Theorem ltsubaddi

Description: 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997) (Proof shortened by Andrew Salmon, 19-Nov-2011)

	Ref	Expression
Hypotheses	lt2.1	$$⊢A\in ℝ$$
	lt2.2	$$⊢B\in ℝ$$
	lt2.3	$$⊢C\in ℝ$$
Assertion	ltsubaddi	$$⊢A-B<C\leftrightarrow A<C+B$$

Proof

Step	Hyp	Ref	Expression
1		lt2.1	$$⊢A\in ℝ$$
2		lt2.2	$$⊢B\in ℝ$$
3		lt2.3	$$⊢C\in ℝ$$
4		ltsubadd	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge C\in ℝ\right)\to \left(A-B<C\leftrightarrow A<C+B\right)$$
5	1 2 3 4	mp3an	$$⊢A-B<C\leftrightarrow A<C+B$$