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Theorem hfmmval 26431
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 9574 . . 3  |-  CC  e.  _V
2 ax-hilex 25689 . . 3  |-  ~H  e.  _V
31, 2elmap 7448 . 2  |-  ( T  e.  ( CC  ^m  ~H )  <->  T : ~H --> CC )
4 oveq1 6292 . . . 4  |-  ( f  =  A  ->  (
f  x.  ( g `
 x ) )  =  ( A  x.  ( g `  x
) ) )
54mpteq2dv 4534 . . 3  |-  ( f  =  A  ->  (
x  e.  ~H  |->  ( f  x.  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( A  x.  ( g `  x ) ) ) )
6 fveq1 5865 . . . . 5  |-  ( g  =  T  ->  (
g `  x )  =  ( T `  x ) )
76oveq2d 6301 . . . 4  |-  ( g  =  T  ->  ( A  x.  ( g `  x ) )  =  ( A  x.  ( T `  x )
) )
87mpteq2dv 4534 . . 3  |-  ( g  =  T  ->  (
x  e.  ~H  |->  ( A  x.  ( g `
 x ) ) )  =  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) ) )
9 df-hfmul 26426 . . 3  |-  .fn  =  ( f  e.  CC ,  g  e.  ( CC  ^m  ~H )  |->  ( x  e.  ~H  |->  ( f  x.  ( g `
 x ) ) ) )
102mptex 6132 . . 3  |-  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) )  e.  _V
115, 8, 9, 10ovmpt2 6423 . 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 1379    e. wcel 1767    |-> cmpt 4505   -->wf 5584   ` cfv 5588  (class class class)co 6285    ^m cmap 7421   CCcc 9491    x. cmul 9498   ~Hchil 25609    .fn chft 25632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-hilex 25689
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-map 7423  df-hfmul 26426
This theorem is referenced by:  hfmval  26436  brafnmul  26643  kbass2  26809
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