Theorem lt2addd

Description: Adding both side of two inequalities. Theorem I.25 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 )
	lt2addd.4	|- ( ph -> D e. RR )
	lt2addd.5	|- ( ph -> A < C )
	lt2addd.6	|- ( ph -> B < D )
Assertion	lt2addd	|- ( ph -> ( A + B ) < ( C + D ) )

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		lt2addd.4	|- ( ph -> D e. RR )
5		lt2addd.5	|- ( ph -> A < C )
6		lt2addd.6	|- ( ph -> B < D )
7	2 4 6	ltled	|- ( ph -> B <_ D )
8	1 2 3 4 5 7	ltleaddd	|- ( ph -> ( A + B ) < ( C + D ) )