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Metamath Proof Explorer

Theorem addge02d

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 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion addge02d ( 𝜑 → ( 0 ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐵 + 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 addge02 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐵 + 𝐴 ) ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 0 ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐵 + 𝐴 ) ) )