MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  expgt1 Unicode version

Theorem expgt1 11187
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 8882 . . 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 10065 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  N  e.  NN0 )
6 reexpcl 11167 . . 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 10052 . . . . . 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 8955 . . . . . . 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 11186 . . . . 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 11167 . . . . . 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 8883 . . . . . . 7  |-  0  e.  RR
1918a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  e.  RR )
20 0lt1 9341 . . . . . . 7  |-  0  <  1
2120a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  <  1 )
2219, 2, 3, 21, 8lttrd 9022 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  <  A )
23 lemul1 9653 . . . . 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 8872 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
27263ad2ant1 976 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  e.  CC )
2827mulid2d 8898 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  (
1  x.  A )  =  A )
2928eqcomd 2321 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  =  ( 1  x.  A ) )
30 expm1t 11177 . . . 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 4090 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  <_  ( A ^ N
) )
332, 3, 7, 8, 32ltletrd 9021 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 1633    e. wcel 1701   class class class wbr 4060  (class class class)co 5900   CCcc 8780   RRcr 8781   0cc0 8782   1c1 8783    x. cmul 8787    < clt 8912    <_ cle 8913    - cmin 9082   NNcn 9791   NN0cn0 10012   ^cexp 11151
This theorem is referenced by:  ltexp2a  11200  perfectlem1  20521  perfectlem2  20522  dchrisum0flblem2  20711  stirlinglem10  26980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-n0 10013  df-z 10072  df-uz 10278  df-seq 11094  df-exp 11152
  Copyright terms: Public domain W3C validator