Theorem ltadd2i

Description: Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997) (Proof shortened by OpenAI, 25-Mar-2020)

	Ref	Expression
Hypotheses	lt.1	$$⊢A\in ℝ$$
	lt.2	$$⊢B\in ℝ$$
	lt.3	$$⊢C\in ℝ$$
Assertion	ltadd2i	$$⊢A<B\leftrightarrow C+A<C+B$$

Proof

Step	Hyp	Ref	Expression
1		lt.1	$$⊢A\in ℝ$$
2		lt.2	$$⊢B\in ℝ$$
3		lt.3	$$⊢C\in ℝ$$
4		ltadd2	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge C\in ℝ\right)\to \left(A<B\leftrightarrow C+A<C+B\right)$$
5	1 2 3 4	mp3an	$$⊢A<B\leftrightarrow C+A<C+B$$