Theorem div11i

Description: One-to-one relationship for division. (Contributed by NM, 20-Aug-2001)

	Ref	Expression
Hypotheses	divclz.1	$$⊢A\in ℂ$$
	divclz.2	$$⊢B\in ℂ$$
	divmulz.3	$$⊢C\in ℂ$$
	divass.4	$$⊢C\ne 0$$
Assertion	div11i	$$⊢\frac{A}{C}=\frac{B}{C}\leftrightarrow A=B$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divmulz.3	$$⊢C\in ℂ$$
4		divass.4	$$⊢C\ne 0$$
5	3 4	pm3.2i	$$⊢\left(C\in ℂ\wedge C\ne 0\right)$$
6		div11	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge \left(C\in ℂ\wedge C\ne 0\right)\right)\to \left(\frac{A}{C}=\frac{B}{C}\leftrightarrow A=B\right)$$
7	1 2 5 6	mp3an	$$⊢\frac{A}{C}=\frac{B}{C}\leftrightarrow A=B$$