Theorem int-ineq2ndprincd

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

	Ref	Expression
Hypotheses	int-ineq2ndprincd.1	$$⊢\phi \to A\in ℝ$$
	int-ineq2ndprincd.2	$$⊢\phi \to B\in ℝ$$
	int-ineq2ndprincd.3	$$⊢\phi \to C\in ℝ$$
	int-ineq2ndprincd.4	$$⊢\phi \to B\le A$$
	int-ineq2ndprincd.5	$$⊢\phi \to 0\le C$$
Assertion	int-ineq2ndprincd	$$⊢\phi \to BC\le AC$$

Proof

Step	Hyp	Ref	Expression
1		int-ineq2ndprincd.1	$$⊢\phi \to A\in ℝ$$
2		int-ineq2ndprincd.2	$$⊢\phi \to B\in ℝ$$
3		int-ineq2ndprincd.3	$$⊢\phi \to C\in ℝ$$
4		int-ineq2ndprincd.4	$$⊢\phi \to B\le A$$
5		int-ineq2ndprincd.5	$$⊢\phi \to 0\le C$$
6	2 1 3 5 4	lemul1ad	$$⊢\phi \to BC\le AC$$