Theorem lt2halvesd

Description: A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	rehalfcld.1	|- ( ph -> A e. RR )
	lt2halvesd.2	|- ( ph -> B e. RR )
	lt2halvesd.3	|- ( ph -> C e. RR )
	lt2halvesd.4	|- ( ph -> A < ( C / 2 ) )
	lt2halvesd.5	|- ( ph -> B < ( C / 2 ) )
Assertion	lt2halvesd	|- ( ph -> ( A + B ) < C )

Proof

Step	Hyp	Ref	Expression
1		rehalfcld.1	|- ( ph -> A e. RR )
2		lt2halvesd.2	|- ( ph -> B e. RR )
3		lt2halvesd.3	|- ( ph -> C e. RR )
4		lt2halvesd.4	|- ( ph -> A < ( C / 2 ) )
5		lt2halvesd.5	|- ( ph -> B < ( C / 2 ) )
6		lt2halves	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < ( C / 2 ) /\ B < ( C / 2 ) ) -> ( A + B ) < C ) )
7	1 2 3 6	syl3anc	|- ( ph -> ( ( A < ( C / 2 ) /\ B < ( C / 2 ) ) -> ( A + B ) < C ) )
8	4 5 7	mp2and	|- ( ph -> ( A + B ) < C )