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Metamath Proof Explorer

Theorem addge0i

Description: Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)

Ref Expression
Hypotheses lt2.1 ⊢ 𝐴 ∈ ℝ
lt2.2 ⊢ 𝐵 ∈ ℝ
Assertion addge0i ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → 0 ≤ ( 𝐴 + 𝐵 ) )

Proof

Step Hyp Ref Expression
1 lt2.1 ⊢ 𝐴 ∈ ℝ
2 lt2.2 ⊢ 𝐵 ∈ ℝ
3 addge0 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) ) → 0 ≤ ( 𝐴 + 𝐵 ) )
4 1 2 3 mpanl12 ⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → 0 ≤ ( 𝐴 + 𝐵 ) )