43anassrs 629 . . 3  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( T `  x ) )  e.  ~H )
5 eqid 2366 . . 3  |-  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( A  .h  ( T `  x ) ) )
64, 5fmptd 5795 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( x  e.  ~H  |->  ( A  .h  ( T `  x )
) ) : ~H --> ~H )
7 hommval 22750 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
)  =  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) )
87feq1d 5484 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( ( A  .op  T ) : ~H --> ~H  <->  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) : ~H --> ~H ) )
96, 8mpbird 223 1  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1715    e. cmpt 4179   -->wf 5354   ` cfv 5358  (class class class)co 5981   CCcc 8882   ~Hchil 21933    .h csm 21935    .op chot 21953
This theorem is referenced by:  honegsubi  22810  homulid2  22814  homco1  22815  homulass  22816  hoadddi  22817  hoadddir  22818  hosubneg  22821  hosubdi  22822  honegsubdi  22824  honegsubdi2  22825  hosub4  22827  hosubsub4  22832  hosubeq0i  22840  nmopnegi  22979  homco2  22991  lnopmi  23014  hmopm  23035  nmophmi  23045  adjmul  23106  opsqrlem1  23154  opsqrlem6  23159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-hilex 22013  ax-hfvmul 22019
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-map 6917  df-homul 22745
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