Metamath Proof Explorer


Theorem ipcni

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 ) ) )

Proof

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 ) ) )