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Theorem qexpz 12333
Description: If a power of a rational number is an integer, then the number is an integer. (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 0z 9253 . . . 4  |-  0  e.  ZZ
2 eleq1 2240 . . . 4  |-  ( A  =  0  ->  ( A  e.  ZZ  <->  0  e.  ZZ ) )
31, 2mpbiri 168 . . 3  |-  ( A  =  0  ->  A  e.  ZZ )
43adantl 277 . 2  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =  0 )  ->  A  e.  ZZ )
5 simpll2 1037 . . . . . . . 8  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  NN )
65nncnd 8922 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  CC )
76mul01d 8340 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( N  x.  0 )  =  0 )
8 simpr 110 . . . . . . . . 9  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  p  e.  Prime )
9 simpll3 1038 . . . . . . . . 9  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( A ^ N )  e.  ZZ )
10 simpll1 1036 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  A  e.  QQ )
11 qcn 9623 . . . . . . . . . . . 12  |-  ( A  e.  QQ  ->  A  e.  CC )
1210, 11syl 14 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  A  e.  CC )
13 simplr 528 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  A  =/=  0 )
14 zq 9615 . . . . . . . . . . . . . 14  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
151, 14ax-mp 5 . . . . . . . . . . . . 13  |-  0  e.  QQ
16 qapne 9628 . . . . . . . . . . . . 13  |-  ( ( A  e.  QQ  /\  0  e.  QQ )  ->  ( A #  0  <->  A  =/=  0 ) )
1710, 15, 16sylancl 413 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( A #  0  <->  A  =/=  0
) )
1813, 17mpbird 167 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  A #  0 )
195nnzd 9363 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  ZZ )
2012, 18, 19expap0d 10645 . . . . . . . . . 10  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( A ^ N ) #  0 )
21 0zd 9254 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  e.  ZZ )
22 zapne 9316 . . . . . . . . . . 11  |-  ( ( ( A ^ N
)  e.  ZZ  /\  0  e.  ZZ )  ->  ( ( A ^ N ) #  0  <->  ( A ^ N )  =/=  0
) )
239, 21, 22syl2anc 411 . . . . . . . . . 10  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
( A ^ N
) #  0  <->  ( A ^ N )  =/=  0
) )
2420, 23mpbid 147 . . . . . . . . 9  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( A ^ N )  =/=  0 )
25 pczcl 12281 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  (
( A ^ N
)  e.  ZZ  /\  ( A ^ N )  =/=  0 ) )  ->  ( p  pCnt  ( A ^ N ) )  e.  NN0 )
268, 9, 24, 25syl12anc 1236 . . . . . . . 8  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  ( A ^ N ) )  e. 
NN0 )
2726nn0ge0d 9221 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <_  ( p  pCnt  ( A ^ N ) ) )
28 pcexp 12292 . . . . . . . 8  |-  ( ( p  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 )  /\  N  e.  ZZ )  ->  (
p  pCnt  ( A ^ N ) )  =  ( N  x.  (
p  pCnt  A )
) )
298, 10, 13, 19, 28syl121anc 1243 . . . . . . 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 )
) )
3027, 29breqtrd 4026 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <_  ( N  x.  (
p  pCnt  A )
) )
317, 30eqbrtrd 4022 . . . . 5  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( N  x.  0 )  <_  ( N  x.  ( p  pCnt  A ) ) )
32 0red 7949 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  e.  RR )
33 pcqcl 12289 . . . . . . . 8  |-  ( ( p  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( p  pCnt  A
)  e.  ZZ )
348, 10, 13, 33syl12anc 1236 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  ZZ )
3534zred 9364 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  RR )
365nnred 8921 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  RR )
375nngt0d 8952 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <  N )
38 lemul2 8803 . . . . . 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 ) ) ) )
3932, 35, 36, 37, 38syl112anc 1242 . . . . 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 ) ) ) )
4031, 39mpbird 167 . . . 4  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <_  ( p  pCnt  A
) )
4140ralrimiva 2550 . . 3  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  ->  A. p  e.  Prime  0  <_  ( p  pCnt  A ) )
42 simpl1 1000 . . . 4  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  ->  A  e.  QQ )
43 pcz 12314 . . . 4  |-  ( A  e.  QQ  ->  ( A  e.  ZZ  <->  A. p  e.  Prime  0  <_  (
p  pCnt  A )
) )
4442, 43syl 14 . . 3  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  -> 
( A  e.  ZZ  <->  A. p  e.  Prime  0  <_  ( p  pCnt  A
) ) )
4541, 44mpbird 167 . 2  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  ->  A  e.  ZZ )
46 simp1 997 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  ->  A  e.  QQ )
47 qdceq 10233 . . . 4  |-  ( ( A  e.  QQ  /\  0  e.  QQ )  -> DECID  A  =  0 )
4846, 15, 47sylancl 413 . . 3  |-  ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  -> DECID  A  =  0
)
49 dcne 2358 . . 3  |-  (DECID  A  =  0  <->  ( A  =  0  \/  A  =/=  0 ) )
5048, 49sylib 122 . 2  |-  ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  ->  ( A  =  0  \/  A  =/=  0 ) )
514, 45, 50mpjaodan 798 1  |-  ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  ->  A  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708  DECID wdc 834    /\ w3a 978    = wceq 1353    e. wcel 2148    =/= wne 2347   A.wral 2455   class class class wbr 4000  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802    x. cmul 7807    < clt 7982    <_ cle 7983   # cap 8528   NNcn 8908   NN0cn0 9165   ZZcz 9242   QQcq 9608   ^cexp 10505   Primecprime 12090    pCnt cpc 12267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-1o 6411  df-2o 6412  df-er 6529  df-en 6735  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-xnn0 9229  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927  df-prm 12091  df-pc 12268
This theorem is referenced by: (None)
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