Theorem cjdivd

Description: Complex conjugate distributes over division. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	recld.1	$$⊢\phi \to A\in ℂ$$
	readdd.2	$$⊢\phi \to B\in ℂ$$
	cjdivd.2	$$⊢\phi \to B\ne 0$$
Assertion	cjdivd	$$⊢\phi \to \overline{\frac{A}{B}}=\frac{\bar{A}}{\bar{B}}$$

Proof

Step	Hyp	Ref	Expression
1		recld.1	$$⊢\phi \to A\in ℂ$$
2		readdd.2	$$⊢\phi \to B\in ℂ$$
3		cjdivd.2	$$⊢\phi \to B\ne 0$$
4		cjdiv	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \overline{\frac{A}{B}}=\frac{\bar{A}}{\bar{B}}$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \overline{\frac{A}{B}}=\frac{\bar{A}}{\bar{B}}$$