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Theorem expnprm 14075
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 11032 . . . 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 10974 . . . . . . . 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 13877 . . . . . . . . . . . 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 10470 . . . . . . . . . 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 12134 . . . . . . . . . 10  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
0 ^ N )  =  0 )
1410, 13neeqtrrd 2748 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  =/=  ( 0 ^ N
) )
15 oveq1 6200 . . . . . . . . . 10  |-  ( A  =  0  ->  ( A ^ N )  =  ( 0 ^ N
) )
1615necon3i 2688 . . . . . . . . 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 14034 . . . . . . . 8  |-  ( ( ( A ^ N
)  e.  Prime  /\  ( A  e.  QQ  /\  A  =/=  0 ) )  -> 
( ( A ^ N )  pCnt  A
)  e.  ZZ )
196, 7, 17, 18syl12anc 1217 . . . . . . 7  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
( A ^ N
)  pCnt  A )  e.  ZZ )
20 dvdsmul1 13665 . . . . . . 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 10442 . . . . . . . . 9  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( A ^ N )  e.  CC )
2322exp1d 12113 . . . . . . . 8  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  (
( A ^ N
) ^ 1 )  =  ( A ^ N ) )
2423oveq2d 6209 . . . . . . 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 10780 . . . . . . . 8  |-  1  e.  ZZ
26 pcid 14050 . . . . . . . 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 14037 . . . . . . . 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 1224 . . . . . . 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 2501 . . . . . 6  |-  ( ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( A ^ N )  e.  Prime )  ->  ( N  x.  ( ( A ^ N )  pCnt  A ) )  =  1 )
3121, 30breqtrd 4417 . . . . 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 10740 . . . . 5  |-  ( ( A  e.  QQ  /\  N  e.  ( ZZ>= ` 
2 ) )  ->  N  e.  NN0 )
35 dvds1 13692 . . . . 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 2658 . 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 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   0cc0 9386   1c1 9387    x. cmul 9391   NNcn 10426   2c2 10475   NN0cn0 10683   ZZcz 10750   ZZ>=cuz 10965   QQcq 11057   ^cexp 11975    || cdivides 13646   Primecprime 13874    pCnt cpc 14014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-q 11058  df-rp 11096  df-fl 11752  df-mod 11819  df-seq 11917  df-exp 11976  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-dvds 13647  df-gcd 13802  df-prm 13875  df-pc 14015
This theorem is referenced by:  rplogsumlem2  22860
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