Description: Real part of a product. (Contributed by NM, 28-Jul-1999)
Ref | Expression | ||
---|---|---|---|
Hypotheses | recl.1 | |- A e. CC |
|
readdi.2 | |- B e. CC |
||
Assertion | remuli | |- ( 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 | remul | |- ( ( 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 ) ) ) |