Metamath Proof Explorer
Description: Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999) (Revised by Mario Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
lt2.1 |
⊢ 𝐴 ∈ ℝ |
|
|
lt2.2 |
⊢ 𝐵 ∈ ℝ |
|
Assertion |
addgegt0i |
⊢ ( ( 0 ≤ 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 + 𝐵 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
lt2.1 |
⊢ 𝐴 ∈ ℝ |
2 |
|
lt2.2 |
⊢ 𝐵 ∈ ℝ |
3 |
|
addgegt0 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 0 < 𝐵 ) ) → 0 < ( 𝐴 + 𝐵 ) ) |
4 |
1 2 3
|
mpanl12 |
⊢ ( ( 0 ≤ 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 + 𝐵 ) ) |