Theorem divreci

Description: Relationship between division and reciprocal. Theorem I.9 of Apostol p. 18. (Contributed by NM, 9-Feb-1995)

	Ref	Expression
Hypotheses	divclz.1	$$⊢A\in ℂ$$
	divclz.2	$$⊢B\in ℂ$$
	divcl.3	$$⊢B\ne 0$$
Assertion	divreci	$$⊢\frac{A}{B}=A\left(\frac{1}{B}\right)$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divcl.3	$$⊢B\ne 0$$
4	1 2	divreczi	$$⊢B\ne 0\to \frac{A}{B}=A\left(\frac{1}{B}\right)$$
5	3 4	ax-mp	$$⊢\frac{A}{B}=A\left(\frac{1}{B}\right)$$