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Theorem expgt1 12159
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 9584 . . 3  |-  1  e.  RR
21a1i 11 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  e.  RR )
3 simp1 991 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  e.  RR )
4 simp2 992 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  N  e.  NN )
54nnnn0d 10841 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  N  e.  NN0 )
6 reexpcl 12139 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  RR )
73, 5, 6syl2anc 661 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( A ^ N )  e.  RR )
8 simp3 993 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <  A )
9 nnm1nn0 10826 . . . . . 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 9662 . . . . . . 7  |-  ( ( 1  e.  RR  /\  A  e.  RR )  ->  ( 1  <  A  ->  1  <_  A )
)
121, 3, 11sylancr 663 . . . . . 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 12158 . . . . 5  |-  ( ( A  e.  RR  /\  ( N  -  1
)  e.  NN0  /\  1  <_  A )  -> 
1  <_  ( A ^ ( N  - 
1 ) ) )
153, 10, 13, 14syl3anc 1223 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  1  <_  ( A ^ ( N  -  1 ) ) )
16 reexpcl 12139 . . . . . 6  |-  ( ( A  e.  RR  /\  ( N  -  1
)  e.  NN0 )  ->  ( A ^ ( N  -  1 ) )  e.  RR )
173, 10, 16syl2anc 661 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( A ^ ( N  - 
1 ) )  e.  RR )
18 0red 9586 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  e.  RR )
19 0lt1 10064 . . . . . . 7  |-  0  <  1
2019a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  <  1 )
2118, 2, 3, 20, 8lttrd 9731 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  0  <  A )
22 lemul1 10383 . . . . 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 1227 . . . 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 9571 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
26253ad2ant1 1012 . . . . 5  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  e.  CC )
2726mulid2d 9603 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  (
1  x.  A )  =  A )
2827eqcomd 2468 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  =  ( 1  x.  A ) )
29 expm1t 12149 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( A ^ N
)  =  ( ( A ^ ( N  -  1 ) )  x.  A ) )
3026, 4, 29syl2anc 661 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  ( A ^ N )  =  ( ( A ^
( N  -  1 ) )  x.  A
) )
3124, 28, 303brtr4d 4470 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN  /\  1  <  A )  ->  A  <_  ( A ^ N
) )
322, 3, 7, 8, 31ltletrd 9730 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 968    = wceq 1374    e. wcel 1762   class class class wbr 4440  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    x. cmul 9486    < clt 9617    <_ cle 9618    - cmin 9794   NNcn 10525   NN0cn0 10784   ^cexp 12122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-seq 12064  df-exp 12123
This theorem is referenced by:  ltexp2a  12172  perfectlem1  23225  perfectlem2  23226  dchrisum0flblem2  23415  stirlinglem10  31202
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