Theorem subreci

Description: Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017)

	Ref	Expression
Hypotheses	subreci.1	$$⊢A\in ℂ$$
	subreci.2	$$⊢B\in ℂ$$
	subreci.3	$$⊢A\ne 0$$
	subreci.4	$$⊢B\ne 0$$
Assertion	subreci	$$⊢\left(\frac{1}{A}\right)-\left(\frac{1}{B}\right)=\frac{B-A}{AB}$$

Proof

Step	Hyp	Ref	Expression
1		subreci.1	$$⊢A\in ℂ$$
2		subreci.2	$$⊢B\in ℂ$$
3		subreci.3	$$⊢A\ne 0$$
4		subreci.4	$$⊢B\ne 0$$
5		subrec	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \left(\frac{1}{A}\right)-\left(\frac{1}{B}\right)=\frac{B-A}{AB}$$
6	1 3 2 4 5	mp4an	$$⊢\left(\frac{1}{A}\right)-\left(\frac{1}{B}\right)=\frac{B-A}{AB}$$