Theorem expnprm 12491

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 not rational. 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

Step	Hyp	Ref	Expression
1		eluz2b3 9669	. . . 4 |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) )
2	1	simprbi 275	. . 3 |- ( N e. ( ZZ>= ` 2 ) -> N =/= 1 )
3	2	adantl 277	. 2 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> N =/= 1 )
4		eluzelz 9601	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> N e. ZZ )
5	4	ad2antlr 489	. . . . . . 7 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> N e. ZZ )
6		simpr 110	. . . . . . . 8 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) e. Prime )
7		simpll 527	. . . . . . . 8 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> A e. QQ )
8		prmnn 12248	. . . . . . . . . . . 12 |- ( ( A ^ N ) e. Prime -> ( A ^ N ) e. NN )
9	8	adantl 277	. . . . . . . . . . 11 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) e. NN )
10	9	nnne0d 9027	. . . . . . . . . 10 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) =/= 0 )
11		eluz2nn 9631	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN )
12	11	ad2antlr 489	. . . . . . . . . . 11 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> N e. NN )
13	12	0expd 10760	. . . . . . . . . 10 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( 0 ^ N ) = 0 )
14	10, 13	neeqtrrd 2394	. . . . . . . . 9 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) =/= ( 0 ^ N ) )
15		oveq1 5925	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ N ) = ( 0 ^ N ) )
16	15	necon3i 2412	. . . . . . . . 9 |- ( ( A ^ N ) =/= ( 0 ^ N ) -> A =/= 0 )
17	14, 16	syl 14	. . . . . . . 8 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> A =/= 0 )
18		pcqcl 12444	. . . . . . . 8 |- ( ( ( A ^ N ) e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( ( A ^ N ) pCnt A ) e. ZZ )
19	6, 7, 17, 18	syl12anc 1247	. . . . . . 7 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( ( A ^ N ) pCnt A ) e. ZZ )
20		dvdsmul1 11956	. . . . . . 7 |- ( ( N e. ZZ /\ ( ( A ^ N ) pCnt A ) e. ZZ ) -> N || ( N x. ( ( A ^ N ) pCnt A ) ) )
21	5, 19, 20	syl2anc 411	. . . . . 6 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> N || ( N x. ( ( A ^ N ) pCnt A ) ) )
22	9	nncnd 8996	. . . . . . . . 9 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) e. CC )
23	22	exp1d 10739	. . . . . . . 8 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( ( A ^ N ) ^ 1 ) = ( A ^ N ) )
24	23	oveq2d 5934	. . . . . . 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 9343	. . . . . . . 8 |- 1 e. ZZ
26		pcid 12462	. . . . . . . 8 |- ( ( ( A ^ N ) e. Prime /\ 1 e. ZZ ) -> ( ( A ^ N ) pCnt ( ( A ^ N ) ^ 1 ) ) = 1 )
27	6, 25, 26	sylancl 413	. . . . . . 7 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( ( A ^ N ) pCnt ( ( A ^ N ) ^ 1 ) ) = 1 )
28		pcexp 12447	. . . . . . . 8 |- ( ( ( A ^ N ) e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ N e. ZZ ) -> ( ( A ^ N ) pCnt ( A ^ N ) ) = ( N x. ( ( A ^ N ) pCnt A ) ) )
29	6, 7, 17, 5, 28	syl121anc 1254	. . . . . . 7 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( ( A ^ N ) pCnt ( A ^ N ) ) = ( N x. ( ( A ^ N ) pCnt A ) ) )
30	24, 27, 29	3eqtr3rd 2235	. . . . . 6 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( N x. ( ( A ^ N ) pCnt A ) ) = 1 )
31	21, 30	breqtrd 4055	. . . . 5 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> N || 1 )
32	31	ex 115	. . . 4 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( A ^ N ) e. Prime -> N || 1 ) )
33	11	adantl 277	. . . . . 6 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> N e. NN )
34	33	nnnn0d 9293	. . . . 5 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> N e. NN0 )
35		dvds1 11995	. . . . 5 |- ( N e. NN0 -> ( N || 1 <-> N = 1 ) )
36	34, 35	syl 14	. . . 4 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> ( N || 1 <-> N = 1 ) )
37	32, 36	sylibd 149	. . 3 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( A ^ N ) e. Prime -> N = 1 ) )
38	37	necon3ad 2406	. 2 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> ( N =/= 1 -> -. ( A ^ N ) e. Prime ) )
39	3, 38	mpd 13	1 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> -. ( A ^ N ) e. Prime )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   =/= wne 2364   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   0cc0 7872   1c1 7873   x. cmul 7877   NNcn 8982   2c2 9033   NN0cn0 9240   ZZcz 9317   ZZ>=cuz 9592   QQcq 9684   ^cexp 10609   || cdvds 11930   Primecprime 12245   pCnt cpc 12422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-er 6587  df-en 6795  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080  df-prm 12246  df-pc 12423

This theorem is referenced by: (None)