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Theorem braval 22540
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 22539 . . 3  |-  ( A  e.  ~H  ->  ( bra `  A )  =  ( x  e.  ~H  |->  ( x  .ih  A ) ) )
21fveq1d 5543 . 2  |-  ( A  e.  ~H  ->  (
( bra `  A
) `  B )  =  ( ( x  e.  ~H  |->  ( x 
.ih  A ) ) `
 B ) )
3 oveq1 5881 . . 3  |-  ( x  =  B  ->  (
x  .ih  A )  =  ( B  .ih  A ) )
4 eqid 2296 . . 3  |-  ( x  e.  ~H  |->  ( x 
.ih  A ) )  =  ( x  e. 
~H  |->  ( x  .ih  A ) )
5 ovex 5899 . . 3  |-  ( B 
.ih  A )  e. 
_V
63, 4, 5fvmpt 5618 . 2  |-  ( B  e.  ~H  ->  (
( x  e.  ~H  |->  ( x  .ih  A ) ) `  B )  =  ( B  .ih  A ) )
72, 6sylan9eq 2348 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  =  ( B  .ih  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   ~Hchil 21515    .ih csp 21518   bracbr 21552
This theorem is referenced by:  braadd  22541  bramul  22542  brafnmul  22547  branmfn  22701  rnbra  22703  bra11  22704  cnvbraval  22706  kbass1  22712  kbass2  22713  kbass6  22717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-bra 22446
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