Theorem addgtge0d

Description: Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021)

	Ref	Expression
Hypotheses	leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	addgtge0d.3	⊢ ( 𝜑 → 0 < 𝐴 )
	addgtge0d.4	⊢ ( 𝜑 → 0 ≤ 𝐵 )
Assertion	addgtge0d	⊢ ( 𝜑 → 0 < ( 𝐴 + 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		addgtge0d.3	⊢ ( 𝜑 → 0 < 𝐴 )
4		addgtge0d.4	⊢ ( 𝜑 → 0 ≤ 𝐵 )
5		addgtge0	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 < 𝐴 ∧ 0 ≤ 𝐵 ) ) → 0 < ( 𝐴 + 𝐵 ) )
6	1 2 3 4 5	syl22anc	⊢ ( 𝜑 → 0 < ( 𝐴 + 𝐵 ) )