Theorem root1eq1 20592

Description: The only powers of an N-th root of unity that equal 1 are the multiples of N. In other words, -u 1 ^ c ( 2 / N ) has order N in the multiplicative group of nonzero complex numbers. (In fact, these and their powers are the only elements of finite order in the complexes.) (Contributed by Mario Carneiro, 28-Apr-2016.)

Assertion

Ref	Expression
root1eq1	|- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( -u 1 ^ c ( 2 / N ) ) ^ K ) = 1 <-> N || K ) )

Proof of Theorem root1eq1

Step	Hyp	Ref	Expression
1		2re 10025	. . . . . . . 8 |- 2 e. RR
2		simpl 444	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> N e. NN )
3		nndivre 9991	. . . . . . . 8 |- ( ( 2 e. RR /\ N e. NN ) -> ( 2 / N ) e. RR )
4	1, 2, 3	sylancr 645	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> ( 2 / N ) e. RR )
5	4	recnd 9070	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( 2 / N ) e. CC )
6		ax-icn 9005	. . . . . . . 8 |- _i e. CC
7		pire 20325	. . . . . . . . 9 |- pi e. RR
8	7	recni 9058	. . . . . . . 8 |- pi e. CC
9	6, 8	mulcli 9051	. . . . . . 7 |- ( _i x. pi ) e. CC
10	9	a1i 11	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( _i x. pi ) e. CC )
11	5, 10	mulcld 9064	. . . . 5 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( 2 / N ) x. ( _i x. pi ) ) e. CC )
12		efexp 12657	. . . . 5 |- ( ( ( ( 2 / N ) x. ( _i x. pi ) ) e. CC /\ K e. ZZ ) -> ( exp ` ( K x. ( ( 2 / N ) x. ( _i x. pi ) ) ) ) = ( ( exp ` ( ( 2 / N ) x. ( _i x. pi ) ) ) ^ K ) )
13	11, 12	sylancom 649	. . . 4 |- ( ( N e. NN /\ K e. ZZ ) -> ( exp ` ( K x. ( ( 2 / N ) x. ( _i x. pi ) ) ) ) = ( ( exp ` ( ( 2 / N ) x. ( _i x. pi ) ) ) ^ K ) )
14		zcn 10243	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
15	14	adantl 453	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> K e. CC )
16		nncn 9964	. . . . . . . . 9 |- ( N e. NN -> N e. CC )
17	16	adantr 452	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> N e. CC )
18		2cn 10026	. . . . . . . . 9 |- 2 e. CC
19	18	a1i 11	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> 2 e. CC )
20		nnne0 9988	. . . . . . . . 9 |- ( N e. NN -> N =/= 0 )
21	20	adantr 452	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> N =/= 0 )
22	15, 17, 19, 21	div32d 9769	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( K / N ) x. 2 ) = ( K x. ( 2 / N ) ) )
23	22	oveq1d 6055	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( K / N ) x. 2 ) x. ( _i x. pi ) ) = ( ( K x. ( 2 / N ) ) x. ( _i x. pi ) ) )
24	15, 17, 21	divcld 9746	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> ( K / N ) e. CC )
25	24, 19, 10	mulassd 9067	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( K / N ) x. 2 ) x. ( _i x. pi ) ) = ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) )
26	15, 5, 10	mulassd 9067	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( K x. ( 2 / N ) ) x. ( _i x. pi ) ) = ( K x. ( ( 2 / N ) x. ( _i x. pi ) ) ) )
27	23, 25, 26	3eqtr3d 2444	. . . . 5 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) = ( K x. ( ( 2 / N ) x. ( _i x. pi ) ) ) )
