Theorem ltadd1dd

Description: Addition to both sides of 'less than'. Theorem I.18 of Apostol p. 20. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	leidd.1	|- ( ph -> A e. RR )
	ltnegd.2	|- ( ph -> B e. RR )
	ltadd1d.3	|- ( ph -> C e. RR )
	ltadd1dd.4	|- ( ph -> A < B )
Assertion	ltadd1dd	|- ( ph -> ( A + C ) < ( B + C ) )

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		ltadd1dd.4	|- ( ph -> A < B )
5	1 2 3	ltadd1d	|- ( ph -> ( A < B <-> ( A + C ) < ( B + C ) ) )
6	4 5	mpbid	|- ( ph -> ( A + C ) < ( B + C ) )