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Theorem kbfval 23486
Description: The outer product of two vectors, expressed as  |  A >.  <. B  | in Dirac notation. See df-kb 23385. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
kbfval  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  ketbra  B )  =  ( x  e. 
~H  |->  ( ( x 
.ih  B )  .h  A ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem kbfval
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6118 . . 3  |-  ( y  =  A  ->  (
( x  .ih  z
)  .h  y )  =  ( ( x 
.ih  z )  .h  A ) )
21mpteq2dv 4321 . 2  |-  ( y  =  A  ->  (
x  e.  ~H  |->  ( ( x  .ih  z
)  .h  y ) )  =  ( x  e.  ~H  |->  ( ( x  .ih  z )  .h  A ) ) )
3 oveq2 6118 . . . 4  |-  ( z  =  B  ->  (
x  .ih  z )  =  ( x  .ih  B ) )
43oveq1d 6125 . . 3  |-  ( z  =  B  ->  (
( x  .ih  z
)  .h  A )  =  ( ( x 
.ih  B )  .h  A ) )
54mpteq2dv 4321 . 2  |-  ( z  =  B  ->  (
x  e.  ~H  |->  ( ( x  .ih  z
)  .h  A ) )  =  ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) ) )
6 df-kb 23385 . 2  |-  ketbra  =  ( y  e.  ~H , 
z  e.  ~H  |->  ( x  e.  ~H  |->  ( ( x  .ih  z
)  .h  y ) ) )
7 ax-hilex 22533 . . 3  |-  ~H  e.  _V
87mptex 5995 . 2  |-  ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) )  e.  _V
92, 5, 6, 8ovmpt2 6238 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  ketbra  B )  =  ( x  e. 
~H  |->  ( ( x 
.ih  B )  .h  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727    e. cmpt 4291  (class class class)co 6110   ~Hchil 22453    .h csm 22455    .ih csp 22456    ketbra ck 22491
This theorem is referenced by:  kbop  23487  kbval  23488  kbmul  23489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-hilex 22533
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-kb 23385
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