Theorem int-ineq2ndprincd

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

	Ref	Expression
Hypotheses	int-ineq2ndprincd.1	|- ( ph -> A e. RR )
	int-ineq2ndprincd.2	|- ( ph -> B e. RR )
	int-ineq2ndprincd.3	|- ( ph -> C e. RR )
	int-ineq2ndprincd.4	|- ( ph -> B <_ A )
	int-ineq2ndprincd.5	|- ( ph -> 0 <_ C )
Assertion	int-ineq2ndprincd	|- ( ph -> ( B x. C ) <_ ( A x. C ) )

Proof

Step	Hyp	Ref	Expression
1		int-ineq2ndprincd.1	|- ( ph -> A e. RR )
2		int-ineq2ndprincd.2	|- ( ph -> B e. RR )
3		int-ineq2ndprincd.3	|- ( ph -> C e. RR )
4		int-ineq2ndprincd.4	|- ( ph -> B <_ A )
5		int-ineq2ndprincd.5	|- ( ph -> 0 <_ C )
6	2 1 3 5 4	lemul1ad	|- ( ph -> ( B x. C ) <_ ( A x. C ) )