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Theorem braval 22470
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
StepHypRef Expression
1 brafval 22469 . . 3  |-  ( A  e.  ~H  ->  ( bra `  A )  =  ( x  e.  ~H  |->  ( x  .ih  A ) ) )
21fveq1d 5446 . 2  |-  ( A  e.  ~H  ->  (
( bra `  A
) `  B )  =  ( ( x  e.  ~H  |->  ( x 
.ih  A ) ) `
 B ) )
3 oveq1 5785 . . 3  |-  ( x  =  B  ->  (
x  .ih  A )  =  ( B  .ih  A ) )
4 eqid 2256 . . 3  |-  ( x  e.  ~H  |->  ( x 
.ih  A ) )  =  ( x  e. 
~H  |->  ( x  .ih  A ) )
5 ovex 5803 . . 3  |-  ( B 
.ih  A )  e. 
_V
63, 4, 5fvmpt 5522 . 2  |-  ( B  e.  ~H  ->  (
( x  e.  ~H  |->  ( x  .ih  A ) ) `  B )  =  ( B  .ih  A ) )
72, 6sylan9eq 2308 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  =  ( B  .ih  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    e. cmpt 4037   ` cfv 4659  (class class class)co 5778   ~Hchil 21445    .ih csp 21448   bracbr 21482
This theorem is referenced by:  braadd  22471  bramul  22472  brafnmul  22477  branmfn  22631  rnbra  22633  bra11  22634  cnvbraval  22636  kbass1  22642  kbass2  22643  kbass6  22647
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470  ax-hilex 21525
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-bra 22376
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