Theorem recreci

Description: A number is equal to the reciprocal of its reciprocal. Theorem I.10 of Apostol p. 18. (Contributed by NM, 9-Feb-1995)

	Ref	Expression
Hypotheses	divclz.1	$$⊢A\in ℂ$$
	reccl.2	$$⊢A\ne 0$$
Assertion	recreci	$$⊢\frac{1}{\frac{1}{A}}=A$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		reccl.2	$$⊢A\ne 0$$
3		recrec	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \frac{1}{\frac{1}{A}}=A$$
4	1 2 3	mp2an	$$⊢\frac{1}{\frac{1}{A}}=A$$