Theorem add20i

Description: Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999)

	Ref	Expression
Hypotheses	lt2.1	$$⊢A\in ℝ$$
	lt2.2	$$⊢B\in ℝ$$
Assertion	add20i	$$⊢\left(0\le A\wedge 0\le B\right)\to \left(A+B=0\leftrightarrow \left(A=0\wedge B=0\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		lt2.1	$$⊢A\in ℝ$$
2		lt2.2	$$⊢B\in ℝ$$
3		add20	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge \left(B\in ℝ\wedge 0\le B\right)\right)\to \left(A+B=0\leftrightarrow \left(A=0\wedge B=0\right)\right)$$
4	3	an4s	$$⊢\left(\left(A\in ℝ\wedge B\in ℝ\right)\wedge \left(0\le A\wedge 0\le B\right)\right)\to \left(A+B=0\leftrightarrow \left(A=0\wedge B=0\right)\right)$$
5	1 2 4	mpanl12	$$⊢\left(0\le A\wedge 0\le B\right)\to \left(A+B=0\leftrightarrow \left(A=0\wedge B=0\right)\right)$$