Theorem divid

Description: A number divided by itself is one. (Contributed by NM, 1-Aug-2004)

	Ref	Expression
Assertion	divid	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \frac{A}{A}=1$$

Proof

Step	Hyp	Ref	Expression
1		divrec	$$⊢\left(A\in ℂ\wedge A\in ℂ\wedge A\ne 0\right)\to \frac{A}{A}=A\left(\frac{1}{A}\right)$$
2	1	3anidm12	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \frac{A}{A}=A\left(\frac{1}{A}\right)$$
3		recid	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to A\left(\frac{1}{A}\right)=1$$
4	2 3	eqtrd	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \frac{A}{A}=1$$