Theorem ltnegd

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

	Ref	Expression
Hypotheses	leidd.1	|- ( ph -> A e. RR )
	ltnegd.2	|- ( ph -> B e. RR )
Assertion	ltnegd	|- ( ph -> ( A < B <-> -u B < -u A ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	|- ( ph -> A e. RR )
2		ltnegd.2	|- ( ph -> B e. RR )
3		ltneg	|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -u B < -u A ) )
4	1 2 3	syl2anc	|- ( ph -> ( A < B <-> -u B < -u A ) )