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Theorem kbfval 26644
Description: The outer product of two vectors, expressed as  |  A >.  <. B  | in Dirac notation. See df-kb 26543. (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 6293 . . 3  |-  ( y  =  A  ->  (
( x  .ih  z
)  .h  y )  =  ( ( x 
.ih  z )  .h  A ) )
21mpteq2dv 4534 . 2  |-  ( y  =  A  ->  (
x  e.  ~H  |->  ( ( x  .ih  z
)  .h  y ) )  =  ( x  e.  ~H  |->  ( ( x  .ih  z )  .h  A ) ) )
3 oveq2 6293 . . . 4  |-  ( z  =  B  ->  (
x  .ih  z )  =  ( x  .ih  B ) )
43oveq1d 6300 . . 3  |-  ( z  =  B  ->  (
( x  .ih  z
)  .h  A )  =  ( ( x 
.ih  B )  .h  A ) )
54mpteq2dv 4534 . 2  |-  ( z  =  B  ->  (
x  e.  ~H  |->  ( ( x  .ih  z
)  .h  A ) )  =  ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) ) )
6 df-kb 26543 . 2  |-  ketbra  =  ( y  e.  ~H , 
z  e.  ~H  |->  ( x  e.  ~H  |->  ( ( x  .ih  z
)  .h  y ) ) )
7 ax-hilex 25689 . . 3  |-  ~H  e.  _V
87mptex 6132 . 2  |-  ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) )  e.  _V
92, 5, 6, 8ovmpt2 6423 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 369    = wceq 1379    e. wcel 1767    |-> cmpt 4505  (class class class)co 6285   ~Hchil 25609    .h csm 25611    .ih csp 25612    ketbra ck 25647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-hilex 25689
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-kb 26543
This theorem is referenced by:  kbop  26645  kbval  26646  kbmul  26647
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