Theorem rec11ii

Description: Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999)

	Ref	Expression
Hypotheses	divclz.1	$$⊢A\in ℂ$$
	divclz.2	$$⊢B\in ℂ$$
	divneq0.3	$$⊢A\ne 0$$
	divneq0.4	$$⊢B\ne 0$$
Assertion	rec11ii	$$⊢\frac{1}{A}=\frac{1}{B}\leftrightarrow A=B$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divneq0.3	$$⊢A\ne 0$$
4		divneq0.4	$$⊢B\ne 0$$
5	1 2	rec11i	$$⊢\left(A\ne 0\wedge B\ne 0\right)\to \left(\frac{1}{A}=\frac{1}{B}\leftrightarrow A=B\right)$$
6	3 4 5	mp2an	$$⊢\frac{1}{A}=\frac{1}{B}\leftrightarrow A=B$$