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 ∈ ℝ
	lt2.2	⊢ 𝐵 ∈ ℝ
Assertion	add20i	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( ( 𝐴 + 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		lt2.2	⊢ 𝐵 ∈ ℝ
3		add20	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 + 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) )
4	3	an4s	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 + 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) )
5	1 2 4	mpanl12	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( ( 𝐴 + 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) )