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Theorem expnprm 13264
Description: A second or higher power of a rational number is not a prime number. Or by contraposition, the n-th root of a prime number is irrational. Suggested by Norm Megill. (Contributed by Mario Carneiro, 10-Aug-2015.)
Assertion
Ref Expression
expnprm  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  -.  ( A ^ N
)  e.  Prime )

Proof of Theorem expnprm
StepHypRef Expression
1 eluz2b3 10542 . . . 4  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  N  =/=  1 ) )
21simprbi 451 . . 3  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  =/=  1 )
32adantl 453 . 2  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  =/=  1 )
4 eluzelz 10489 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  ZZ )
54ad2antlr 708 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  N  e.  ZZ )
6 simpr 448 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  e. 
Prime )
7 simpll 731 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  A  e.  QQ )
8 prmnn 13075 . . . . . . . . . . . 12  |-  ( ( A ^ N )  e.  Prime  ->  ( A ^ N )  e.  NN )
98adantl 453 . . . . . . . . . . 11  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  e.  NN )
109nnne0d 10037 . . . . . . . . . 10  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  =/=  0 )
111simplbi 447 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN )
1211ad2antlr 708 . . . . . . . . . . 11  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  N  e.  NN )
13120expd 11532 . . . . . . . . . 10  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
0 ^ N )  =  0 )
1410, 13neeqtrrd 2623 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  =/=  ( 0 ^ N
) )
15 oveq1 6081 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ N )  =  ( 0 ^ N
) )
1615necon3i 2638 . . . . . . . . 9  |-  ( ( A ^ N )  =/=  ( 0 ^ N )  ->  A  =/=  0 )
1714, 16syl 16 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  A  =/=  0 )
18 pcqcl 13223 . . . . . . . 8  |-  ( ( ( A ^ N
)  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( ( A ^ N )  pCnt  A
)  e.  ZZ )
196, 7, 17, 18syl12anc 1182 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
( A ^ N
)  pCnt  A )  e.  ZZ )
20 dvdsmul1 12864 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  ( ( A ^ N )  pCnt  A
)  e.  ZZ )  ->  N  ||  ( N  x.  ( ( A ^ N )  pCnt  A ) ) )
215, 19, 20syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  N  ||  ( N  x.  (
( A ^ N
)  pCnt  A )
) )
229nncnd 10009 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  e.  CC )
2322exp1d 11511 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
( A ^ N
) ^ 1 )  =  ( A ^ N ) )
2423oveq2d 6090 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
( A ^ N
)  pCnt  ( ( A ^ N ) ^
1 ) )  =  ( ( A ^ N )  pCnt  ( A ^ N ) ) )
25 1z 10304 . . . . . . . 8  |-  1  e.  ZZ
26 pcid 13239 . . . . . . . 8  |-  ( ( ( A ^ N
)  e.  Prime  /\  1  e.  ZZ )  ->  (
( A ^ N
)  pCnt  ( ( A ^ N ) ^
1 ) )  =  1 )
276, 25, 26sylancl 644 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
( A ^ N
)  pCnt  ( ( A ^ N ) ^
1 ) )  =  1 )
28 pcexp 13226 . . . . . . . 8  |-  ( ( ( A ^ N
)  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 )  /\  N  e.  ZZ )  ->  (
( A ^ N
)  pCnt  ( A ^ N ) )  =  ( N  x.  (
( A ^ N
)  pCnt  A )
) )
296, 7, 17, 5, 28syl121anc 1189 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
( A ^ N
)  pCnt  ( A ^ N ) )  =  ( N  x.  (
( A ^ N
)  pCnt  A )
) )
3024, 27, 293eqtr3rd 2477 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( N  x.  ( ( A ^ N )  pCnt  A ) )  =  1 )
3121, 30breqtrd 4229 . . . . 5  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  N  ||  1 )
3231ex 424 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( A ^ N )  e.  Prime  ->  N  ||  1 ) )
3311adantl 453 . . . . . 6  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  NN )
3433nnnn0d 10267 . . . . 5  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  NN0 )
35 dvds1 12891 . . . . 5  |-  ( N  e.  NN0  ->  ( N 
||  1  <->  N  = 
1 ) )
3634, 35syl 16 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( N  ||  1  <->  N  =  1 ) )
3732, 36sylibd 206 . . 3  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( A ^ N )  e.  Prime  ->  N  =  1 ) )
3837necon3ad 2635 . 2  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( N  =/=  1  ->  -.  ( A ^ N )  e.  Prime ) )
393, 38mpd 15 1  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  -.  ( A ^ N
)  e.  Prime )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4205   ` cfv 5447  (class class class)co 6074   0cc0 8983   1c1 8984    x. cmul 8988   NNcn 9993   2c2 10042   NN0cn0 10214   ZZcz 10275   ZZ>=cuz 10481   QQcq 10567   ^cexp 11375    || cdivides 12845   Primecprime 13072    pCnt cpc 13203
This theorem is referenced by:  rplogsumlem2  21172
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-2o 6718  df-oadd 6721  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-sup 7439  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-3 10052  df-n0 10215  df-z 10276  df-uz 10482  df-q 10568  df-rp 10606  df-fl 11195  df-mod 11244  df-seq 11317  df-exp 11376  df-cj 11897  df-re 11898  df-im 11899  df-sqr 12033  df-abs 12034  df-dvds 12846  df-gcd 13000  df-prm 13073  df-pc 13204
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