Theorem le2halvesd

Description: A sum is less than the whole if each term is less than half. (Contributed by Thierry Arnoux, 29-Nov-2017)

	Ref	Expression
Hypotheses	le2halvesd.1	|- ( ph -> A e. RR )
	le2halvesd.2	|- ( ph -> B e. RR )
	le2halvesd.3	|- ( ph -> C e. RR )
	le2halvesd.4	|- ( ph -> A <_ ( C / 2 ) )
	le2halvesd.5	|- ( ph -> B <_ ( C / 2 ) )
Assertion	le2halvesd	|- ( ph -> ( A + B ) <_ C )

Proof

Step	Hyp	Ref	Expression
1		le2halvesd.1	|- ( ph -> A e. RR )
2		le2halvesd.2	|- ( ph -> B e. RR )
3		le2halvesd.3	|- ( ph -> C e. RR )
4		le2halvesd.4	|- ( ph -> A <_ ( C / 2 ) )
5		le2halvesd.5	|- ( ph -> B <_ ( C / 2 ) )
6	3	rehalfcld	|- ( ph -> ( C / 2 ) e. RR )
7	1 2 6 6 4 5	le2addd	|- ( ph -> ( A + B ) <_ ( ( C / 2 ) + ( C / 2 ) ) )
8	3	recnd	|- ( ph -> C e. CC )
9	8	2halvesd	|- ( ph -> ( ( C / 2 ) + ( C / 2 ) ) = C )
10	7 9	breqtrd	|- ( ph -> ( A + B ) <_ C )