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Theorem hfmval 22379
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 22374 . . . 4  |-  ( ( A  e.  CC  /\  T : ~H --> CC )  ->  ( A  .fn  T )  =  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) ) )
21fveq1d 5565 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> CC )  ->  ( ( A 
.fn  T ) `  B )  =  ( ( x  e.  ~H  |->  ( A  x.  ( T `  x )
) ) `  B
) )
3 fveq2 5563 . . . . 5  |-  ( x  =  B  ->  ( T `  x )  =  ( T `  B ) )
43oveq2d 5916 . . . 4  |-  ( x  =  B  ->  ( A  x.  ( T `  x ) )  =  ( A  x.  ( T `  B )
) )
5 eqid 2316 . . . 4  |-  ( x  e.  ~H  |->  ( A  x.  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( A  x.  ( T `  x ) ) )
6 ovex 5925 . . . 4  |-  ( A  x.  ( T `  B ) )  e. 
_V
74, 5, 6fvmpt 5640 . . 3  |-  ( B  e.  ~H  ->  (
( x  e.  ~H  |->  ( A  x.  ( T `  x )
) ) `  B
)  =  ( A  x.  ( T `  B ) ) )
82, 7sylan9eq 2368 . 2  |-  ( ( ( A  e.  CC  /\  T : ~H --> CC )  /\  B  e.  ~H )  ->  ( ( A 
.fn  T ) `  B )  =  ( A  x.  ( T `
 B ) ) )
983impa 1146 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 358    /\ w3a 934    = wceq 1633    e. wcel 1701    e. cmpt 4114   -->wf 5288   ` cfv 5292  (class class class)co 5900   CCcc 8780    x. cmul 8787   ~Hchil 21554    .fn chft 21577
This theorem is referenced by:  kbass2  22752  kbass3  22753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-hilex 21634
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-map 6817  df-hfmul 22369
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