Theorem recdiv2

Description: Division into a reciprocal. (Contributed by NM, 19-Oct-2007)

	Ref	Expression
Assertion	recdiv2	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \frac{\frac{1}{A}}{B}=\frac{1}{AB}$$

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	$$⊢1\in ℂ$$
2		divdiv1	$$⊢\left(1\in ℂ\wedge \left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \frac{\frac{1}{A}}{B}=\frac{1}{AB}$$
3	1 2	mp3an1	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \frac{\frac{1}{A}}{B}=\frac{1}{AB}$$