Theorem ltmul2dd

Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of Apostol p. 20. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	ltmul1d.1	$$⊢\phi \to A\in ℝ$$
	ltmul1d.2	$$⊢\phi \to B\in ℝ$$
	ltmul1d.3	$$⊢\phi \to C\in {ℝ}^{+}$$
	ltdiv1dd.4	$$⊢\phi \to A<B$$
Assertion	ltmul2dd	$$⊢\phi \to CA<CB$$

Proof

Step	Hyp	Ref	Expression
1		ltmul1d.1	$$⊢\phi \to A\in ℝ$$
2		ltmul1d.2	$$⊢\phi \to B\in ℝ$$
3		ltmul1d.3	$$⊢\phi \to C\in {ℝ}^{+}$$
4		ltdiv1dd.4	$$⊢\phi \to A<B$$
5	1 2 3	ltmul2d	$$⊢\phi \to \left(A<B\leftrightarrow CA<CB\right)$$
6	4 5	mpbid	$$⊢\phi \to CA<CB$$