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Theorem kbval 22364
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
StepHypRef Expression
1 kbfval 22362 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  ketbra  B )  =  ( x  e. 
~H  |->  ( ( x 
.ih  B )  .h  A ) ) )
21fveq1d 5379 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  ketbra  B ) `  C )  =  ( ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) ) `
 C ) )
3 oveq1 5717 . . . . 5  |-  ( x  =  C  ->  (
x  .ih  B )  =  ( C  .ih  B ) )
43oveq1d 5725 . . . 4  |-  ( x  =  C  ->  (
( x  .ih  B
)  .h  A )  =  ( ( C 
.ih  B )  .h  A ) )
5 eqid 2253 . . . 4  |-  ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) )  =  ( x  e. 
~H  |->  ( ( x 
.ih  B )  .h  A ) )
6 ovex 5735 . . . 4  |-  ( ( C  .ih  B )  .h  A )  e. 
_V
74, 5, 6fvmpt 5454 . . 3  |-  ( C  e.  ~H  ->  (
( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) ) `  C )  =  ( ( C 
.ih  B )  .h  A ) )
82, 7sylan9eq 2305 . 2  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( A 
ketbra  B ) `  C
)  =  ( ( C  .ih  B )  .h  A ) )
983impa 1151 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    e. cmpt 3974   ` cfv 4592  (class class class)co 5710   ~Hchil 21329    .h csm 21331    .ih csp 21332    ketbra ck 21367
This theorem is referenced by:  kbpj  22366  kbass1  22526  kbass2  22527  kbass5  22530  kbass6  22531
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403  ax-hilex 21409
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-kb 22261
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