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Theorem expnprm 13956
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 10920 . . . 4  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  N  =/=  1 ) )
21simprbi 464 . . 3  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  =/=  1 )
32adantl 466 . 2  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  =/=  1 )
4 eluzelz 10862 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  ZZ )
54ad2antlr 726 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  N  e.  ZZ )
6 simpr 461 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  e. 
Prime )
7 simpll 753 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  A  e.  QQ )
8 prmnn 13758 . . . . . . . . . . . 12  |-  ( ( A ^ N )  e.  Prime  ->  ( A ^ N )  e.  NN )
98adantl 466 . . . . . . . . . . 11  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  e.  NN )
109nnne0d 10358 . . . . . . . . . 10  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  =/=  0 )
111simplbi 460 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN )
1211ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  N  e.  NN )
13120expd 12016 . . . . . . . . . 10  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
0 ^ N )  =  0 )
1410, 13neeqtrrd 2627 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  =/=  ( 0 ^ N
) )
15 oveq1 6093 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ N )  =  ( 0 ^ N
) )
1615necon3i 2645 . . . . . . . . 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 13915 . . . . . . . 8  |-  ( ( ( A ^ N
)  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( ( A ^ N )  pCnt  A
)  e.  ZZ )
196, 7, 17, 18syl12anc 1216 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
( A ^ N
)  pCnt  A )  e.  ZZ )
20 dvdsmul1 13546 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  ( ( A ^ N )  pCnt  A
)  e.  ZZ )  ->  N  ||  ( N  x.  ( ( A ^ N )  pCnt  A ) ) )
215, 19, 20syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  N  ||  ( N  x.  (
( A ^ N
)  pCnt  A )
) )
229nncnd 10330 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  e.  CC )
2322exp1d 11995 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
( A ^ N
) ^ 1 )  =  ( A ^ N ) )
2423oveq2d 6102 . . . . . . 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 10668 . . . . . . . 8  |-  1  e.  ZZ
26 pcid 13931 . . . . . . . 8  |-  ( ( ( A ^ N
)  e.  Prime  /\  1  e.  ZZ )  ->  (
( A ^ N
)  pCnt  ( ( A ^ N ) ^
1 ) )  =  1 )
276, 25, 26sylancl 662 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
( A ^ N
)  pCnt  ( ( A ^ N ) ^
1 ) )  =  1 )
28 pcexp 13918 . . . . . . . 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 1223 . . . . . . 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 2479 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( N  x.  ( ( A ^ N )  pCnt  A ) )  =  1 )
3121, 30breqtrd 4311 . . . . 5  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  N  ||  1 )
3231ex 434 . . . 4  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( A ^ N )  e.  Prime  ->  N  ||  1 ) )
3311adantl 466 . . . . . 6  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  NN )
3433nnnn0d 10628 . . . . 5  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  NN0 )
35 dvds1 13573 . . . . 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 214 . . 3  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( ( A ^ N )  e.  Prime  ->  N  =  1 ) )
3837necon3ad 2639 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   0cc0 9274   1c1 9275    x. cmul 9279   NNcn 10314   2c2 10363   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   QQcq 10945   ^cexp 11857    || cdivides 13527   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-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:  rplogsumlem2  22709
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