Theorem addgt0d

Description: Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	addgt0d.3	⊢ ( 𝜑 → 0 < 𝐴 )
	addgt0d.4	⊢ ( 𝜑 → 0 < 𝐵 )
Assertion	addgt0d	⊢ ( 𝜑 → 0 < ( 𝐴 + 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		addgt0d.3	⊢ ( 𝜑 → 0 < 𝐴 )
4		addgt0d.4	⊢ ( 𝜑 → 0 < 𝐵 )
5		0red	⊢ ( 𝜑 → 0 ∈ ℝ )
6	5 1 3	ltled	⊢ ( 𝜑 → 0 ≤ 𝐴 )
7	1 2 6 4	addgegt0d	⊢ ( 𝜑 → 0 < ( 𝐴 + 𝐵 ) )