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Metamath Proof Explorer

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 )