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 ) )