Theorem lt2addi

Description: Adding both side of two inequalities. Theorem I.25 of Apostol p. 20. (Contributed by NM, 14-May-1999)

	Ref	Expression
Hypotheses	lt2.1	$$⊢A\in ℝ$$
	lt2.2	$$⊢B\in ℝ$$
	lt2.3	$$⊢C\in ℝ$$
	lt.4	$$⊢D\in ℝ$$
Assertion	lt2addi	$$⊢\left(A<C\wedge B<D\right)\to A+B<C+D$$

Proof

Step	Hyp	Ref	Expression
1		lt2.1	$$⊢A\in ℝ$$
2		lt2.2	$$⊢B\in ℝ$$
3		lt2.3	$$⊢C\in ℝ$$
4		lt.4	$$⊢D\in ℝ$$
5		lt2add	$$⊢\left(\left(A\in ℝ\wedge B\in ℝ\right)\wedge \left(C\in ℝ\wedge D\in ℝ\right)\right)\to \left(\left(A<C\wedge B<D\right)\to A+B<C+D\right)$$
6	1 2 3 4 5	mp4an	$$⊢\left(A<C\wedge B<D\right)\to A+B<C+D$$