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Metamath Proof Explorer

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 )