Theorem recdiv2d

Description: Division into a reciprocal. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	$$⊢\phi \to A\in ℂ$$
	divcld.2	$$⊢\phi \to B\in ℂ$$
	divne0d.3	$$⊢\phi \to A\ne 0$$
	divne0d.4	$$⊢\phi \to B\ne 0$$
Assertion	recdiv2d	$$⊢\phi \to \frac{\frac{1}{A}}{B}=\frac{1}{AB}$$

Proof

Step	Hyp	Ref	Expression
1		div1d.1	$$⊢\phi \to A\in ℂ$$
2		divcld.2	$$⊢\phi \to B\in ℂ$$
3		divne0d.3	$$⊢\phi \to A\ne 0$$
4		divne0d.4	$$⊢\phi \to B\ne 0$$
5		recdiv2	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \frac{\frac{1}{A}}{B}=\frac{1}{AB}$$
6	1 3 2 4 5	syl22anc	$$⊢\phi \to \frac{\frac{1}{A}}{B}=\frac{1}{AB}$$