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Theorem kbfval 27590
Description: The outer product of two vectors, expressed as  |  A >.  <. B  | in Dirac notation. See df-kb 27489. (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 6309 . . 3  |-  ( y  =  A  ->  (
( x  .ih  z
)  .h  y )  =  ( ( x 
.ih  z )  .h  A ) )
21mpteq2dv 4508 . 2  |-  ( y  =  A  ->  (
x  e.  ~H  |->  ( ( x  .ih  z
)  .h  y ) )  =  ( x  e.  ~H  |->  ( ( x  .ih  z )  .h  A ) ) )
3 oveq2 6309 . . . 4  |-  ( z  =  B  ->  (
x  .ih  z )  =  ( x  .ih  B ) )
43oveq1d 6316 . . 3  |-  ( z  =  B  ->  (
( x  .ih  z
)  .h  A )  =  ( ( x 
.ih  B )  .h  A ) )
54mpteq2dv 4508 . 2  |-  ( z  =  B  ->  (
x  e.  ~H  |->  ( ( x  .ih  z
)  .h  A ) )  =  ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) ) )
6 df-kb 27489 . 2  |-  ketbra  =  ( y  e.  ~H , 
z  e.  ~H  |->  ( x  e.  ~H  |->  ( ( x  .ih  z
)  .h  y ) ) )
7 ax-hilex 26637 . . 3  |-  ~H  e.  _V
87mptex 6147 . 2  |-  ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) )  e.  _V
92, 5, 6, 8ovmpt2 6442 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  ketbra  B )  =  ( x  e. 
~H  |->  ( ( x 
.ih  B )  .h  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868    |-> cmpt 4479  (class class class)co 6301   ~Hchil 26557    .h csm 26559    .ih csp 26560    ketbra ck 26595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pr 4656  ax-hilex 26637
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-kb 27489
This theorem is referenced by:  kbop  27591  kbval  27592  kbmul  27593
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