Theorem expnprm 12916

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 9828	. . . 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 9755	. . . . . . . 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 12672	. . . . . . . . . . . 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 9178	. . . . . . . . . 10 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) =/= 0 )
11		eluz2nn 9790	. . . . . . . . . . . 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 10941	. . . . . . . . . 10 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( 0 ^ N ) = 0 )
14	10, 13	neeqtrrd 2430	. . . . . . . . 9 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) =/= ( 0 ^ N ) )
15		oveq1 6020	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ N ) = ( 0 ^ N ) )
16	15	necon3i 2448	. . . . . . . . 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 12869	. . . . . . . 8 |- ( ( ( A ^ N ) e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( ( A ^ N ) pCnt A ) e. ZZ )
19	6, 7, 17, 18	syl12anc 1269	. . . . . . 7 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( ( A ^ N ) pCnt A ) e. ZZ )
20		dvdsmul1 12364	. . . . . . 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 9147	. . . . . . . . 9 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) e. CC )
23	22	exp1d 10920	. . . . . . . 8 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( ( A ^ N ) ^ 1 ) = ( A ^ N ) )
24	23	oveq2d 6029	. . . . . . 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 9495	. . . . . . . 8 |- 1 e. ZZ
26		pcid 12887	. . . . . . . 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 12872	. . . . . . . 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 1276	. . . . . . 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 2271	. . . . . 6 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( N x. ( ( A ^ N ) pCnt A ) ) = 1 )
31	21, 30	breqtrd 4112	. . . . 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 9445	. . . . 5 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> N e. NN0 )
35		dvds1 12404	. . . . 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 2442	. 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 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   0cc0 8022   1c1 8023   x. cmul 8027   NNcn 9133   2c2 9184   NN0cn0 9392   ZZcz 9469   ZZ>=cuz 9745   QQcq 9843   ^cexp 10790   || cdvds 12338   Primecprime 12669   pCnt cpc 12847

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-2o 6578  df-er 6697  df-en 6905  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-dvds 12339  df-gcd 12515  df-prm 12670  df-pc 12848

This theorem is referenced by: (None)