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Theorem expnprm 13198
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 10481 . . . 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 10428 . . . . . . . 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 13009 . . . . . . . . . . . 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 9976 . . . . . . . . . 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 11466 . . . . . . . . . 10  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
0 ^ N )  =  0 )
1410, 13neeqtrrd 2574 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  =/=  ( 0 ^ N
) )
15 oveq1 6027 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ N )  =  ( 0 ^ N
) )
1615necon3i 2589 . . . . . . . . 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 13157 . . . . . . . 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 12798 . . . . . . 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 9948 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  e.  CC )
2322exp1d 11445 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
( A ^ N
) ^ 1 )  =  ( A ^ N ) )
2423oveq2d 6036 . . . . . . 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 10243 . . . . . . . 8  |-  1  e.  ZZ
26 pcid 13173 . . . . . . . 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 13160 . . . . . . . 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 2428 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( N  x.  ( ( A ^ N )  pCnt  A ) )  =  1 )
3121, 30breqtrd 4177 . . . . 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 10206 . . . . 5  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  NN0 )
35 dvds1 12825 . . . . 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 2586 . 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 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   0cc0 8923   1c1 8924    x. cmul 8928   NNcn 9932   2c2 9981   NN0cn0 10153   ZZcz 10214   ZZ>=cuz 10420   QQcq 10506   ^cexp 11309    || cdivides 12779   Primecprime 13006    pCnt cpc 13137
This theorem is referenced by:  rplogsumlem2  21046
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-q 10507  df-rp 10545  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-dvds 12780  df-gcd 12934  df-prm 13007  df-pc 13138
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