Theorem ltadd2d

Description: Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	|- ( ph -> A e. RR )
	ltd.2	|- ( ph -> B e. RR )
	letrd.3	|- ( ph -> C e. RR )
Assertion	ltadd2d	|- ( ph -> ( A < B <-> ( C + A ) < ( C + B ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	|- ( ph -> A e. RR )
2		ltd.2	|- ( ph -> B e. RR )
3		letrd.3	|- ( ph -> C e. RR )
4		ltadd2	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C + A ) < ( C + B ) ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( A < B <-> ( C + A ) < ( C + B ) ) )