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Metamath Proof Explorer

Theorem addgtge0d

Description: Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021)

Ref Expression
Hypotheses leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
addgtge0d.3 ⊢ ( 𝜑 → 0 < 𝐴 )
addgtge0d.4 ⊢ ( 𝜑 → 0 ≤ 𝐵 )
Assertion addgtge0d ( 𝜑 → 0 < ( 𝐴 + 𝐵 ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 addgtge0d.3 ⊢ ( 𝜑 → 0 < 𝐴 )
4 addgtge0d.4 ⊢ ( 𝜑 → 0 ≤ 𝐵 )
5 addgtge0 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 < 𝐴 ∧ 0 ≤ 𝐵 ) ) → 0 < ( 𝐴 + 𝐵 ) )
6 1 2 3 4 5 syl22anc ⊢ ( 𝜑 → 0 < ( 𝐴 + 𝐵 ) )