Theorem divreczi

Description: Relationship between division and reciprocal. Theorem I.9 of Apostol p. 18. (Contributed by NM, 11-Oct-1999)

	Ref	Expression
Hypotheses	divclz.1	$$⊢A\in ℂ$$
	divclz.2	$$⊢B\in ℂ$$
Assertion	divreczi	$$⊢B\ne 0\to \frac{A}{B}=A\left(\frac{1}{B}\right)$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divrec	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \frac{A}{B}=A\left(\frac{1}{B}\right)$$
4	1 2 3	mp3an12	$$⊢B\ne 0\to \frac{A}{B}=A\left(\frac{1}{B}\right)$$