Metamath Proof Explorer


Theorem remuld

Description: Real part of a product. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses recld.1
|- ( ph -> A e. CC )
readdd.2
|- ( ph -> B e. CC )
Assertion remuld
|- ( ph -> ( Re ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) - ( ( Im ` A ) x. ( Im ` B ) ) ) )

Proof

Step Hyp Ref Expression
1 recld.1
 |-  ( ph -> A e. CC )
2 readdd.2
 |-  ( ph -> 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 syl2anc
 |-  ( ph -> ( Re ` ( A x. B ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) - ( ( Im ` A ) x. ( Im ` B ) ) ) )