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Theorem hommval 27224
Description: Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hommval  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
)  =  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) )
Distinct variable groups:    x, A    x, T

Proof of Theorem hommval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hilex 26487 . . 3  |-  ~H  e.  _V
21, 1elmap 7508 . 2  |-  ( T  e.  ( ~H  ^m  ~H )  <->  T : ~H --> ~H )
3 oveq1 6312 . . . 4  |-  ( f  =  A  ->  (
f  .h  ( g `
 x ) )  =  ( A  .h  ( g `  x
) ) )
43mpteq2dv 4513 . . 3  |-  ( f  =  A  ->  (
x  e.  ~H  |->  ( f  .h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( A  .h  ( g `  x ) ) ) )
5 fveq1 5880 . . . . 5  |-  ( g  =  T  ->  (
g `  x )  =  ( T `  x ) )
65oveq2d 6321 . . . 4  |-  ( g  =  T  ->  ( A  .h  ( g `  x ) )  =  ( A  .h  ( T `  x )
) )
76mpteq2dv 4513 . . 3  |-  ( g  =  T  ->  (
x  e.  ~H  |->  ( A  .h  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) )
8 df-homul 27219 . . 3  |-  .op  =  ( f  e.  CC ,  g  e.  ( ~H  ^m  ~H )  |->  ( x  e.  ~H  |->  ( f  .h  ( g `
 x ) ) ) )
91mptex 6151 . . 3  |-  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) )  e.  _V
104, 7, 8, 9ovmpt2 6446 . 2  |-  ( ( A  e.  CC  /\  T  e.  ( ~H  ^m 
~H ) )  -> 
( A  .op  T
)  =  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) )
112, 10sylan2br 478 1  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
)  =  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    |-> cmpt 4484   -->wf 5597   ` cfv 5601  (class class class)co 6305    ^m cmap 7480   CCcc 9536   ~Hchil 26407    .h csm 26409    .op chot 26427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-hilex 26487
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-homul 27219
This theorem is referenced by:  homval  27229  homulcl  27247
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