Theorem divmuli

Description: Relationship between division and multiplication. (Contributed by NM, 2-Feb-1995) (Revised by Mario Carneiro, 17-Feb-2014)

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

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divmulz.3	$$⊢C\in ℂ$$
4		divmul.4	$$⊢B\ne 0$$
5	1 2 3	divmulzi	$$⊢B\ne 0\to \left(\frac{A}{B}=C\leftrightarrow BC=A\right)$$
6	4 5	ax-mp	$$⊢\frac{A}{B}=C\leftrightarrow BC=A$$