Theorem divdirzi

Description: Distribution of division over addition. (Contributed by NM, 31-Jul-2004)

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

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divmulz.3	$$⊢C\in ℂ$$
4		divdir	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge \left(C\in ℂ\wedge C\ne 0\right)\right)\to \frac{A+B}{C}=\left(\frac{A}{C}\right)+\left(\frac{B}{C}\right)$$
5	1 2 4	mp3an12	$$⊢\left(C\in ℂ\wedge C\ne 0\right)\to \frac{A+B}{C}=\left(\frac{A}{C}\right)+\left(\frac{B}{C}\right)$$
6	3 5	mpan	$$⊢C\ne 0\to \frac{A+B}{C}=\left(\frac{A}{C}\right)+\left(\frac{B}{C}\right)$$