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Theorem qexpz 14432
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 2529 . 2  |-  ( A  =  0  ->  ( A  e.  ZZ  <->  0  e.  ZZ ) )
2 simpll2 1036 . . . . . . . 8  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  NN )
32nncnd 10572 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  CC )
43mul01d 9796 . . . . . 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 1037 . . . . . . . . 9  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( A ^ N )  e.  ZZ )
7 simpll1 1035 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  A  e.  QQ )
8 qcn 11221 . . . . . . . . . . 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 755 . . . . . . . . . 10  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  A  =/=  0 )
112nnzd 10989 . . . . . . . . . 10  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  ZZ )
129, 10, 11expne0d 12319 . . . . . . . . 9  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  ( A ^ N )  =/=  0 )
13 pczcl 14384 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  (
( A ^ N
)  e.  ZZ  /\  ( A ^ N )  =/=  0 ) )  ->  ( p  pCnt  ( A ^ N ) )  e.  NN0 )
145, 6, 12, 13syl12anc 1226 . . . . . . . 8  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  ( A ^ N ) )  e. 
NN0 )
1514nn0ge0d 10876 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <_  ( p  pCnt  ( A ^ N ) ) )
16 pcexp 14395 . . . . . . . 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 1233 . . . . . . 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 4480 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <_  ( N  x.  (
p  pCnt  A )
) )
194, 18eqbrtrd 4476 . . . . 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 9614 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  e.  RR )
21 pcqcl 14392 . . . . . . . 8  |-  ( ( p  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( p  pCnt  A
)  e.  ZZ )
225, 7, 10, 21syl12anc 1226 . . . . . . 7  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  ZZ )
2322zred 10990 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  RR )
242nnred 10571 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  N  e.  RR )
252nngt0d 10600 . . . . . 6  |-  ( ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  /\  p  e.  Prime )  ->  0  <  N )
26 lemul2 10416 . . . . . 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 1232 . . . . 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 2871 . . 3  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  ->  A. p  e.  Prime  0  <_  ( p  pCnt  A ) )
30 simpl1 999 . . . 4  |-  ( ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  /\  A  =/=  0 )  ->  A  e.  QQ )
31 pcz 14416 . . . 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 10897 . 2  |-  ( ( A  e.  QQ  /\  N  e.  NN  /\  ( A ^ N )  e.  ZZ )  ->  0  e.  ZZ )
351, 33, 34pm2.61ne 2772 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   class class class wbr 4456  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509    x. cmul 9514    < clt 9645    <_ cle 9646   NNcn 10556   NN0cn0 10816   ZZcz 10885   QQcq 11207   ^cexp 12169   Primecprime 14229    pCnt cpc 14372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-fz 11698  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-prm 14230  df-pc 14373
This theorem is referenced by: (None)
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