Theorem divdiv3d

Description: Division into a fraction. (Contributed by Glauco Siliprandi, 24-Dec-2020)

	Ref	Expression
Hypotheses	divdiv3d.1	$$⊢\phi \to A\in ℂ$$
	divdiv3d.2	$$⊢\phi \to B\in ℂ$$
	divdiv3d.3	$$⊢\phi \to C\in ℂ$$
	divdiv3d.4	$$⊢\phi \to B\ne 0$$
	divdiv3d.5	$$⊢\phi \to C\ne 0$$
Assertion	divdiv3d	$$⊢\phi \to \frac{\frac{A}{B}}{C}=\frac{A}{CB}$$

Proof

Step	Hyp	Ref	Expression
1		divdiv3d.1	$$⊢\phi \to A\in ℂ$$
2		divdiv3d.2	$$⊢\phi \to B\in ℂ$$
3		divdiv3d.3	$$⊢\phi \to C\in ℂ$$
4		divdiv3d.4	$$⊢\phi \to B\ne 0$$
5		divdiv3d.5	$$⊢\phi \to C\ne 0$$
6	1 2 3 4 5	divdiv1d	$$⊢\phi \to \frac{\frac{A}{B}}{C}=\frac{A}{BC}$$
7	2 3	mulcomd	$$⊢\phi \to BC=CB$$
8	7	oveq2d	$$⊢\phi \to \frac{A}{BC}=\frac{A}{CB}$$
9	6 8	eqtrd	$$⊢\phi \to \frac{\frac{A}{B}}{C}=\frac{A}{CB}$$