Theorem divdiri

Description: Distribution of division over addition. (Contributed by NM, 16-Feb-1995)

	Ref	Expression
Hypotheses	divclz.1	$$⊢A\in ℂ$$
	divclz.2	$$⊢B\in ℂ$$
	divmulz.3	$$⊢C\in ℂ$$
	divass.4	$$⊢C\ne 0$$
Assertion	divdiri	$$⊢\frac{A+B}{C}=\left(\frac{A}{C}\right)+\left(\frac{B}{C}\right)$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divmulz.3	$$⊢C\in ℂ$$
4		divass.4	$$⊢C\ne 0$$
5	1 2 3	divdirzi	$$⊢C\ne 0\to \frac{A+B}{C}=\left(\frac{A}{C}\right)+\left(\frac{B}{C}\right)$$
6	4 5	ax-mp	$$⊢\frac{A+B}{C}=\left(\frac{A}{C}\right)+\left(\frac{B}{C}\right)$$