Theorem divcan1i

Description: A cancellation law for division. (Contributed by NM, 18-May-1999)

	Ref	Expression
Hypotheses	divclz.1	$$⊢A\in ℂ$$
	divclz.2	$$⊢B\in ℂ$$
	divcl.3	$$⊢B\ne 0$$
Assertion	divcan1i	$$⊢\left(\frac{A}{B}\right)B=A$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divcl.3	$$⊢B\ne 0$$
4	1 2 3	divcli	$$⊢\frac{A}{B}\in ℂ$$
5	1 2 3	divcan2i	$$⊢B\left(\frac{A}{B}\right)=A$$
6	2 4 5	mulcomli	$$⊢\left(\frac{A}{B}\right)B=A$$