Theorem divcan4

Description: A cancellation law for division. (Contributed by NM, 8-Feb-2005)

	Ref	Expression
Assertion	divcan4	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \frac{AB}{B}=A$$

Proof

Step	Hyp	Ref	Expression
1		mulcom	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to AB=BA$$
2	1	3adant3	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to AB=BA$$
3	2	oveq1d	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \frac{AB}{B}=\frac{BA}{B}$$
4		divcan3	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \frac{BA}{B}=A$$
5	3 4	eqtrd	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \frac{AB}{B}=A$$