Theorem recid

Description: Multiplication of a number and its reciprocal. (Contributed by NM, 25-Oct-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	$$⊢1\in ℂ$$
2		divcan2	$$⊢\left(1\in ℂ\wedge A\in ℂ\wedge A\ne 0\right)\to A\left(\frac{1}{A}\right)=1$$
3	1 2	mp3an1	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to A\left(\frac{1}{A}\right)=1$$