Theorem divsubdird

Description: Distribution of division over subtraction. (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	$$⊢\phi \to C\in ℂ$$
	divassd.4	$$⊢\phi \to C\ne 0$$
Assertion	divsubdird	$$⊢\phi \to \frac{A-B}{C}=\left(\frac{A}{C}\right)-\left(\frac{B}{C}\right)$$

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		divassd.4	$$⊢\phi \to C\ne 0$$
5		divsubdir	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge \left(C\in ℂ\wedge C\ne 0\right)\right)\to \frac{A-B}{C}=\left(\frac{A}{C}\right)-\left(\frac{B}{C}\right)$$
6	1 2 3 4 5	syl112anc	$$⊢\phi \to \frac{A-B}{C}=\left(\frac{A}{C}\right)-\left(\frac{B}{C}\right)$$