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Theorem homulcl 23267
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 5871 . . . . 5  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
2 hvmulcl 22521 . . . . 5  |-  ( ( A  e.  CC  /\  ( T `  x )  e.  ~H )  -> 
( A  .h  ( T `  x )
)  e.  ~H )
31, 2sylan2 462 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  x  e.  ~H )
)  ->  ( A  .h  ( T `  x
) )  e.  ~H )
43anassrs 631 . . 3  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( T `  x ) )  e.  ~H )
5 eqid 2438 . . 3  |-  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( A  .h  ( T `  x ) ) )
64, 5fmptd 5896 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( x  e.  ~H  |->  ( A  .h  ( T `  x )
) ) : ~H --> ~H )
7 hommval 23244 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
)  =  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) )
87feq1d 5583 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( ( A  .op  T ) : ~H --> ~H  <->  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) : ~H --> ~H ) )
96, 8mpbird 225 1  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726    e. cmpt 4269   -->wf 5453   ` cfv 5457  (class class class)co 6084   CCcc 8993   ~Hchil 22427    .h csm 22429    .op chot 22447
This theorem is referenced by:  honegsubi  23304  homulid2  23308  homco1  23309  homulass  23310  hoadddi  23311  hoadddir  23312  hosubneg  23315  hosubdi  23316  honegsubdi  23318  honegsubdi2  23319  hosub4  23321  hosubsub4  23326  hosubeq0i  23334  nmopnegi  23473  homco2  23485  lnopmi  23508  hmopm  23529  nmophmi  23539  adjmul  23600  opsqrlem1  23648  opsqrlem6  23653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-hilex 22507  ax-hfvmul 22513
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-homul 23239
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