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Metamath Proof Explorer

Theorem divgt0d

Description: The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses ltp1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
divgt0d.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
divgt0d.3 ⊢ ( 𝜑 → 0 < 𝐴 )
divgt0d.4 ⊢ ( 𝜑 → 0 < 𝐵 )
Assertion divgt0d ( 𝜑 → 0 < ( 𝐴 / 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ltp1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 divgt0d.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 divgt0d.3 ⊢ ( 𝜑 → 0 < 𝐴 )
4 divgt0d.4 ⊢ ( 𝜑 → 0 < 𝐵 )
5 divgt0 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 0 < ( 𝐴 / 𝐵 ) )
6 1 3 2 4 5 syl22anc ⊢ ( 𝜑 → 0 < ( 𝐴 / 𝐵 ) )