Theorem subrecd

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

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

Proof

Step	Hyp	Ref	Expression
1		subrecd.1	$$⊢\phi \to A\in ℂ$$
2		subrecd.2	$$⊢\phi \to B\in ℂ$$
3		subrecd.3	$$⊢\phi \to A\ne 0$$
4		subrecd.4	$$⊢\phi \to B\ne 0$$
5		1cnd	$$⊢\phi \to 1\in ℂ$$
6	5 1 5 2 3 4	divsubdivd	$$⊢\phi \to \left(\frac{1}{A}\right)-\left(\frac{1}{B}\right)=\frac{1B-1A}{AB}$$
7	2	mullidd	$$⊢\phi \to 1B=B$$
8	1	mullidd	$$⊢\phi \to 1A=A$$
9	7 8	oveq12d	$$⊢\phi \to 1B-1A=B-A$$
10	9	oveq1d	$$⊢\phi \to \frac{1B-1A}{AB}=\frac{B-A}{AB}$$
11	6 10	eqtrd	$$⊢\phi \to \left(\frac{1}{A}\right)-\left(\frac{1}{B}\right)=\frac{B-A}{AB}$$