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Metamath Proof Explorer

Theorem divgt0i2i

Description: The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999)

Ref Expression
Hypotheses ltplus1.1 ⊢ A ∈ ℝ
prodgt0.2 ⊢ B ∈ ℝ
divgt0i2.3 ⊢ 0 < B
Assertion divgt0i2i ⊢ 0 < A → 0 < A B

Proof

Step Hyp Ref Expression
1 ltplus1.1 ⊢ A ∈ ℝ
2 prodgt0.2 ⊢ B ∈ ℝ
3 divgt0i2.3 ⊢ 0 < B
4 1 2 divgt0i ⊢ 0 < A ∧ 0 < B → 0 < A B
5 3 4 mpan2 ⊢ 0 < A → 0 < A B