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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 ) )