Theorem divmuld

Description: Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	$$⊢\phi \to A\in ℂ$$
	divcld.2	$$⊢\phi \to B\in ℂ$$
	divmuld.3	$$⊢\phi \to C\in ℂ$$
	divmuld.4	$$⊢\phi \to B\ne 0$$
Assertion	divmuld	$$⊢\phi \to \left(\frac{A}{B}=C\leftrightarrow BC=A\right)$$

Proof

Step	Hyp	Ref	Expression
1		div1d.1	$$⊢\phi \to A\in ℂ$$
2		divcld.2	$$⊢\phi \to B\in ℂ$$
3		divmuld.3	$$⊢\phi \to C\in ℂ$$
4		divmuld.4	$$⊢\phi \to B\ne 0$$
5		divmul	$$⊢\left(A\in ℂ\wedge C\in ℂ\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \left(\frac{A}{B}=C\leftrightarrow BC=A\right)$$
6	1 3 2 4 5	syl112anc	$$⊢\phi \to \left(\frac{A}{B}=C\leftrightarrow BC=A\right)$$