This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem mulgt0i

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