Theorem ge0p1rpd

Description: A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpgecld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ge0p1rp.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
Assertion	ge0p1rpd	⊢ ( 𝜑 → ( 𝐴 + 1 ) ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ge0p1rp.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
3		ge0p1rp	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( 𝐴 + 1 ) ∈ ℝ+ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 + 1 ) ∈ ℝ+ )