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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