Theorem expnprm 13110

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 9983	. . . 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 9910	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> N e. ZZ )
5	4	ad2antlr 493	. . . . . . 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 531	. . . . . . . 8 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> A e. QQ )
8		prmnn 12866	. . . . . . . . . . . 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 9328	. . . . . . . . . 10 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) =/= 0 )
11		eluz2nn 9945	. . . . . . . . . . . 12 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN )
12	11	ad2antlr 493	. . . . . . . . . . 11 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> N e. NN )
13	12	0expd 11105	. . . . . . . . . 10 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( 0 ^ N ) = 0 )
14	10, 13	neeqtrrd 2450	. . . . . . . . 9 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) =/= ( 0 ^ N ) )
15		oveq1 6082	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ N ) = ( 0 ^ N ) )
16	15	necon3i 2468	. . . . . . . . 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 13063	. . . . . . . 8 |- ( ( ( A ^ N ) e. Prime /\ ( A e. QQ /\ A =/= 0 ) ) -> ( ( A ^ N ) pCnt A ) e. ZZ )
19	6, 7, 17, 18	syl12anc 1276	. . . . . . 7 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( ( A ^ N ) pCnt A ) e. ZZ )
20		dvdsmul1 12558	. . . . . . 7 |- ( ( N e. ZZ /\ ( ( A ^ N ) pCnt A ) e. ZZ ) -> N || ( N x. ( ( A ^ N ) pCnt A ) ) )
21	5, 19, 20	syl2anc 415	. . . . . 6 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> N || ( N x. ( ( A ^ N ) pCnt A ) ) )
22	9	nncnd 9297	. . . . . . . . 9 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( A ^ N ) e. CC )
23	22	exp1d 11084	. . . . . . . 8 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( ( A ^ N ) ^ 1 ) = ( A ^ N ) )
24	23	oveq2d 6091	. . . . . . 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 9649	. . . . . . . 8 |- 1 e. ZZ
26		pcid 13081	. . . . . . . 8 |- ( ( ( A ^ N ) e. Prime /\ 1 e. ZZ ) -> ( ( A ^ N ) pCnt ( ( A ^ N ) ^ 1 ) ) = 1 )
27	6, 25, 26	sylancl 417	. . . . . . 7 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( ( A ^ N ) pCnt ( ( A ^ N ) ^ 1 ) ) = 1 )
28		pcexp 13066	. . . . . . . 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 1283	. . . . . . 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 2280	. . . . . 6 |- ( ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) /\ ( A ^ N ) e. Prime ) -> ( N x. ( ( A ^ N ) pCnt A ) ) = 1 )
31	21, 30	breqtrd 4151	. . . . 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 9599	. . . . 5 |- ( ( A e. QQ /\ N e. ( ZZ>= ` 2 ) ) -> N e. NN0 )
35		dvds1 12598	. . . . 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 2462	. 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 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   0cc0 8169   1c1 8170   x. cmul 8174   NNcn 9283   2c2 9334   NN0cn0 9542   ZZcz 9623   ZZ>=cuz 9900   QQcq 9998   ^cexp 10953   || cdvds 12532   Primecprime 12863   pCnt cpc 13041

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-2o 6678  df-er 6797  df-en 7013  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-dvds 12533  df-gcd 12709  df-prm 12864  df-pc 13042

This theorem is referenced by: (None)