# Metamath Proof Explorer

## Theorem kbval

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.)

Ref Expression
Assertion kbval
`|- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A ketbra B ) ` C ) = ( ( C .ih B ) .h A ) )`

### Proof

Step Hyp Ref Expression
1 kbfval
` |-  ( ( A e. ~H /\ B e. ~H ) -> ( A ketbra B ) = ( x e. ~H |-> ( ( x .ih B ) .h A ) ) )`
2 1 fveq1d
` |-  ( ( A e. ~H /\ B e. ~H ) -> ( ( A ketbra B ) ` C ) = ( ( x e. ~H |-> ( ( x .ih B ) .h A ) ) ` C ) )`
3 oveq1
` |-  ( x = C -> ( x .ih B ) = ( C .ih B ) )`
4 3 oveq1d
` |-  ( x = C -> ( ( x .ih B ) .h A ) = ( ( C .ih B ) .h A ) )`
5 eqid
` |-  ( x e. ~H |-> ( ( x .ih B ) .h A ) ) = ( x e. ~H |-> ( ( x .ih B ) .h A ) )`
6 ovex
` |-  ( ( C .ih B ) .h A ) e. _V`
7 4 5 6 fvmpt
` |-  ( C e. ~H -> ( ( x e. ~H |-> ( ( x .ih B ) .h A ) ) ` C ) = ( ( C .ih B ) .h A ) )`
8 2 7 sylan9eq
` |-  ( ( ( A e. ~H /\ B e. ~H ) /\ C e. ~H ) -> ( ( A ketbra B ) ` C ) = ( ( C .ih B ) .h A ) )`
9 8 3impa
` |-  ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A ketbra B ) ` C ) = ( ( C .ih B ) .h A ) )`