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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