Theorem divmulzi

Description: Relationship between division and multiplication. (Contributed by NM, 8-May-1999) (Revised by Mario Carneiro, 17-Feb-2014)

	Ref	Expression
Hypotheses	divclz.1	$$⊢A\in ℂ$$
	divclz.2	$$⊢B\in ℂ$$
	divmulz.3	$$⊢C\in ℂ$$
Assertion	divmulzi	$$⊢B\ne 0\to \left(\frac{A}{B}=C\leftrightarrow BC=A\right)$$

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	$$⊢\left(A\in ℂ\wedge C\in ℂ\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \left(\frac{A}{B}=C\leftrightarrow BC=A\right)$$
5	1 3 4	mp3an12	$$⊢\left(B\in ℂ\wedge B\ne 0\right)\to \left(\frac{A}{B}=C\leftrightarrow BC=A\right)$$
6	2 5	mpan	$$⊢B\ne 0\to \left(\frac{A}{B}=C\leftrightarrow BC=A\right)$$