Theorem lesubaddd

Description: 'Less than or equal to' relationship between subtraction and addition. (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	$$⊢\phi \to C\in ℝ$$
Assertion	lesubaddd	$$⊢\phi \to \left(A-B\le C\leftrightarrow A\le C+B\right)$$

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		lesubadd	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge C\in ℝ\right)\to \left(A-B\le C\leftrightarrow A\le C+B\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(A-B\le C\leftrightarrow A\le C+B\right)$$