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Theorem qexpz 13225
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 2464 . 2  |-  ( A  =  0  ->  ( A  e.  ZZ  <->  0  e.  ZZ ) )
2 simpll2 997 . . . . . . . 8  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  NN )
32nncnd 9972 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  CC )
43mul01d 9221 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( N  x.  0 )  =  0 )
5 simpr 448 . . . . . . . . 9  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  p  e.  Prime )
6 simpll3 998 . . . . . . . . 9  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( A ^ N )  e.  ZZ )
7 simpll1 996 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  A  e.  QQ )
8 qcn 10544 . . . . . . . . . . 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 732 . . . . . . . . . 10  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  A  =/=  0 )
112nnzd 10330 . . . . . . . . . 10  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  ZZ )
129, 10, 11expne0d 11484 . . . . . . . . 9  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( A ^ N )  =/=  0 )
13 pczcl 13177 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  (
( A ^ N
)  e.  ZZ  /\  ( A ^ N )  =/=  0 ) )  ->  ( p  pCnt  ( A ^ N ) )  e.  NN0 )
145, 6, 12, 13syl12anc 1182 . . . . . . . 8  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  ( A ^ N ) )  e. 
NN0 )
1514nn0ge0d 10233 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <_  ( p  pCnt  ( A ^ N ) ) )
16 pcexp 13188 . . . . . . . 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 1189 . . . . . . 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 4196 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <_  ( N  x.  (
p  pCnt  A )
) )
194, 18eqbrtrd 4192 . . . . 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 0re 9047 . . . . . . 7  |-  0  e.  RR
2120a1i 11 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  e.  RR )
22 pcqcl 13185 . . . . . . . 8  |-  ( ( p  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( p  pCnt  A
)  e.  ZZ )
235, 7, 10, 22syl12anc 1182 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  ZZ )
2423zred 10331 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  RR )
252nnred 9971 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  RR )
262nngt0d 9999 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <  N )
27 lemul2 9819 . . . . . 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 ) ) ) )
2821, 24, 25, 26, 27syl112anc 1188 . . . . 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 ) ) ) )
2919, 28mpbird 224 . . . 4  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <_  ( p  pCnt  A
) )
3029ralrimiva 2749 . . 3  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  ->  A. p  e.  Prime  0  <_  ( p  pCnt  A ) )
31 simpl1 960 . . . 4  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  ->  A  e.  QQ )
32 pcz 13209 . . . 4  |-  ( A  e.  QQ  ->  ( A  e.  ZZ  <->  A. p  e.  Prime  0  <_  (
p  pCnt  A )
) )
3331, 32syl 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
) ) )
3430, 33mpbird 224 . 2  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  ->  A  e.  ZZ )
35 0z 10249 . . 3  |-  0  e.  ZZ
3635a1i 11 . 2  |-  ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  ->  0  e.  ZZ )
371, 34, 36pm2.61ne 2642 1  |-  ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  ->  A  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   class class class wbr 4172  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    x. cmul 8951    < clt 9076    <_ cle 9077   NNcn 9956   NN0cn0 10177   ZZcz 10238   QQcq 10530   ^cexp 11337   Primecprime 13034    pCnt cpc 13165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166
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