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Metamath Proof Explorer

Theorem expgt1

Description: A real greater than 1 raised to a positive integer is greater than 1. (Contributed by NM, 13-Feb-2005) (Revised by Mario Carneiro, 4-Jun-2014)

Ref Expression
Assertion expgt1 
|- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < ( A ^ N ) )

Proof

Step Hyp Ref Expression
1 1re 
 |-  1 e. RR
2 1 a1i 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 e. RR )
3 simp1 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A e. RR )
4 simp2 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> N e. NN )
5 4 nnnn0d 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> N e. NN0 )
6 reexpcl 
 |-  ( ( A e. RR /\ N e. NN0 ) -> ( A ^ N ) e. RR )
7 3 5 6 syl2anc 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( A ^ N ) e. RR )
8 simp3 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < A )
9 nnm1nn0 
 |-  ( N e. NN -> ( N - 1 ) e. NN0 )
10 4 9 syl 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( N - 1 ) e. NN0 )
11 ltle 
 |-  ( ( 1 e. RR /\ A e. RR ) -> ( 1 < A -> 1 <_ A ) )
12 1 3 11 sylancr 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 < A -> 1 <_ A ) )
13 8 12 mpd 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 <_ A )
14 expge1 
 |-  ( ( A e. RR /\ ( N - 1 ) e. NN0 /\ 1 <_ A ) -> 1 <_ ( A ^ ( N - 1 ) ) )
15 3 10 13 14 syl3anc 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 <_ ( A ^ ( N - 1 ) ) )
16 reexpcl 
 |-  ( ( A e. RR /\ ( N - 1 ) e. NN0 ) -> ( A ^ ( N - 1 ) ) e. RR )
17 3 10 16 syl2anc 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( A ^ ( N - 1 ) ) e. RR )
18 0red 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 e. RR )
19 0lt1 
 |-  0 < 1
20 19 a1i 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 < 1 )
21 18 2 3 20 8 lttrd 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 < A )
22 lemul1 
 |-  ( ( 1 e. RR /\ ( A ^ ( N - 1 ) ) e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 <_ ( A ^ ( N - 1 ) ) <-> ( 1 x. A ) <_ ( ( A ^ ( N - 1 ) ) x. A ) ) )
23 2 17 3 21 22 syl112anc 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 <_ ( A ^ ( N - 1 ) ) <-> ( 1 x. A ) <_ ( ( A ^ ( N - 1 ) ) x. A ) ) )
24 15 23 mpbid 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 x. A ) <_ ( ( A ^ ( N - 1 ) ) x. A ) )
25 recn 
 |-  ( A e. RR -> A e. CC )
26 25 3ad2ant1 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A e. CC )
27 26 mullidd 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 x. A ) = A )
28 27 eqcomd 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A = ( 1 x. A ) )
29 expm1t 
 |-  ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( ( A ^ ( N - 1 ) ) x. A ) )
30 26 4 29 syl2anc 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( A ^ N ) = ( ( A ^ ( N - 1 ) ) x. A ) )
31 24 28 30 3brtr4d 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A <_ ( A ^ N ) )
32 2 3 7 8 31 ltletrd 
 |-  ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < ( A ^ N ) )