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

Theorem expgt1 11418
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 9090 . . 3  |-  1  e.  RR
21a1i 11 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  e.  RR )
3 simp1 957 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  e.  RR )
4 simp2 958 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  N  e.  NN )
54nnnn0d 10274 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  N  e.  NN0 )
6 reexpcl 11398 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  RR )
73, 5, 6syl2anc 643 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( A ^ N )  e.  RR )
8 simp3 959 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <  A )
9 nnm1nn0 10261 . . . . . 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 9163 . . . . . . 7  |-  ( ( 1  e.  RR  /\  A  e.  RR )  ->  ( 1  <  A  ->  1  <_  A )
)
121, 3, 11sylancr 645 . . . . . 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 11417 . . . . 5  |-  ( ( A  e.  RR  /\  ( N  -  1
)  e.  NN0  /\  1  <_  A )  -> 
1  <_  ( A ^ ( N  - 
1 ) ) )
153, 10, 13, 14syl3anc 1184 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <_  ( A ^ ( N  -  1 ) ) )
16 reexpcl 11398 . . . . . 6  |-  ( ( A  e.  RR  /\  ( N  -  1
)  e.  NN0 )  ->  ( A ^ ( N  -  1 ) )  e.  RR )
173, 10, 16syl2anc 643 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( A ^ ( N  - 
1 ) )  e.  RR )
18 0re 9091 . . . . . . 7  |-  0  e.  RR
1918a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  e.  RR )
20 0lt1 9550 . . . . . . 7  |-  0  <  1
2120a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  <  1 )
2219, 2, 3, 21, 8lttrd 9231 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  <  A )
23 lemul1 9862 . . . . 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 1188 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  (
1  <_  ( A ^ ( N  - 
1 ) )  <->  ( 1  x.  A )  <_ 
( ( A ^
( N  -  1 ) )  x.  A
) ) )
2515, 24mpbid 202 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  (
1  x.  A )  <_  ( ( A ^ ( N  - 
1 ) )  x.  A ) )
26 recn 9080 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
27263ad2ant1 978 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  e.  CC )
2827mulid2d 9106 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  (
1  x.  A )  =  A )
2928eqcomd 2441 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  =  ( 1  x.  A ) )
30 expm1t 11408 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  ( ( A ^ ( N  -  1 ) )  x.  A ) )
3127, 4, 30syl2anc 643 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( A ^ N )  =  ( ( A ^
( N  -  1 ) )  x.  A
) )
3225, 29, 313brtr4d 4242 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  <_  ( A ^ N
) )
332, 3, 7, 8, 32ltletrd 9230 1  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <  ( A ^ N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    < clt 9120    <_ cle 9121    - cmin 9291   NNcn 10000   NN0cn0 10221   ^cexp 11382
This theorem is referenced by:  ltexp2a  11431  perfectlem1  21013  perfectlem2  21014  dchrisum0flblem2  21203  stirlinglem10  27808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-seq 11324  df-exp 11383
  Copyright terms: Public domain W3C validator