Theorem recdivd

Description: The reciprocal of a ratio. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	$$⊢\phi \to A\in ℂ$$
	divcld.2	$$⊢\phi \to B\in ℂ$$
	divne0d.3	$$⊢\phi \to A\ne 0$$
	divne0d.4	$$⊢\phi \to B\ne 0$$
Assertion	recdivd	$$⊢\phi \to \frac{1}{\frac{A}{B}}=\frac{B}{A}$$

Proof

Step	Hyp	Ref	Expression
1		div1d.1	$$⊢\phi \to A\in ℂ$$
2		divcld.2	$$⊢\phi \to B\in ℂ$$
3		divne0d.3	$$⊢\phi \to A\ne 0$$
4		divne0d.4	$$⊢\phi \to B\ne 0$$
5		recdiv	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \frac{1}{\frac{A}{B}}=\frac{B}{A}$$
6	1 3 2 4 5	syl22anc	$$⊢\phi \to \frac{1}{\frac{A}{B}}=\frac{B}{A}$$