Theorem cjmuld

Description: Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		recld.1	|- ( ph -> A e. CC )
2		readdd.2	|- ( ph -> B e. CC )
3		cjmul	|- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( * ` ( A x. B ) ) = ( ( * ` A ) x. ( * ` B ) ) )