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Theorem expgt1 11885
Description: Positive integer exponentiation with a mantissa greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
expgt1  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <  ( A ^ N
) )

Proof of Theorem expgt1
StepHypRef Expression
1 1re 9372 . . 3  |-  1  e.  RR
21a1i 11 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  e.  RR )
3 simp1 981 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  e.  RR )
4 simp2 982 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  N  e.  NN )
54nnnn0d 10623 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  N  e.  NN0 )
6 reexpcl 11865 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  RR )
73, 5, 6syl2anc 654 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( A ^ N )  e.  RR )
8 simp3 983 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <  A )
9 nnm1nn0 10608 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
104, 9syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( N  -  1 )  e.  NN0 )
11 ltle 9450 . . . . . . 7  |-  ( ( 1  e.  RR  /\  A  e.  RR )  ->  ( 1  <  A  ->  1  <_  A )
)
121, 3, 11sylancr 656 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  (
1  <  A  ->  1  <_  A ) )
138, 12mpd 15 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <_  A )
14 expge1 11884 . . . . 5  |-  ( ( A  e.  RR  /\  ( N  -  1
)  e.  NN0  /\  1  <_  A )  -> 
1  <_  ( A ^ ( N  - 
1 ) ) )
153, 10, 13, 14syl3anc 1211 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <_  ( A ^ ( N  -  1 ) ) )
16 reexpcl 11865 . . . . . 6  |-  ( ( A  e.  RR  /\  ( N  -  1
)  e.  NN0 )  ->  ( A ^ ( N  -  1 ) )  e.  RR )
173, 10, 16syl2anc 654 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( A ^ ( N  - 
1 ) )  e.  RR )
18 0red 9374 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  e.  RR )
19 0lt1 9849 . . . . . . 7  |-  0  <  1
2019a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  <  1 )
2118, 2, 3, 20, 8lttrd 9519 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  <  A )
22 lemul1 10168 . . . . 5  |-  ( ( 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
) ) )
232, 17, 3, 21, 22syl112anc 1215 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  (
1  <_  ( A ^ ( N  - 
1 ) )  <->  ( 1  x.  A )  <_ 
( ( A ^
( N  -  1 ) )  x.  A
) ) )
2415, 23mpbid 210 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  (
1  x.  A )  <_  ( ( A ^ ( N  - 
1 ) )  x.  A ) )
25 recn 9359 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
26253ad2ant1 1002 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  e.  CC )
2726mulid2d 9391 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  (
1  x.  A )  =  A )
2827eqcomd 2438 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  =  ( 1  x.  A ) )
29 expm1t 11875 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  ( ( A ^ ( N  -  1 ) )  x.  A ) )
3026, 4, 29syl2anc 654 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( A ^ N )  =  ( ( A ^
( N  -  1 ) )  x.  A
) )
3124, 28, 303brtr4d 4310 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  <_  ( A ^ N
) )
322, 3, 7, 8, 31ltletrd 9518 1  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <  ( A ^ N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 958    = wceq 1362    e. wcel 1755   class class class wbr 4280  (class class class)co 6080   CCcc 9267   RRcr 9268   0cc0 9269   1c1 9270    x. cmul 9274    < clt 9405    <_ cle 9406    - cmin 9582   NNcn 10309   NN0cn0 10566   ^cexp 11848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-nn 10310  df-n0 10567  df-z 10634  df-uz 10849  df-seq 11790  df-exp 11849
This theorem is referenced by:  ltexp2a  11898  perfectlem1  22452  perfectlem2  22453  dchrisum0flblem2  22642  stirlinglem10  29721
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