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Theorem kbass1 22712
Description: Dirac bra-ket associative law  (  |  A >.  <. B  |  )  |  C >.  =  |  A >. (
<. B  |  C >. ) i.e. the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since  <. B  |  C >. is a complex number, it is the first argument in the inner product  .h that it is mapped to, although in Dirac notation it is placed after the ket. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
kbass1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  ketbra  B ) `
 C )  =  ( ( ( bra `  B ) `  C
)  .h  A ) )

Proof of Theorem kbass1
StepHypRef Expression
1 kbval 22550 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  ketbra  B ) `
 C )  =  ( ( C  .ih  B )  .h  A ) )
2 braval 22540 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( bra `  B
) `  C )  =  ( C  .ih  B ) )
323adant1 973 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( bra `  B
) `  C )  =  ( C  .ih  B ) )
43oveq1d 5889 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( ( bra `  B
) `  C )  .h  A )  =  ( ( C  .ih  B
)  .h  A ) )
51, 4eqtr4d 2331 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  ketbra  B ) `
 C )  =  ( ( ( bra `  B ) `  C
)  .h  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   ~Hchil 21515    .h csm 21517    .ih csp 21518   bracbr 21552    ketbra ck 21553
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-oprab 5878  df-mpt2 5879  df-bra 22446  df-kb 22447
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