Theorem leexp2ad

Description: Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	sqgt0d.1	|- ( ph -> A e. RR )
	leexp2ad.2	|- ( ph -> 1 <_ A )
	leexp2ad.3	|- ( ph -> N e. ( ZZ>= ` M ) )
Assertion	leexp2ad	|- ( ph -> ( A ^ M ) <_ ( A ^ N ) )

Proof

Step	Hyp	Ref	Expression
1		sqgt0d.1	|- ( ph -> A e. RR )
2		leexp2ad.2	|- ( ph -> 1 <_ A )
3		leexp2ad.3	|- ( ph -> N e. ( ZZ>= ` M ) )
4		leexp2a	|- ( ( A e. RR /\ 1 <_ A /\ N e. ( ZZ>= ` M ) ) -> ( A ^ M ) <_ ( A ^ N ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( A ^ M ) <_ ( A ^ N ) )