Metamath Proof Explorer


Theorem exple1

Description: A real between 0 and 1 inclusive raised to a nonnegative integer 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 )