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Theorem hfmmval 25143
Description: Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hfmmval  |-  ( ( A  e.  CC  /\  T : ~H --> CC )  ->  ( A  .fn  T )  =  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) ) )
Distinct variable groups:    x, A    x, T

Proof of Theorem hfmmval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnex 9363 . . 3  |-  CC  e.  _V
2 ax-hilex 24401 . . 3  |-  ~H  e.  _V
31, 2elmap 7241 . 2  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
4 oveq1 6098 . . . 4  |-  ( f  =  A  ->  (
f  x.  ( g `
 x ) )  =  ( A  x.  ( g `  x
) ) )
54mpteq2dv 4379 . . 3  |-  ( f  =  A  ->  (
x  e.  ~H  |->  ( f  x.  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( A  x.  ( g `  x ) ) ) )
6 fveq1 5690 . . . . 5  |-  ( g  =  T  ->  (
g `  x )  =  ( T `  x ) )
76oveq2d 6107 . . . 4  |-  ( g  =  T  ->  ( A  x.  ( g `  x ) )  =  ( A  x.  ( T `  x )
) )
87mpteq2dv 4379 . . 3  |-  ( g  =  T  ->  (
x  e.  ~H  |->  ( A  x.  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) ) )
9 df-hfmul 25138 . . 3  |-  .fn  =  ( f  e.  CC ,  g  e.  ( CC  ^m  ~H )  |->  ( x  e.  ~H  |->  ( f  x.  ( g `
 x ) ) ) )
102mptex 5948 . . 3  |-  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) )  e.  _V
115, 8, 9, 10ovmpt2 6226 . 2  |-  ( ( A  e.  CC  /\  T  e.  ( CC  ^m 
~H ) )  -> 
( A  .fn  T
)  =  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) ) )
123, 11sylan2br 476 1  |-  ( ( A  e.  CC  /\  T : ~H --> CC )  ->  ( A  .fn  T )  =  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    e. cmpt 4350   -->wf 5414   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   CCcc 9280    x. cmul 9287   ~Hchil 24321    .fn chft 24344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-hilex 24401
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-map 7216  df-hfmul 25138
This theorem is referenced by:  hfmval  25148  brafnmul  25355  kbass2  25521
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