Theorem addgt0ii

Description: Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999)

	Ref	Expression
Hypotheses	lt2.1	⊢ 𝐴 ∈ ℝ
	lt2.2	⊢ 𝐵 ∈ ℝ
	addgt0i.3	⊢ 0 < 𝐴
	addgt0i.4	⊢ 0 < 𝐵
Assertion	addgt0ii	⊢ 0 < ( 𝐴 + 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		lt2.2	⊢ 𝐵 ∈ ℝ
3		addgt0i.3	⊢ 0 < 𝐴
4		addgt0i.4	⊢ 0 < 𝐵
5	1 2	addgt0i	⊢ ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 + 𝐵 ) )
6	3 4 5	mp2an	⊢ 0 < ( 𝐴 + 𝐵 )