Theorem ltmulgt12

Description: Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005)

	Ref	Expression
Assertion	ltmulgt12	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge 0<A\right)\to \left(1<B\leftrightarrow A<BA\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltmulgt11	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge 0<A\right)\to \left(1<B\leftrightarrow A<AB\right)$$
2		recn	$$⊢A\in ℝ\to A\in ℂ$$
3		recn	$$⊢B\in ℝ\to B\in ℂ$$
4		mulcom	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to AB=BA$$
5	2 3 4	syl2an	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to AB=BA$$
6	5	3adant3	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge 0<A\right)\to AB=BA$$
7	6	breq2d	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge 0<A\right)\to \left(A<AB\leftrightarrow A<BA\right)$$
8	1 7	bitrd	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge 0<A\right)\to \left(1<B\leftrightarrow A<BA\right)$$