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Theorem kbfval 23416
Description: The outer product of two vectors, expressed as  |  A >.  <. B  | in Dirac notation. See df-kb 23315. (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 6056 . . 3  |-  ( y  =  A  ->  (
( x  .ih  z
)  .h  y )  =  ( ( x 
.ih  z )  .h  A ) )
21mpteq2dv 4264 . 2  |-  ( y  =  A  ->  (
x  e.  ~H  |->  ( ( x  .ih  z
)  .h  y ) )  =  ( x  e.  ~H  |->  ( ( x  .ih  z )  .h  A ) ) )
3 oveq2 6056 . . . 4  |-  ( z  =  B  ->  (
x  .ih  z )  =  ( x  .ih  B ) )
43oveq1d 6063 . . 3  |-  ( z  =  B  ->  (
( x  .ih  z
)  .h  A )  =  ( ( x 
.ih  B )  .h  A ) )
54mpteq2dv 4264 . 2  |-  ( z  =  B  ->  (
x  e.  ~H  |->  ( ( x  .ih  z
)  .h  A ) )  =  ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) ) )
6 df-kb 23315 . 2  |-  ketbra  =  ( y  e.  ~H , 
z  e.  ~H  |->  ( x  e.  ~H  |->  ( ( x  .ih  z
)  .h  y ) ) )
7 ax-hilex 22463 . . 3  |-  ~H  e.  _V
87mptex 5933 . 2  |-  ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) )  e.  _V
92, 5, 6, 8ovmpt2 6176 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 359    = wceq 1649    e. wcel 1721    e. cmpt 4234  (class class class)co 6048   ~Hchil 22383    .h csm 22385    .ih csp 22386    ketbra ck 22421
This theorem is referenced by:  kbop  23417  kbval  23418  kbmul  23419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-hilex 22463
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-kb 23315
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