Theorem int-ineq1stprincd

Description: FirstPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-ineq1stprincd.1	|- ( ph -> A e. RR )
	int-ineq1stprincd.2	|- ( ph -> B e. RR )
	int-ineq1stprincd.3	|- ( ph -> C e. RR )
	int-ineq1stprincd.4	|- ( ph -> D e. RR )
	int-ineq1stprincd.5	|- ( ph -> B <_ A )
	int-ineq1stprincd.6	|- ( ph -> D <_ C )
Assertion	int-ineq1stprincd	|- ( ph -> ( B + D ) <_ ( A + C ) )

Proof

Step	Hyp	Ref	Expression
1		int-ineq1stprincd.1	|- ( ph -> A e. RR )
2		int-ineq1stprincd.2	|- ( ph -> B e. RR )
3		int-ineq1stprincd.3	|- ( ph -> C e. RR )
4		int-ineq1stprincd.4	|- ( ph -> D e. RR )
5		int-ineq1stprincd.5	|- ( ph -> B <_ A )
6		int-ineq1stprincd.6	|- ( ph -> D <_ C )
7	2 4 1 3 5 6	le2addd	|- ( ph -> ( B + D ) <_ ( A + C ) )