Theorem leltaddd

Description: Adding both sides of two orderings. (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	$$⊢\phi \to D\in ℝ$$
	leltaddd.5	$$⊢\phi \to A\le C$$
	leltaddd.6	$$⊢\phi \to B<D$$
Assertion	leltaddd	$$⊢\phi \to A+B<C+D$$

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		leltaddd.5	$$⊢\phi \to A\le C$$
6		leltaddd.6	$$⊢\phi \to B<D$$
7		leltadd	$$⊢\left(\left(A\in ℝ\wedge B\in ℝ\right)\wedge \left(C\in ℝ\wedge D\in ℝ\right)\right)\to \left(\left(A\le C\wedge B<D\right)\to A+B<C+D\right)$$
8	1 2 3 4 7	syl22anc	$$⊢\phi \to \left(\left(A\le C\wedge B<D\right)\to A+B<C+D\right)$$
9	5 6 8	mp2and	$$⊢\phi \to A+B<C+D$$