Metamath Proof Explorer
Description: A number greater than or equal to a positive real is positive real.
(Contributed by Mario Carneiro, 28-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
rpgecld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
rpgecld.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
|
|
rpgecld.3 |
⊢ ( 𝜑 → 𝐵 ≤ 𝐴 ) |
|
Assertion |
rpgecld |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
rpgecld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
rpgecld.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
3 |
|
rpgecld.3 |
⊢ ( 𝜑 → 𝐵 ≤ 𝐴 ) |
4 |
|
rpgecl |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ≤ 𝐴 ) → 𝐴 ∈ ℝ+ ) |
5 |
2 1 3 4
|
syl3anc |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |