Theorem sublt0d

Description: When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	sublt0d.1	$$⊢\phi \to A\in ℝ$$
	sublt0d.2	$$⊢\phi \to B\in ℝ$$
Assertion	sublt0d	$$⊢\phi \to \left(A-B<0\leftrightarrow A<B\right)$$

Proof

Step	Hyp	Ref	Expression
1		sublt0d.1	$$⊢\phi \to A\in ℝ$$
2		sublt0d.2	$$⊢\phi \to B\in ℝ$$
3		0red	$$⊢\phi \to 0\in ℝ$$
4	1 2 3	ltsubaddd	$$⊢\phi \to \left(A-B<0\leftrightarrow A<0+B\right)$$
5	2	recnd	$$⊢\phi \to B\in ℂ$$
6	5	addlidd	$$⊢\phi \to 0+B=B$$
7	6	breq2d	$$⊢\phi \to \left(A<0+B\leftrightarrow A<B\right)$$
8	4 7	bitrd	$$⊢\phi \to \left(A-B<0\leftrightarrow A<B\right)$$