Metamath Proof Explorer


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 ) )