Theorem le2addd

Description: Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016) (Proof shortened by Glauco Siliprandi, 5-Apr-2020)

	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	$$⊢\phi \to D\in ℝ$$
	le2addd.5	$$⊢\phi \to A\le C$$
	le2addd.6	$$⊢\phi \to B\le D$$
Assertion	le2addd	$$⊢\phi \to A+B\le C+D$$

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		le2addd.5	$$⊢\phi \to A\le C$$
6		le2addd.6	$$⊢\phi \to B\le D$$
7	1 2	readdcld	$$⊢\phi \to A+B\in ℝ$$
8	3 2	readdcld	$$⊢\phi \to C+B\in ℝ$$
9	3 4	readdcld	$$⊢\phi \to C+D\in ℝ$$
10	1 3 2 5	leadd1dd	$$⊢\phi \to A+B\le C+B$$
11	2 4 3 6	leadd2dd	$$⊢\phi \to C+B\le C+D$$
12	7 8 9 10 11	letrd	$$⊢\phi \to A+B\le C+D$$