Description: A bra-ket juxtaposition, expressed as <. A | B >. in Dirac notation, equals the inner product of the vectors. Based on definition of bra in Prugovecki p. 186. (Contributed by NM, 15-May-2006) (Revised by Mario Carneiro, 17-Nov-2013) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | braval | |- ( ( A e. ~H /\ B e. ~H ) -> ( ( bra ` A ) ` B ) = ( B .ih A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brafval | |- ( A e. ~H -> ( bra ` A ) = ( x e. ~H |-> ( x .ih A ) ) ) |
|
| 2 | 1 | fveq1d | |- ( A e. ~H -> ( ( bra ` A ) ` B ) = ( ( x e. ~H |-> ( x .ih A ) ) ` B ) ) |
| 3 | oveq1 | |- ( x = B -> ( x .ih A ) = ( B .ih A ) ) |
|
| 4 | eqid | |- ( x e. ~H |-> ( x .ih A ) ) = ( x e. ~H |-> ( x .ih A ) ) |
|
| 5 | ovex | |- ( B .ih A ) e. _V |
|
| 6 | 3 4 5 | fvmpt | |- ( B e. ~H -> ( ( x e. ~H |-> ( x .ih A ) ) ` B ) = ( B .ih A ) ) |
| 7 | 2 6 | sylan9eq | |- ( ( A e. ~H /\ B e. ~H ) -> ( ( bra ` A ) ` B ) = ( B .ih A ) ) |