Theorem ge0p1rp

Description: A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015)

	Ref	Expression
Assertion	ge0p1rp	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( 𝐴 + 1 ) ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		peano2re	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 + 1 ) ∈ ℝ )
2	1	adantr	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( 𝐴 + 1 ) ∈ ℝ )
3		0red	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → 0 ∈ ℝ )
4		simpl	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → 𝐴 ∈ ℝ )
5		simpr	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → 0 ≤ 𝐴 )
6		ltp1	⊢ ( 𝐴 ∈ ℝ → 𝐴 < ( 𝐴 + 1 ) )
7	6	adantr	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → 𝐴 < ( 𝐴 + 1 ) )
8	3 4 2 5 7	lelttrd	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → 0 < ( 𝐴 + 1 ) )
9		elrp	⊢ ( ( 𝐴 + 1 ) ∈ ℝ+ ↔ ( ( 𝐴 + 1 ) ∈ ℝ ∧ 0 < ( 𝐴 + 1 ) ) )
10	2 8 9	sylanbrc	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( 𝐴 + 1 ) ∈ ℝ+ )