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Metamath Proof Explorer

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 ) )