Metamath Proof Explorer


Theorem addge01d

Description: A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1 ( 𝜑𝐴 ∈ ℝ )
ltnegd.2 ( 𝜑𝐵 ∈ ℝ )
Assertion addge01d ( 𝜑 → ( 0 ≤ 𝐵𝐴 ≤ ( 𝐴 + 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ( 𝜑𝐴 ∈ ℝ )
2 ltnegd.2 ( 𝜑𝐵 ∈ ℝ )
3 addge01 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ 𝐵𝐴 ≤ ( 𝐴 + 𝐵 ) ) )
4 1 2 3 syl2anc ( 𝜑 → ( 0 ≤ 𝐵𝐴 ≤ ( 𝐴 + 𝐵 ) ) )