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Theorem hfmval 23239
Description: Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
hfmval  |-  ( ( A  e.  CC  /\  T : ~H --> CC  /\  B  e.  ~H )  ->  ( ( A  .fn  T ) `  B )  =  ( A  x.  ( T `  B ) ) )

Proof of Theorem hfmval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hfmmval 23234 . . . 4  |-  ( ( A  e.  CC  /\  T : ~H --> CC )  ->  ( A  .fn  T )  =  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) ) )
21fveq1d 5722 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> CC )  ->  ( ( A 
.fn  T ) `  B )  =  ( ( x  e.  ~H  |->  ( A  x.  ( T `  x )
) ) `  B
) )
3 fveq2 5720 . . . . 5  |-  ( x  =  B  ->  ( T `  x )  =  ( T `  B ) )
43oveq2d 6089 . . . 4  |-  ( x  =  B  ->  ( A  x.  ( T `  x ) )  =  ( A  x.  ( T `  B )
) )
5 eqid 2435 . . . 4  |-  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( A  x.  ( T `  x ) ) )
6 ovex 6098 . . . 4  |-  ( A  x.  ( T `  B ) )  e. 
_V
74, 5, 6fvmpt 5798 . . 3  |-  ( B  e.  ~H  ->  (
( x  e.  ~H  |->  ( A  x.  ( T `  x )
) ) `  B
)  =  ( A  x.  ( T `  B ) ) )
82, 7sylan9eq 2487 . 2  |-  ( ( ( A  e.  CC  /\  T : ~H --> CC )  /\  B  e.  ~H )  ->  ( ( A 
.fn  T ) `  B )  =  ( A  x.  ( T `
 B ) ) )
983impa 1148 1  |-  ( ( A  e.  CC  /\  T : ~H --> CC  /\  B  e.  ~H )  ->  ( ( A  .fn  T ) `  B )  =  ( A  x.  ( T `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    e. cmpt 4258   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980    x. cmul 8987   ~Hchil 22414    .fn chft 22437
This theorem is referenced by:  kbass2  23612  kbass3  23613
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-hilex 22494
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-hfmul 23229
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