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	|- ( ph -> A e. RR )
	divgt0d.2	|- ( ph -> B e. RR )
	mulgt1d.3	|- ( ph -> 1 < A )
	mulgt1d.4	|- ( ph -> 1 < B )
Assertion	mulgt1d	|- ( ph -> 1 < ( A x. B ) )

Proof

Step	Hyp	Ref	Expression
1		ltp1d.1	|- ( ph -> A e. RR )
2		divgt0d.2	|- ( ph -> B e. RR )
3		mulgt1d.3	|- ( ph -> 1 < A )
4		mulgt1d.4	|- ( ph -> 1 < B )
5		mulgt1	|- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < A /\ 1 < B ) ) -> 1 < ( A x. B ) )
6	1 2 3 4 5	syl22anc	|- ( ph -> 1 < ( A x. B ) )