Theorem ltmul1i

Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of Apostol p. 20. (Contributed by NM, 16-May-1999)

	Ref	Expression
Hypotheses	ltplus1.1	$$⊢A\in ℝ$$
	prodgt0.2	$$⊢B\in ℝ$$
	ltmul1.3	$$⊢C\in ℝ$$
Assertion	ltmul1i	$$⊢0<C\to \left(A<B\leftrightarrow AC<BC\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	$$⊢A\in ℝ$$
2		prodgt0.2	$$⊢B\in ℝ$$
3		ltmul1.3	$$⊢C\in ℝ$$
4		ltmul1	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge \left(C\in ℝ\wedge 0<C\right)\right)\to \left(A<B\leftrightarrow AC<BC\right)$$
5	1 2 4	mp3an12	$$⊢\left(C\in ℝ\wedge 0<C\right)\to \left(A<B\leftrightarrow AC<BC\right)$$
6	3 5	mpan	$$⊢0<C\to \left(A<B\leftrightarrow AC<BC\right)$$