Theorem lesubadd2i

Description: 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999)

	Ref	Expression
Hypotheses	lt2.1	⊢ 𝐴 ∈ ℝ
	lt2.2	⊢ 𝐵 ∈ ℝ
	lt2.3	⊢ 𝐶 ∈ ℝ
Assertion	lesubadd2i	⊢ ( ( 𝐴 − 𝐵 ) ≤ 𝐶 ↔ 𝐴 ≤ ( 𝐵 + 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		lt2.2	⊢ 𝐵 ∈ ℝ
3		lt2.3	⊢ 𝐶 ∈ ℝ
4		lesubadd2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) ≤ 𝐶 ↔ 𝐴 ≤ ( 𝐵 + 𝐶 ) ) )
5	1 2 3 4	mp3an	⊢ ( ( 𝐴 − 𝐵 ) ≤ 𝐶 ↔ 𝐴 ≤ ( 𝐵 + 𝐶 ) )