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	$$⊢\phi \to A\in ℝ$$
	rpgecld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
	rpgecld.3	$$⊢\phi \to B\le A$$
Assertion	rpgecld	$$⊢\phi \to A\in {ℝ}^{+}$$

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	$$⊢\phi \to A\in ℝ$$
2		rpgecld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
3		rpgecld.3	$$⊢\phi \to B\le A$$
4		rpgecl	$$⊢\left(B\in {ℝ}^{+}\wedge A\in ℝ\wedge B\le A\right)\to A\in {ℝ}^{+}$$
5	2 1 3 4	syl3anc	$$⊢\phi \to A\in {ℝ}^{+}$$