Theorem lt2addd

Description: Adding both side of two inequalities. Theorem I.25 of Apostol p. 20. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	$$⊢\phi \to A\in ℝ$$
	ltnegd.2	$$⊢\phi \to B\in ℝ$$
	ltadd1d.3	$$⊢\phi \to C\in ℝ$$
	lt2addd.4	$$⊢\phi \to D\in ℝ$$
	lt2addd.5	$$⊢\phi \to A<C$$
	lt2addd.6	$$⊢\phi \to B<D$$
Assertion	lt2addd	$$⊢\phi \to A+B<C+D$$

Proof

Step	Hyp	Ref	Expression
1		leidd.1	$$⊢\phi \to A\in ℝ$$
2		ltnegd.2	$$⊢\phi \to B\in ℝ$$
3		ltadd1d.3	$$⊢\phi \to C\in ℝ$$
4		lt2addd.4	$$⊢\phi \to D\in ℝ$$
5		lt2addd.5	$$⊢\phi \to A<C$$
6		lt2addd.6	$$⊢\phi \to B<D$$
7	2 4 6	ltled	$$⊢\phi \to B\le D$$
8	1 2 3 4 5 7	ltleaddd	$$⊢\phi \to A+B<C+D$$