Description: Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999)
Ref | Expression | ||
---|---|---|---|
Hypotheses | recl.1 | |- A e. CC |
|
readdi.2 | |- B e. CC |
||
Assertion | ipcni | |- ( Re ` ( A x. ( * ` B ) ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recl.1 | |- A e. CC |
|
2 | readdi.2 | |- B e. CC |
|
3 | ipcnval | |- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A x. ( * ` B ) ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) ) ) |
|
4 | 1 2 3 | mp2an | |- ( Re ` ( A x. ( * ` B ) ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) ) |