Metamath Proof Explorer
Description: A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
rpgecld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
ge0p1rp.2 |
⊢ ( 𝜑 → 0 ≤ 𝐴 ) |
|
Assertion |
ge0p1rpd |
⊢ ( 𝜑 → ( 𝐴 + 1 ) ∈ ℝ+ ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
rpgecld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
ge0p1rp.2 |
⊢ ( 𝜑 → 0 ≤ 𝐴 ) |
3 |
|
ge0p1rp |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( 𝐴 + 1 ) ∈ ℝ+ ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 + 1 ) ∈ ℝ+ ) |