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Theorem kbval 22459
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 22457 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  ketbra  B )  =  ( x  e. 
~H  |->  ( ( x 
.ih  B )  .h  A ) ) )
21fveq1d 5425 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  ketbra  B ) `  C )  =  ( ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) ) `
 C ) )
3 oveq1 5764 . . . . 5  |-  ( x  =  C  ->  (
x  .ih  B )  =  ( C  .ih  B ) )
43oveq1d 5772 . . . 4  |-  ( x  =  C  ->  (
( x  .ih  B
)  .h  A )  =  ( ( C 
.ih  B )  .h  A ) )
5 eqid 2256 . . . 4  |-  ( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) )  =  ( x  e. 
~H  |->  ( ( x 
.ih  B )  .h  A ) )
6 ovex 5782 . . . 4  |-  ( ( C  .ih  B )  .h  A )  e. 
_V
74, 5, 6fvmpt 5501 . . 3  |-  ( C  e.  ~H  ->  (
( x  e.  ~H  |->  ( ( x  .ih  B )  .h  A ) ) `  C )  =  ( ( C 
.ih  B )  .h  A ) )
82, 7sylan9eq 2308 . 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 4017   ` cfv 4638  (class class class)co 5757   ~Hchil 21424    .h csm 21426    .ih csp 21427    ketbra ck 21462
This theorem is referenced by:  kbpj  22461  kbass1  22621  kbass2  22622  kbass5  22625  kbass6  22626
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449  ax-hilex 21504
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-kb 22356
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