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Theorem qexpz 14080
Description: If a power of a rational number is an integer, then the number is an integer. In other words, all n-th roots are irrational unless they are integers (so that the original number is an n-th power). (Contributed by Mario Carneiro, 10-Aug-2015.)
Assertion
Ref Expression
qexpz  |-  ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  ->  A  e.  ZZ )

Proof of Theorem qexpz
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 eleq1 2526 . 2  |-  ( A  =  0  ->  ( A  e.  ZZ  <->  0  e.  ZZ ) )
2 simpll2 1028 . . . . . . . 8  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  NN )
32nncnd 10448 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  CC )
43mul01d 9678 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( N  x.  0 )  =  0 )
5 simpr 461 . . . . . . . . 9  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  p  e.  Prime )
6 simpll3 1029 . . . . . . . . 9  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( A ^ N )  e.  ZZ )
7 simpll1 1027 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  A  e.  QQ )
8 qcn 11077 . . . . . . . . . . 11  |-  ( A  e.  QQ  ->  A  e.  CC )
97, 8syl 16 . . . . . . . . . 10  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  A  e.  CC )
10 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  A  =/=  0 )
112nnzd 10856 . . . . . . . . . 10  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  ZZ )
129, 10, 11expne0d 12130 . . . . . . . . 9  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( A ^ N )  =/=  0 )
13 pczcl 14032 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  (
( A ^ N
)  e.  ZZ  /\  ( A ^ N )  =/=  0 ) )  ->  ( p  pCnt  ( A ^ N ) )  e.  NN0 )
145, 6, 12, 13syl12anc 1217 . . . . . . . 8  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  ( A ^ N ) )  e. 
NN0 )
1514nn0ge0d 10749 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <_  ( p  pCnt  ( A ^ N ) ) )
16 pcexp 14043 . . . . . . . 8  |-  ( ( p  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 )  /\  N  e.  ZZ )  ->  (
p  pCnt  ( A ^ N ) )  =  ( N  x.  (
p  pCnt  A )
) )
175, 7, 10, 11, 16syl121anc 1224 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  ( A ^ N ) )  =  ( N  x.  (
p  pCnt  A )
) )
1815, 17breqtrd 4423 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <_  ( N  x.  (
p  pCnt  A )
) )
194, 18eqbrtrd 4419 . . . . 5  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( N  x.  0 )  <_  ( N  x.  ( p  pCnt  A ) ) )
20 0red 9497 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  e.  RR )
21 pcqcl 14040 . . . . . . . 8  |-  ( ( p  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( p  pCnt  A
)  e.  ZZ )
225, 7, 10, 21syl12anc 1217 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  ZZ )
2322zred 10857 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  RR )
242nnred 10447 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  RR )
252nngt0d 10475 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <  N )
26 lemul2 10292 . . . . . 6  |-  ( ( 0  e.  RR  /\  ( p  pCnt  A )  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( 0  <_  (
p  pCnt  A )  <->  ( N  x.  0 )  <_  ( N  x.  ( p  pCnt  A ) ) ) )
2720, 23, 24, 25, 26syl112anc 1223 . . . . 5  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
0  <_  ( p  pCnt  A )  <->  ( N  x.  0 )  <_  ( N  x.  ( p  pCnt  A ) ) ) )
2819, 27mpbird 232 . . . 4  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <_  ( p  pCnt  A
) )
2928ralrimiva 2829 . . 3  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  ->  A. p  e.  Prime  0  <_  ( p  pCnt  A ) )
30 simpl1 991 . . . 4  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  ->  A  e.  QQ )
31 pcz 14064 . . . 4  |-  ( A  e.  QQ  ->  ( A  e.  ZZ  <->  A. p  e.  Prime  0  <_  (
p  pCnt  A )
) )
3230, 31syl 16 . . 3  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  -> 
( A  e.  ZZ  <->  A. p  e.  Prime  0  <_  ( p  pCnt  A
) ) )
3329, 32mpbird 232 . 2  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  ->  A  e.  ZZ )
34 0zd 10768 . 2  |-  ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  ->  0  e.  ZZ )
351, 33, 34pm2.61ne 2766 1  |-  ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  ->  A  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   A.wral 2798   class class class wbr 4399  (class class class)co 6199   CCcc 9390   RRcr 9391   0cc0 9392    x. cmul 9397    < clt 9528    <_ cle 9529   NNcn 10432   NN0cn0 10689   ZZcz 10756   QQcq 11063   ^cexp 11981   Primecprime 13880    pCnt cpc 14020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-n0 10690  df-z 10757  df-uz 10972  df-q 11064  df-rp 11102  df-fz 11554  df-fl 11758  df-mod 11825  df-seq 11923  df-exp 11982  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-dvds 13653  df-gcd 13808  df-prm 13881  df-pc 14021
This theorem is referenced by: (None)
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