Theorem ltadd1i

Description: Addition to both sides of 'less than'. Theorem I.18 of Apostol p. 20. (Contributed by NM, 21-Jan-1997)

	Ref	Expression
Hypotheses	lt2.1	$$⊢A\in ℝ$$
	lt2.2	$$⊢B\in ℝ$$
	lt2.3	$$⊢C\in ℝ$$
Assertion	ltadd1i	$$⊢A<B\leftrightarrow A+C<B+C$$

Proof

Step	Hyp	Ref	Expression
1		lt2.1	$$⊢A\in ℝ$$
2		lt2.2	$$⊢B\in ℝ$$
3		lt2.3	$$⊢C\in ℝ$$
4		ltadd1	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge C\in ℝ\right)\to \left(A<B\leftrightarrow A+C<B+C\right)$$
5	1 2 3 4	mp3an	$$⊢A<B\leftrightarrow A+C<B+C$$