Theorem ltsubposd

Description: Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	|- ( ph -> A e. RR )
	ltnegd.2	|- ( ph -> B e. RR )
Assertion	ltsubposd	|- ( ph -> ( 0 < A <-> ( B - A ) < B ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	|- ( ph -> A e. RR )
2		ltnegd.2	|- ( ph -> B e. RR )
3		ltsubpos	|- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> ( B - A ) < B ) )
4	1 2 3	syl2anc	|- ( ph -> ( 0 < A <-> ( B - A ) < B ) )