Theorem lemul1ad

Description: Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltp1d.1	$$⊢\phi \to A\in ℝ$$
	divgt0d.2	$$⊢\phi \to B\in ℝ$$
	lemul1ad.3	$$⊢\phi \to C\in ℝ$$
	lemul1ad.4	$$⊢\phi \to 0\le C$$
	lemul1ad.5	$$⊢\phi \to A\le B$$
Assertion	lemul1ad	$$⊢\phi \to AC\le BC$$

Proof

Step	Hyp	Ref	Expression
1		ltp1d.1	$$⊢\phi \to A\in ℝ$$
2		divgt0d.2	$$⊢\phi \to B\in ℝ$$
3		lemul1ad.3	$$⊢\phi \to C\in ℝ$$
4		lemul1ad.4	$$⊢\phi \to 0\le C$$
5		lemul1ad.5	$$⊢\phi \to A\le B$$
6	3 4	jca	$$⊢\phi \to \left(C\in ℝ\wedge 0\le C\right)$$
7		lemul1a	$$⊢\left(\left(A\in ℝ\wedge B\in ℝ\wedge \left(C\in ℝ\wedge 0\le C\right)\right)\wedge A\le B\right)\to AC\le BC$$
8	1 2 6 5 7	syl31anc	$$⊢\phi \to AC\le BC$$