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Metamath Proof Explorer

Theorem divgt0ii

Description: The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999)

Ref Expression
Hypotheses ltplus1.1 ⊢ A ∈ ℝ
prodgt0.2 ⊢ B ∈ ℝ
ltreci.3 ⊢ 0 < A
ltreci.4 ⊢ 0 < B
Assertion divgt0ii ⊢ 0 < A B

Proof

Step Hyp Ref Expression
1 ltplus1.1 ⊢ A ∈ ℝ
2 prodgt0.2 ⊢ B ∈ ℝ
3 ltreci.3 ⊢ 0 < A
4 ltreci.4 ⊢ 0 < B
5 1 2 4 divgt0i2i ⊢ 0 < A → 0 < A B
6 3 5 ax-mp ⊢ 0 < A B