Theorem ltmul12ad

Description: Comparison of product of two positive numbers. (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	$$⊢\phi \to C\in ℝ$$
	ltmul12ad.3	$$⊢\phi \to D\in ℝ$$
	ltmul12ad.4	$$⊢\phi \to 0\le A$$
	ltmul12ad.5	$$⊢\phi \to A<B$$
	ltmul12ad.6	$$⊢\phi \to 0\le C$$
	ltmul12ad.7	$$⊢\phi \to C<D$$
Assertion	ltmul12ad	$$⊢\phi \to AC<BD$$

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		ltmul12ad.3	$$⊢\phi \to D\in ℝ$$
5		ltmul12ad.4	$$⊢\phi \to 0\le A$$
6		ltmul12ad.5	$$⊢\phi \to A<B$$
7		ltmul12ad.6	$$⊢\phi \to 0\le C$$
8		ltmul12ad.7	$$⊢\phi \to C<D$$
9	1 2	jca	$$⊢\phi \to \left(A\in ℝ\wedge B\in ℝ\right)$$
10	5 6	jca	$$⊢\phi \to \left(0\le A\wedge A<B\right)$$
11	3 4	jca	$$⊢\phi \to \left(C\in ℝ\wedge D\in ℝ\right)$$
12	7 8	jca	$$⊢\phi \to \left(0\le C\wedge C<D\right)$$
13		ltmul12a	$$⊢\left(\left(\left(A\in ℝ\wedge B\in ℝ\right)\wedge \left(0\le A\wedge A<B\right)\right)\wedge \left(\left(C\in ℝ\wedge D\in ℝ\right)\wedge \left(0\le C\wedge C<D\right)\right)\right)\to AC<BD$$
14	9 10 11 12 13	syl22anc	$$⊢\phi \to AC<BD$$