Metamath Proof Explorer
Description: The product of two positive numbers is positive. (Contributed by NM, 16-May-1999)
|
|
Ref |
Expression |
|
Hypotheses |
lt.1 |
⊢ 𝐴 ∈ ℝ |
|
|
lt.2 |
⊢ 𝐵 ∈ ℝ |
|
Assertion |
mulgt0i |
⊢ ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 · 𝐵 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
lt.1 |
⊢ 𝐴 ∈ ℝ |
2 |
|
lt.2 |
⊢ 𝐵 ∈ ℝ |
3 |
|
axmulgt0 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 · 𝐵 ) ) ) |
4 |
1 2 3
|
mp2an |
⊢ ( ( 0 < 𝐴 ∧ 0 < 𝐵 ) → 0 < ( 𝐴 · 𝐵 ) ) |