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Theorem kbval 22530
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 22528 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  ketbra  B )  =  ( x  e. 
~H  |->  ( ( x 
.ih  B )  .h  A ) ) )
21fveq1d 5488 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  ketbra  B ) `  C )  =  ( ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) ) `
 C ) )
3 oveq1 5827 . . . . 5  |-  ( x  =  C  ->  (
x  .ih  B )  =  ( C  .ih  B ) )
43oveq1d 5835 . . . 4  |-  ( x  =  C  ->  (
( x  .ih  B
)  .h  A )  =  ( ( C 
.ih  B )  .h  A ) )
5 eqid 2284 . . . 4  |-  ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) )  =  ( x  e. 
~H  |->  ( ( x 
.ih  B )  .h  A ) )
6 ovex 5845 . . . 4  |-  ( ( C  .ih  B )  .h  A )  e. 
_V
74, 5, 6fvmpt 5564 . . 3  |-  ( C  e.  ~H  ->  (
( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) ) `  C )  =  ( ( C 
.ih  B )  .h  A ) )
82, 7sylan9eq 2336 . 2  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( A 
ketbra  B ) `  C
)  =  ( ( C  .ih  B )  .h  A ) )
983impa 1146 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 358    /\ w3a 934    = wceq 1623    e. wcel 1685    e. cmpt 4078   ` cfv 5221  (class class class)co 5820   ~Hchil 21495    .h csm 21497    .ih csp 21498    ketbra ck 21533
This theorem is referenced by:  kbpj  22532  kbass1  22692  kbass2  22693  kbass5  22696  kbass6  22697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511  ax-hilex 21575
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-kb 22427
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