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Theorem kbass1 22696
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 22534 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  ketbra  B ) `
 C )  =  ( ( C  .ih  B )  .h  A ) )
2 braval 22524 . . . 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 5873 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( ( bra `  B
) `  C )  .h  A )  =  ( ( C  .ih  B
)  .h  A ) )
51, 4eqtr4d 2318 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   ~Hchil 21499    .h csm 21501    .ih csp 21502   bracbr 21536    ketbra ck 21537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-bra 22430  df-kb 22431
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