Theorem recidd

Description: Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	$$⊢\phi \to A\in ℂ$$
	reccld.2	$$⊢\phi \to A\ne 0$$
Assertion	recidd	$$⊢\phi \to A\left(\frac{1}{A}\right)=1$$

Proof

Step	Hyp	Ref	Expression
1		div1d.1	$$⊢\phi \to A\in ℂ$$
2		reccld.2	$$⊢\phi \to A\ne 0$$
3		recid	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to A\left(\frac{1}{A}\right)=1$$
4	1 2 3	syl2anc	$$⊢\phi \to A\left(\frac{1}{A}\right)=1$$