Metamath Proof Explorer


Theorem ltaddpos2d

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

Ref Expression
Hypotheses leidd.1 ( 𝜑𝐴 ∈ ℝ )
ltnegd.2 ( 𝜑𝐵 ∈ ℝ )
Assertion ltaddpos2d ( 𝜑 → ( 0 < 𝐴𝐵 < ( 𝐴 + 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ( 𝜑𝐴 ∈ ℝ )
2 ltnegd.2 ( 𝜑𝐵 ∈ ℝ )
3 ltaddpos2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴𝐵 < ( 𝐴 + 𝐵 ) ) )
4 1 2 3 syl2anc ( 𝜑 → ( 0 < 𝐴𝐵 < ( 𝐴 + 𝐵 ) ) )