Theorem expnprm 12260

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 9533	. . . 4 |- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) )
2	1	simprbi 273	. . 3 |- ( N e. ( ZZ>= ` 2 ) -> N =/= 1 )
3	2	adantl 275	. 2 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> N =/= 1 )
4		eluzelz 9466	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> N e. ZZ )
5	4	ad2antlr 481	. . . . . . 7 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> N e. ZZ )
6		simpr 109	. . . . . . . 8 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) e. Prime )
7		simpll 519	. . . . . . . 8 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> A e. QQ )
8		prmnn 12021	. . . . . . . . . . . 12 |- ( ( A ^ N ) e. Prime -> ( A ^ N ) e. NN )
9	8	adantl 275	. . . . . . . . . . 11 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) e. NN )
10	9	nnne0d 8893	. . . . . . . . . 10 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) =/= 0 )
11		eluz2nn 9495	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN )
12	11	ad2antlr 481	. . . . . . . . . . 11 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> N e. NN )
13	12	0expd 10593	. . . . . . . . . 10 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( 0 ^ N ) = 0 )
14	10, 13	neeqtrrd 2364	. . . . . . . . 9 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) =/= ( 0 ^ N ) )
15		oveq1 5843	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ N ) = ( 0 ^ N ) )
16	15	necon3i 2382	. . . . . . . . 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 12215	. . . . . . . 8 |- ( ( ( A ^ N ) e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( ( A ^ N ) pCnt A ) e. ZZ )
19	6, 7, 17, 18	syl12anc 1225	. . . . . . 7 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( ( A ^ N ) pCnt A ) e. ZZ )
20		dvdsmul1 11739	. . . . . . 7 |- ( ( N e. ZZ /\ ( ( A ^ N ) pCnt A ) e. ZZ ) -> N || ( N x. ( ( A ^ N ) pCnt A ) ) )
21	5, 19, 20	syl2anc 409	. . . . . 6 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> N || ( N x. ( ( A ^ N ) pCnt A ) ) )
22	9	nncnd 8862	. . . . . . . . 9 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) e. CC )
23	22	exp1d 10572	. . . . . . . 8 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( ( A ^ N ) ^ 1 ) = ( A ^ N ) )
24	23	oveq2d 5852	. . . . . . 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 9208	. . . . . . . 8 |- 1 e. ZZ
26		pcid 12232	. . . . . . . 8 |- ( ( ( A ^ N ) e. Prime /\ 1 e. ZZ ) -> ( ( A ^ N ) pCnt ( ( A ^ N ) ^ 1 ) ) = 1 )
27	6, 25, 26	sylancl 410	. . . . . . 7 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( ( A ^ N ) pCnt ( ( A ^ N ) ^ 1 ) ) = 1 )
28		pcexp 12218	. . . . . . . 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 1232	. . . . . . 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 2206	. . . . . 6 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( N x. ( ( A ^ N ) pCnt A ) ) = 1 )
31	21, 30	breqtrd 4002	. . . . 5 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> N || 1 )
32	31	ex 114	. . . 4 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( A ^ N ) e. Prime -> N || 1 ) )
33	11	adantl 275	. . . . . 6 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> N e. NN )
34	33	nnnn0d 9158	. . . . 5 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> N e. NN0 )
35		dvds1 11776	. . . . 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 148	. . 3 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( A ^ N ) e. Prime -> N = 1 ) )
38	37	necon3ad 2376	. 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 103   <-> wb 104   = wceq 1342   e. wcel 2135   =/= wne 2334   class class class wbr 3976   ` cfv 5182  (class class class)co 5836   0cc0 7744   1c1 7745   x. cmul 7749   NNcn 8848   2c2 8899   NN0cn0 9105   ZZcz 9182   ZZ>=cuz 9457   QQcq 9548   ^cexp 10444   || cdvds 11713   Primecprime 12018   pCnt cpc 12193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862  ax-arch 7863  ax-caucvg 7864

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-isom 5191  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-1o 6375  df-2o 6376  df-er 6492  df-en 6698  df-sup 6940  df-inf 6941  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-3 8908  df-4 8909  df-n0 9106  df-z 9183  df-uz 9458  df-q 9549  df-rp 9581  df-fz 9936  df-fzo 10068  df-fl 10195  df-mod 10248  df-seqfrec 10371  df-exp 10445  df-cj 10770  df-re 10771  df-im 10772  df-rsqrt 10926  df-abs 10927  df-dvds 11714  df-gcd 11861  df-prm 12019  df-pc 12194

This theorem is referenced by: (None)