Theorem addgt0i

Description: Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)

	Ref	Expression
Hypotheses	lt2.1	⊢ 𝐴 ∈ ℝ
	lt2.2	⊢ 𝐵 ∈ ℝ
Assertion	addgt0i	⊢ ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 + 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		lt2.2	⊢ 𝐵 ∈ ℝ
3		addgt0	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 < 𝐴 ∧ 0 < 𝐵 ) ) → 0 < ( 𝐴 + 𝐵 ) )
4	1 2 3	mpanl12	⊢ ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 + 𝐵 ) )