Theorem cjdivi

Description: Complex conjugate distributes over division. (Contributed by NM, 29-Apr-2005) (Revised by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	recl.1	$$⊢A\in ℂ$$
	readdi.2	$$⊢B\in ℂ$$
Assertion	cjdivi	$$⊢B\ne 0\to \overline{\frac{A}{B}}=\frac{\bar{A}}{\bar{B}}$$

Proof

Step	Hyp	Ref	Expression
1		recl.1	$$⊢A\in ℂ$$
2		readdi.2	$$⊢B\in ℂ$$
3		cjdiv	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \overline{\frac{A}{B}}=\frac{\bar{A}}{\bar{B}}$$
4	1 2 3	mp3an12	$$⊢B\ne 0\to \overline{\frac{A}{B}}=\frac{\bar{A}}{\bar{B}}$$