Metamath Proof Explorer

Theorem cjmuli

Description: Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of Gleason p. 133. (Contributed by NM, 28-Jul-1999)

Ref Expression
Hypotheses recl.1 
|- A e. CC
readdi.2 
|- B e. CC
Assertion cjmuli 
|- ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) )

Proof

Step Hyp Ref Expression
1 recl.1 
 |-  A e. CC
2 readdi.2 
 |-  B e. CC
3 cjmul 
 |-  ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) )
4 1 2 3 mp2an 
 |-  ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) )