Theorem mulgt0d

Description: The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	mulgt0d.3	⊢ ( 𝜑 → 0 < 𝐴 )
	mulgt0d.4	⊢ ( 𝜑 → 0 < 𝐵 )
Assertion	mulgt0d	⊢ ( 𝜑 → 0 < ( 𝐴 · 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		mulgt0d.3	⊢ ( 𝜑 → 0 < 𝐴 )
4		mulgt0d.4	⊢ ( 𝜑 → 0 < 𝐵 )
5		mulgt0	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 0 < ( 𝐴 · 𝐵 ) )
6	1 3 2 4 5	syl22anc	⊢ ( 𝜑 → 0 < ( 𝐴 · 𝐵 ) )