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Theorem expnprm 14276
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 11151 . . . 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 11087 . . . . . . . 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 14075 . . . . . . . . . . . 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 10576 . . . . . . . . . 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 12290 . . . . . . . . . 10  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
0 ^ N )  =  0 )
1410, 13neeqtrrd 2767 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  =/=  ( 0 ^ N
) )
15 oveq1 6289 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ N )  =  ( 0 ^ N
) )
1615necon3i 2707 . . . . . . . . 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 14235 . . . . . . . 8  |-  ( ( ( A ^ N
)  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( ( A ^ N )  pCnt  A
)  e.  ZZ )
196, 7, 17, 18syl12anc 1226 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
( A ^ N
)  pCnt  A )  e.  ZZ )
20 dvdsmul1 13862 . . . . . . 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 10548 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  e.  CC )
2322exp1d 12269 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
( A ^ N
) ^ 1 )  =  ( A ^ N ) )
2423oveq2d 6298 . . . . . . 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 10890 . . . . . . . 8  |-  1  e.  ZZ
26 pcid 14251 . . . . . . . 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 14238 . . . . . . . 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 1233 . . . . . . 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 2517 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( N  x.  ( ( A ^ N )  pCnt  A ) )  =  1 )
3121, 30breqtrd 4471 . . . . 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 10848 . . . . 5  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  NN0 )
35 dvds1 13889 . . . . 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 2677 . 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 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489    x. cmul 9493   NNcn 10532   2c2 10581   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   QQcq 11178   ^cexp 12130    || cdivides 13843   Primecprime 14072    pCnt cpc 14215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-dvds 13844  df-gcd 14000  df-prm 14073  df-pc 14216
This theorem is referenced by:  rplogsumlem2  23398
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