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Theorem expgt1 11140
Description: Natural number 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 8837 . . 3  |-  1  e.  RR
21a1i 10 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  e.  RR )
3 simp1 955 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  e.  RR )
4 simp2 956 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  N  e.  NN )
54nnnn0d 10018 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  N  e.  NN0 )
6 reexpcl 11120 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  RR )
73, 5, 6syl2anc 642 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( A ^ N )  e.  RR )
8 simp3 957 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <  A )
9 nnm1nn0 10005 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
104, 9syl 15 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( N  -  1 )  e.  NN0 )
11 ltle 8910 . . . . . . 7  |-  ( ( 1  e.  RR  /\  A  e.  RR )  ->  ( 1  <  A  ->  1  <_  A )
)
121, 3, 11sylancr 644 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  (
1  <  A  ->  1  <_  A ) )
138, 12mpd 14 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <_  A )
14 expge1 11139 . . . . 5  |-  ( ( A  e.  RR  /\  ( N  -  1
)  e.  NN0  /\  1  <_  A )  -> 
1  <_  ( A ^ ( N  - 
1 ) ) )
153, 10, 13, 14syl3anc 1182 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <_  ( A ^ ( N  -  1 ) ) )
16 reexpcl 11120 . . . . . 6  |-  ( ( A  e.  RR  /\  ( N  -  1
)  e.  NN0 )  ->  ( A ^ ( N  -  1 ) )  e.  RR )
173, 10, 16syl2anc 642 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( A ^ ( N  - 
1 ) )  e.  RR )
18 0re 8838 . . . . . . 7  |-  0  e.  RR
1918a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  e.  RR )
20 0lt1 9296 . . . . . . 7  |-  0  <  1
2120a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  <  1 )
2219, 2, 3, 21, 8lttrd 8977 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  <  A )
23 lemul1 9608 . . . . 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
) ) )
242, 17, 3, 22, 23syl112anc 1186 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  (
1  <_  ( A ^ ( N  - 
1 ) )  <->  ( 1  x.  A )  <_ 
( ( A ^
( N  -  1 ) )  x.  A
) ) )
2515, 24mpbid 201 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  (
1  x.  A )  <_  ( ( A ^ ( N  - 
1 ) )  x.  A ) )
26 recn 8827 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
27263ad2ant1 976 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  e.  CC )
2827mulid2d 8853 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  (
1  x.  A )  =  A )
2928eqcomd 2288 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  =  ( 1  x.  A ) )
30 expm1t 11130 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  ( ( A ^ ( N  -  1 ) )  x.  A ) )
3127, 4, 30syl2anc 642 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( A ^ N )  =  ( ( A ^
( N  -  1 ) )  x.  A
) )
3225, 29, 313brtr4d 4053 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  <_  ( A ^ N
) )
332, 3, 7, 8, 32ltletrd 8976 1  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <  ( A ^ N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   NN0cn0 9965   ^cexp 11104
This theorem is referenced by:  ltexp2a  11153  perfectlem1  20468  perfectlem2  20469  dchrisum0flblem2  20658  stirlinglem10  27832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105
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