Theorem mulgt1d

Description: The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltp1d.1	$$⊢\phi \to A\in ℝ$$
	divgt0d.2	$$⊢\phi \to B\in ℝ$$
	mulgt1d.3	$$⊢\phi \to 1<A$$
	mulgt1d.4	$$⊢\phi \to 1<B$$
Assertion	mulgt1d	$$⊢\phi \to 1<AB$$

Proof

Step	Hyp	Ref	Expression
1		ltp1d.1	$$⊢\phi \to A\in ℝ$$
2		divgt0d.2	$$⊢\phi \to B\in ℝ$$
3		mulgt1d.3	$$⊢\phi \to 1<A$$
4		mulgt1d.4	$$⊢\phi \to 1<B$$
5		mulgt1	$$⊢\left(\left(A\in ℝ\wedge B\in ℝ\right)\wedge \left(1<A\wedge 1<B\right)\right)\to 1<AB$$
6	1 2 3 4 5	syl22anc	$$⊢\phi \to 1<AB$$