Theorem exple1

Description: A real between 0 and 1 inclusive raised to a nonnegative integer power is less than or equal to 1. (Contributed by Paul Chapman, 29-Dec-2007) (Revised by Mario Carneiro, 5-Jun-2014)

	Ref	Expression
Assertion	exple1	|- ( ( ( A e. RR /\ 0 <_ A /\ A <_ 1 ) /\ N e. NN0 ) -> ( A ^ N ) <_ 1 )

Proof

Step	Hyp	Ref	Expression
1		simpl1	|- ( ( ( A e. RR /\ 0 <_ A /\ A <_ 1 ) /\ N e. NN0 ) -> A e. RR )
2		0nn0	|- 0 e. NN0
3	2	a1i	|- ( ( ( A e. RR /\ 0 <_ A /\ A <_ 1 ) /\ N e. NN0 ) -> 0 e. NN0 )
4		simpr	|- ( ( ( A e. RR /\ 0 <_ A /\ A <_ 1 ) /\ N e. NN0 ) -> N e. NN0 )
5		nn0uz	|- NN0 = ( ZZ>= ` 0 )
6	4 5	eleqtrdi	|- ( ( ( A e. RR /\ 0 <_ A /\ A <_ 1 ) /\ N e. NN0 ) -> N e. ( ZZ>= ` 0 ) )
7		simpl2	|- ( ( ( A e. RR /\ 0 <_ A /\ A <_ 1 ) /\ N e. NN0 ) -> 0 <_ A )
8		simpl3	|- ( ( ( A e. RR /\ 0 <_ A /\ A <_ 1 ) /\ N e. NN0 ) -> A <_ 1 )
9		leexp2r	|- ( ( ( A e. RR /\ 0 e. NN0 /\ N e. ( ZZ>= ` 0 ) ) /\ ( 0 <_ A /\ A <_ 1 ) ) -> ( A ^ N ) <_ ( A ^ 0 ) )
10	1 3 6 7 8 9	syl32anc	|- ( ( ( A e. RR /\ 0 <_ A /\ A <_ 1 ) /\ N e. NN0 ) -> ( A ^ N ) <_ ( A ^ 0 ) )
11	1	recnd	|- ( ( ( A e. RR /\ 0 <_ A /\ A <_ 1 ) /\ N e. NN0 ) -> A e. CC )
12		exp0	|- ( A e. CC -> ( A ^ 0 ) = 1 )
13	11 12	syl	|- ( ( ( A e. RR /\ 0 <_ A /\ A <_ 1 ) /\ N e. NN0 ) -> ( A ^ 0 ) = 1 )
14	10 13	breqtrd	|- ( ( ( A e. RR /\ 0 <_ A /\ A <_ 1 ) /\ N e. NN0 ) -> ( A ^ N ) <_ 1 )