Theorem ltmulgt12d

Description: Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpgecld.1	$$⊢\phi \to A\in ℝ$$
	rpgecld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
Assertion	ltmulgt12d	$$⊢\phi \to \left(1<A\leftrightarrow B<AB\right)$$

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	$$⊢\phi \to A\in ℝ$$
2		rpgecld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
3	2	rpred	$$⊢\phi \to B\in ℝ$$
4	2	rpgt0d	$$⊢\phi \to 0<B$$
5		ltmulgt12	$$⊢\left(B\in ℝ\wedge A\in ℝ\wedge 0<B\right)\to \left(1<A\leftrightarrow B<AB\right)$$
6	3 1 4 5	syl3anc	$$⊢\phi \to \left(1<A\leftrightarrow B<AB\right)$$