Theorem ltmul1d

Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltmul1d.1	$$⊢\phi \to A\in ℝ$$
	ltmul1d.2	$$⊢\phi \to B\in ℝ$$
	ltmul1d.3	$$⊢\phi \to C\in {ℝ}^{+}$$
Assertion	ltmul1d	$$⊢\phi \to \left(A<B\leftrightarrow AC<BC\right)$$

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	3	rpregt0d	$$⊢\phi \to \left(C\in ℝ\wedge 0<C\right)$$
5		ltmul1	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge \left(C\in ℝ\wedge 0<C\right)\right)\to \left(A<B\leftrightarrow AC<BC\right)$$
6	1 2 4 5	syl3anc	$$⊢\phi \to \left(A<B\leftrightarrow AC<BC\right)$$