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Metamath Proof Explorer

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 ) )