Theorem ltneg

Description: Negative of both sides of 'less than'. Theorem I.23 of Apostol p. 20. (Contributed by NM, 27-Aug-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	ltneg	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A<B\leftrightarrow -B<-A\right)$$

Proof

Step	Hyp	Ref	Expression
1		0re	$$⊢0\in ℝ$$
2		ltsub2	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge 0\in ℝ\right)\to \left(A<B\leftrightarrow 0-B<0-A\right)$$
3	1 2	mp3an3	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A<B\leftrightarrow 0-B<0-A\right)$$
4		df-neg	$$⊢-B=0-B$$
5		df-neg	$$⊢-A=0-A$$
6	4 5	breq12i	$$⊢-B<-A\leftrightarrow 0-B<0-A$$
7	3 6	bitr4di	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A<B\leftrightarrow -B<-A\right)$$