Metamath Proof Explorer
Description: Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
rpgecld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
rpgecld.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
|
Assertion |
ge0divd |
⊢ ( 𝜑 → ( 0 ≤ 𝐴 ↔ 0 ≤ ( 𝐴 / 𝐵 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
rpgecld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
rpgecld.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
3 |
2
|
rpred |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
4 |
2
|
rpgt0d |
⊢ ( 𝜑 → 0 < 𝐵 ) |
5 |
|
ge0div |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) → ( 0 ≤ 𝐴 ↔ 0 ≤ ( 𝐴 / 𝐵 ) ) ) |
6 |
1 3 4 5
|
syl3anc |
⊢ ( 𝜑 → ( 0 ≤ 𝐴 ↔ 0 ≤ ( 𝐴 / 𝐵 ) ) ) |