Theorem ltsub13d

Description: 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	|- ( ph -> A e. RR )
	ltnegd.2	|- ( ph -> B e. RR )
	ltadd1d.3	|- ( ph -> C e. RR )
	ltsub13d.4	|- ( ph -> A < ( B - C ) )
Assertion	ltsub13d	|- ( ph -> C < ( B - A ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	|- ( ph -> A e. RR )
2		ltnegd.2	|- ( ph -> B e. RR )
3		ltadd1d.3	|- ( ph -> C e. RR )
4		ltsub13d.4	|- ( ph -> A < ( B - C ) )
5		ltsub13	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < ( B - C ) <-> C < ( B - A ) ) )
6	1 2 3 5	syl3anc	|- ( ph -> ( A < ( B - C ) <-> C < ( B - A ) ) )
7	4 6	mpbid	|- ( ph -> C < ( B - A ) )