Theorem rpregt0d

Description: A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypothesis	rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
Assertion	rpregt0d	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2	1	rpred	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
3	1	rpgt0d	⊢ ( 𝜑 → 0 < 𝐴 )
4	2 3	jca	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) )