Theorem le2subd

Description: Subtracting both sides of two 'less than or equal to' relations. (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	$$⊢\phi \to D\in ℝ$$
	le2addd.5	$$⊢\phi \to A\le C$$
	le2addd.6	$$⊢\phi \to B\le D$$
Assertion	le2subd	$$⊢\phi \to A-D\le C-B$$

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		le2sub	$$⊢\left(\left(A\in ℝ\wedge D\in ℝ\right)\wedge \left(C\in ℝ\wedge B\in ℝ\right)\right)\to \left(\left(A\le C\wedge B\le D\right)\to A-D\le C-B\right)$$
8	1 4 3 2 7	syl22anc	$$⊢\phi \to \left(\left(A\le C\wedge B\le D\right)\to A-D\le C-B\right)$$
9	5 6 8	mp2and	$$⊢\phi \to A-D\le C-B$$