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Theorem braval 23435
Description: A bra-ket juxtaposition, expressed as  <. A  |  B >. in Dirac notation, equals the inner product of the vectors. Based on definition of bra in [Prugovecki] p. 186. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
braval  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  =  ( B  .ih  A ) )

Proof of Theorem braval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 brafval 23434 . . 3  |-  ( A  e.  ~H  ->  ( bra `  A )  =  ( x  e.  ~H  |->  ( x  .ih  A ) ) )
21fveq1d 5721 . 2  |-  ( A  e.  ~H  ->  (
( bra `  A
) `  B )  =  ( ( x  e.  ~H  |->  ( x 
.ih  A ) ) `
 B ) )
3 oveq1 6079 . . 3  |-  ( x  =  B  ->  (
x  .ih  A )  =  ( B  .ih  A ) )
4 eqid 2435 . . 3  |-  ( x  e.  ~H  |->  ( x 
.ih  A ) )  =  ( x  e. 
~H  |->  ( x  .ih  A ) )
5 ovex 6097 . . 3  |-  ( B 
.ih  A )  e. 
_V
63, 4, 5fvmpt 5797 . 2  |-  ( B  e.  ~H  ->  (
( x  e.  ~H  |->  ( x  .ih  A ) ) `  B )  =  ( B  .ih  A ) )
72, 6sylan9eq 2487 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  =  ( B  .ih  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4258   ` cfv 5445  (class class class)co 6072   ~Hchil 22410    .ih csp 22413   bracbr 22447
This theorem is referenced by:  braadd  23436  bramul  23437  brafnmul  23442  branmfn  23596  rnbra  23598  bra11  23599  cnvbraval  23601  kbass1  23607  kbass2  23608  kbass6  23612
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-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-pr 4395  ax-hilex 22490
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-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 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-bra 23341
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