Theorem ltaddposd

Description: Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		ltaddpos	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴 ↔ 𝐵 < ( 𝐵 + 𝐴 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 0 < 𝐴 ↔ 𝐵 < ( 𝐵 + 𝐴 ) ) )