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Theorem kbval 23305
Description: The value of the operator resulting from the outer product  |  A >.  <. B  | of two vectors. Equation 8.1 of [Prugovecki] p. 376. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
kbval  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  ketbra  B ) `
 C )  =  ( ( C  .ih  B )  .h  A ) )

Proof of Theorem kbval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 kbfval 23303 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  ketbra  B )  =  ( x  e. 
~H  |->  ( ( x 
.ih  B )  .h  A ) ) )
21fveq1d 5670 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  ketbra  B ) `  C )  =  ( ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) ) `
 C ) )
3 oveq1 6027 . . . . 5  |-  ( x  =  C  ->  (
x  .ih  B )  =  ( C  .ih  B ) )
43oveq1d 6035 . . . 4  |-  ( x  =  C  ->  (
( x  .ih  B
)  .h  A )  =  ( ( C 
.ih  B )  .h  A ) )
5 eqid 2387 . . . 4  |-  ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) )  =  ( x  e. 
~H  |->  ( ( x 
.ih  B )  .h  A ) )
6 ovex 6045 . . . 4  |-  ( ( C  .ih  B )  .h  A )  e. 
_V
74, 5, 6fvmpt 5745 . . 3  |-  ( C  e.  ~H  ->  (
( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) ) `  C )  =  ( ( C 
.ih  B )  .h  A ) )
82, 7sylan9eq 2439 . 2  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( A 
ketbra  B ) `  C
)  =  ( ( C  .ih  B )  .h  A ) )
983impa 1148 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  ketbra  B ) `
 C )  =  ( ( C  .ih  B )  .h  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   ~Hchil 22270    .h csm 22272    .ih csp 22273    ketbra ck 22308
This theorem is referenced by:  kbpj  23307  kbass1  23467  kbass2  23468  kbass5  23471  kbass6  23472
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-hilex 22350
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-kb 23202
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