Metamath Proof Explorer


Theorem remuli

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 ) ) )

Proof

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 ) ) )