Theorem rpgecld

Description: A number greater than or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpgecld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	rpgecld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
	rpgecld.3	⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
Assertion	rpgecld	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		rpgecld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3		rpgecld.3	⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
4		rpgecl	⊢ ( ( 𝐵 ∈ ℝ+ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ≤ 𝐴 ) → 𝐴 ∈ ℝ+ )
5	2 1 3 4	syl3anc	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )