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Theorem braval 22516
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 ) )
Dummy variable  x is distinct from all other variables.

Proof of Theorem braval
StepHypRef Expression
1 brafval 22515 . . 3  |-  ( A  e.  ~H  ->  ( bra `  A )  =  ( x  e.  ~H  |->  ( x  .ih  A ) ) )
21fveq1d 5487 . 2  |-  ( A  e.  ~H  ->  (
( bra `  A
) `  B )  =  ( ( x  e.  ~H  |->  ( x 
.ih  A ) ) `
 B ) )
3 oveq1 5826 . . 3  |-  ( x  =  B  ->  (
x  .ih  A )  =  ( B  .ih  A ) )
4 eqid 2284 . . 3  |-  ( x  e.  ~H  |->  ( x 
.ih  A ) )  =  ( x  e. 
~H  |->  ( x  .ih  A ) )
5 ovex 5844 . . 3  |-  ( B 
.ih  A )  e. 
_V
63, 4, 5fvmpt 5563 . 2  |-  ( B  e.  ~H  ->  (
( x  e.  ~H  |->  ( x  .ih  A ) ) `  B )  =  ( B  .ih  A ) )
72, 6sylan9eq 2336 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 1624    e. wcel 1685    e. cmpt 4078   ` cfv 5221  (class class class)co 5819   ~Hchil 21491    .ih csp 21494   bracbr 21528
This theorem is referenced by:  braadd  22517  bramul  22518  brafnmul  22523  branmfn  22677  rnbra  22679  bra11  22680  cnvbraval  22682  kbass1  22688  kbass2  22689  kbass6  22693
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511  ax-hilex 21571
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-bra 22422
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