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Theorem qexpz 13955
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 2498 . 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 10330 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  CC )
43mul01d 9560 . . . . . 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 10959 . . . . . . . . . . 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 10738 . . . . . . . . . 10  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  ZZ )
129, 10, 11expne0d 12006 . . . . . . . . 9  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( A ^ N )  =/=  0 )
13 pczcl 13907 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  (
( A ^ N
)  e.  ZZ  /\  ( A ^ N )  =/=  0 ) )  ->  ( p  pCnt  ( A ^ N ) )  e.  NN0 )
145, 6, 12, 13syl12anc 1216 . . . . . . . 8  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  ( A ^ N ) )  e. 
NN0 )
1514nn0ge0d 10631 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <_  ( p  pCnt  ( A ^ N ) ) )
16 pcexp 13918 . . . . . . . 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 1223 . . . . . . 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 4311 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <_  ( N  x.  (
p  pCnt  A )
) )
194, 18eqbrtrd 4307 . . . . 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 9379 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  e.  RR )
21 pcqcl 13915 . . . . . . . 8  |-  ( ( p  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( p  pCnt  A
)  e.  ZZ )
225, 7, 10, 21syl12anc 1216 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  ZZ )
2322zred 10739 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  RR )
242nnred 10329 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  RR )
252nngt0d 10357 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <  N )
26 lemul2 10174 . . . . . 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 1222 . . . . 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 2794 . . 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 13939 . . . 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 10650 . 2  |-  ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  ->  0  e.  ZZ )
351, 33, 34pm2.61ne 2681 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 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   class class class wbr 4287  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274    x. cmul 9279    < clt 9410    <_ cle 9411   NNcn 10314   NN0cn0 10571   ZZcz 10638   QQcq 10945   ^cexp 11857   Primecprime 13755    pCnt cpc 13895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-fz 11430  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896
This theorem is referenced by: (None)
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