Theorem ltadd1d

Description: Addition to both sides of 'less than'. Theorem I.18 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 )
	ltadd1d.3	|- ( ph -> C e. RR )
Assertion	ltadd1d	|- ( ph -> ( A < B <-> ( 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		ltadd1	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( A + C ) < ( B + C ) ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( A < B <-> ( A + C ) < ( B + C ) ) )