Theorem dmdcand

Description: Cancellation law for 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	$$⊢\phi \to B\ne 0$$
	divdiv23d.5	$$⊢\phi \to C\ne 0$$
Assertion	dmdcand	$$⊢\phi \to \left(\frac{B}{C}\right)\left(\frac{A}{B}\right)=\frac{A}{C}$$

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		divdiv23d.5	$$⊢\phi \to C\ne 0$$
6		dmdcan	$$⊢\left(\left(B\in ℂ\wedge B\ne 0\right)\wedge \left(C\in ℂ\wedge C\ne 0\right)\wedge A\in ℂ\right)\to \left(\frac{B}{C}\right)\left(\frac{A}{B}\right)=\frac{A}{C}$$
7	2 4 3 5 1 6	syl221anc	$$⊢\phi \to \left(\frac{B}{C}\right)\left(\frac{A}{B}\right)=\frac{A}{C}$$