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	$$⊢\phi \to A\in ℂ$$
	readdd.2	$$⊢\phi \to B\in ℂ$$
Assertion	cjmuld	$$⊢\phi \to \overline{AB}=\bar{A}\bar{B}$$

Proof

Step	Hyp	Ref	Expression
1		recld.1	$$⊢\phi \to A\in ℂ$$
2		readdd.2	$$⊢\phi \to B\in ℂ$$
3		cjmul	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \overline{AB}=\bar{A}\bar{B}$$
4	1 2 3	syl2anc	$$⊢\phi \to \overline{AB}=\bar{A}\bar{B}$$