Metamath Proof Explorer

Theorem lesubaddi

Description: 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)

Ref Expression
Hypotheses lt2.1 ⊢ A ∈ ℝ
lt2.2 ⊢ B ∈ ℝ
lt2.3 ⊢ C ∈ ℝ
Assertion lesubaddi ⊢ A − B ≤ C ↔ A ≤ C + B

Proof

Step Hyp Ref Expression
1 lt2.1 ⊢ A ∈ ℝ
2 lt2.2 ⊢ B ∈ ℝ
3 lt2.3 ⊢ C ∈ ℝ
4 lesubadd ⊢ A ∈ ℝ ∧ B ∈ ℝ ∧ C ∈ ℝ → A − B ≤ C ↔ A ≤ C + B
5 1 2 3 4 mp3an ⊢ A − B ≤ C ↔ A ≤ C + B