Theorem divgt0

Description: The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999)

	Ref	Expression
Assertion	divgt0	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 0 < ( 𝐴 / 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		gt0div	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) → ( 0 < 𝐴 ↔ 0 < ( 𝐴 / 𝐵 ) ) )
2	1	biimpd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) → ( 0 < 𝐴 → 0 < ( 𝐴 / 𝐵 ) ) )
3	2	3exp	⊢ ( 𝐴 ∈ ℝ → ( 𝐵 ∈ ℝ → ( 0 < 𝐵 → ( 0 < 𝐴 → 0 < ( 𝐴 / 𝐵 ) ) ) ) )
4	3	com34	⊢ ( 𝐴 ∈ ℝ → ( 𝐵 ∈ ℝ → ( 0 < 𝐴 → ( 0 < 𝐵 → 0 < ( 𝐴 / 𝐵 ) ) ) ) )
5	4	com23	⊢ ( 𝐴 ∈ ℝ → ( 0 < 𝐴 → ( 𝐵 ∈ ℝ → ( 0 < 𝐵 → 0 < ( 𝐴 / 𝐵 ) ) ) ) )
6	5	imp43	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 0 < ( 𝐴 / 𝐵 ) )