Metamath Proof Explorer


Theorem ge0p1rpd

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

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

Proof

Step Hyp Ref Expression
1 rpgecld.1 ( 𝜑𝐴 ∈ 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+ )