Theorem div13d

Description: A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	$$⊢\phi \to A\in ℂ$$
	divcld.2	$$⊢\phi \to B\in ℂ$$
	divmuld.3	$$⊢\phi \to C\in ℂ$$
	divmuld.4	$$⊢\phi \to B\ne 0$$
Assertion	div13d	$$⊢\phi \to \left(\frac{A}{B}\right)C=\left(\frac{C}{B}\right)A$$

Proof

Step	Hyp	Ref	Expression
1		div1d.1	$$⊢\phi \to A\in ℂ$$
2		divcld.2	$$⊢\phi \to B\in ℂ$$
3		divmuld.3	$$⊢\phi \to C\in ℂ$$
4		divmuld.4	$$⊢\phi \to B\ne 0$$
5		div13	$$⊢\left(A\in ℂ\wedge \left(B\in ℂ\wedge B\ne 0\right)\wedge C\in ℂ\right)\to \left(\frac{A}{B}\right)C=\left(\frac{C}{B}\right)A$$
6	1 2 4 3 5	syl121anc	$$⊢\phi \to \left(\frac{A}{B}\right)C=\left(\frac{C}{B}\right)A$$