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Theorem homval 22435
Description: Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
homval  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  =  ( A  .h  ( T `  B ) ) )

Proof of Theorem homval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hommval 22430 . . . 4  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
)  =  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) )
21fveq1d 5610 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( ( A  .op  T ) `  B )  =  ( ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) ) `
 B ) )
3 fveq2 5608 . . . . 5  |-  ( x  =  B  ->  ( T `  x )  =  ( T `  B ) )
43oveq2d 5961 . . . 4  |-  ( x  =  B  ->  ( A  .h  ( T `  x ) )  =  ( A  .h  ( T `  B )
) )
5 eqid 2358 . . . 4  |-  ( x  e.  ~H  |->  ( A  .h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( A  .h  ( T `  x ) ) )
6 ovex 5970 . . . 4  |-  ( A  .h  ( T `  B ) )  e. 
_V
74, 5, 6fvmpt 5685 . . 3  |-  ( B  e.  ~H  ->  (
( x  e.  ~H  |->  ( A  .h  ( T `  x )
) ) `  B
)  =  ( A  .h  ( T `  B ) ) )
82, 7sylan9eq 2410 . 2  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  B  e.  ~H )  ->  ( ( A 
.op  T ) `  B )  =  ( A  .h  ( T `
 B ) ) )
983impa 1146 1  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  B  e.  ~H )  ->  ( ( A  .op  T ) `  B )  =  ( A  .h  ( T `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    e. cmpt 4158   -->wf 5333   ` cfv 5337  (class class class)co 5945   CCcc 8825   ~Hchil 21613    .h csm 21615    .op chot 21633
This theorem is referenced by:  homcl  22440  honegsubi  22490  homulid2  22494  homco1  22495  homulass  22496  hoadddi  22497  hoadddir  22498  nmopnegi  22659  homco2  22671  lnopmi  22694  hmopm  22715  nmophmi  22725  adjmul  22786  leopmuli  22827  leopnmid  22832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-hilex 21693
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-map 6862  df-homul 22425
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