Theorem ltaddposi

Description: Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999)

	Ref	Expression
Hypotheses	lt2.1	⊢ 𝐴 ∈ ℝ
	lt2.2	⊢ 𝐵 ∈ ℝ
Assertion	ltaddposi	⊢ ( 0 < 𝐴 ↔ 𝐵 < ( 𝐵 + 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		lt2.2	⊢ 𝐵 ∈ ℝ
3		ltaddpos	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴 ↔ 𝐵 < ( 𝐵 + 𝐴 ) ) )
4	1 2 3	mp2an	⊢ ( 0 < 𝐴 ↔ 𝐵 < ( 𝐵 + 𝐴 ) )