Theorem rec11i

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

	Ref	Expression
Hypotheses	divclz.1	$$⊢A\in ℂ$$
	divclz.2	$$⊢B\in ℂ$$
Assertion	rec11i	$$⊢\left(A\ne 0\wedge B\ne 0\right)\to \left(\frac{1}{A}=\frac{1}{B}\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		rec11	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \left(\frac{1}{A}=\frac{1}{B}\leftrightarrow A=B\right)$$
4	3	an4s	$$⊢\left(\left(A\in ℂ\wedge B\in ℂ\right)\wedge \left(A\ne 0\wedge B\ne 0\right)\right)\to \left(\frac{1}{A}=\frac{1}{B}\leftrightarrow A=B\right)$$
5	1 2 4	mpanl12	$$⊢\left(A\ne 0\wedge B\ne 0\right)\to \left(\frac{1}{A}=\frac{1}{B}\leftrightarrow A=B\right)$$