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Theorem hfmmval 23242
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 9071 . . 3  |-  CC  e.  _V
2 ax-hilex 22502 . . 3  |-  ~H  e.  _V
31, 2elmap 7042 . 2  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
4 oveq1 6088 . . . 4  |-  ( f  =  A  ->  (
f  x.  ( g `
 x ) )  =  ( A  x.  ( g `  x
) ) )
54mpteq2dv 4296 . . 3  |-  ( f  =  A  ->  (
x  e.  ~H  |->  ( f  x.  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( A  x.  ( g `  x ) ) ) )
6 fveq1 5727 . . . . 5  |-  ( g  =  T  ->  (
g `  x )  =  ( T `  x ) )
76oveq2d 6097 . . . 4  |-  ( g  =  T  ->  ( A  x.  ( g `  x ) )  =  ( A  x.  ( T `  x )
) )
87mpteq2dv 4296 . . 3  |-  ( g  =  T  ->  (
x  e.  ~H  |->  ( A  x.  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) ) )
9 df-hfmul 23237 . . 3  |-  .fn  =  ( f  e.  CC ,  g  e.  ( CC  ^m  ~H )  |->  ( x  e.  ~H  |->  ( f  x.  ( g `
 x ) ) ) )
102mptex 5966 . . 3  |-  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) )  e.  _V
115, 8, 9, 10ovmpt2 6209 . 2  |-  ( ( A  e.  CC  /\  T  e.  ( CC  ^m 
~H ) )  -> 
( A  .fn  T
)  =  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) ) )
123, 11sylan2br 463 1  |-  ( ( A  e.  CC  /\  T : ~H --> CC )  ->  ( A  .fn  T )  =  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4266   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   CCcc 8988    x. cmul 8995   ~Hchil 22422    .fn chft 22445
This theorem is referenced by:  hfmval  23247  brafnmul  23454  kbass2  23620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-hilex 22502
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-hfmul 23237
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