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 ) )