28	27	fveq2d 5691	. . . 4 |- ( ( N e. NN /\ K e. ZZ ) -> ( exp ` ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) ) = ( exp ` ( K x. ( ( 2 / N ) x. ( _i x. pi ) ) ) ) )
29		neg1cn 10023	. . . . . . . 8 |- -u 1 e. CC
30	29	a1i 11	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> -u 1 e. CC )
31		ax-1cn 9004	. . . . . . . . 9 |- 1 e. CC
32		ax-1ne0 9015	. . . . . . . . 9 |- 1 =/= 0
33	31, 32	negne0i 9331	. . . . . . . 8 |- -u 1 =/= 0
34	33	a1i 11	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> -u 1 =/= 0 )
35	30, 34, 5	cxpefd 20556	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( -u 1 ^ c ( 2 / N ) ) = ( exp ` ( ( 2 / N ) x. ( log ` -u 1 ) ) ) )
36		logm1 20436	. . . . . . . 8 |- ( log ` -u 1 ) = ( _i x. pi )
37	36	oveq2i 6051	. . . . . . 7 |- ( ( 2 / N ) x. ( log ` -u 1 ) ) = ( ( 2 / N ) x. ( _i x. pi ) )
38	37	fveq2i 5690	. . . . . 6 |- ( exp ` ( ( 2 / N ) x. ( log ` -u 1 ) ) ) = ( exp ` ( ( 2 / N ) x. ( _i x. pi ) ) )
39	35, 38	syl6eq 2452	. . . . 5 |- ( ( N e. NN /\ K e. ZZ ) -> ( -u 1 ^ c ( 2 / N ) ) = ( exp ` ( ( 2 / N ) x. ( _i x. pi ) ) ) )
40	39	oveq1d 6055	. . . 4 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( -u 1 ^ c ( 2 / N ) ) ^ K ) = ( ( exp ` ( ( 2 / N ) x. ( _i x. pi ) ) ) ^ K ) )
41	13, 28, 40	3eqtr4rd 2447	. . 3 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( -u 1 ^ c ( 2 / N ) ) ^ K ) = ( exp ` ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) ) )
42	41	eqeq1d 2412	. 2 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( -u 1 ^ c ( 2 / N ) ) ^ K ) = 1 <-> ( exp ` ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) ) = 1 ) )
43	18, 9	mulcli 9051	. . . 4 |- ( 2 x. ( _i x. pi ) ) e. CC
44		mulcl 9030	. . . 4 |- ( ( ( K / N ) e. CC /\ ( 2 x. ( _i x. pi ) ) e. CC ) -> ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) e. CC )
45	24, 43, 44	sylancl 644	. . 3 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) e. CC )
46		efeq1 20384	. . 3 |- ( ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) e. CC -> ( ( exp ` ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) ) = 1 <-> ( ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) / ( _i x. ( 2 x. pi ) ) ) e. ZZ ) )
47	45, 46	syl 16	. 2 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( exp ` ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) ) = 1 <-> ( ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) / ( _i x. ( 2 x. pi ) ) ) e. ZZ ) )
48	6, 18, 8	mul12i 9217	. . . . . 6 |- ( _i x. ( 2 x. pi ) ) = ( 2 x. ( _i x. pi ) )
49	48	oveq2i 6051	. . . . 5 |- ( ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) / ( _i x. ( 2 x. pi ) ) ) = ( ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) / ( 2 x. ( _i x. pi ) ) )
50	43	a1i 11	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( 2 x. ( _i x. pi ) ) e. CC )
51		2ne0 10039	. . . . . . . 8 |- 2 =/= 0
52		ine0 9425	. . . . . . . . 9 |- _i =/= 0
53		pipos 20326	. . . . . . . . . 10 |- 0 < pi
54	7, 53	gt0ne0ii 9519	. . . . . . . . 9 |- pi =/= 0
55	6, 8, 52, 54	mulne0i 9621	. . . . . . . 8 |- ( _i x. pi ) =/= 0
56	18, 9, 51, 55	mulne0i 9621	. . . . . . 7 |- ( 2 x. ( _i x. pi ) ) =/= 0
57	56	a1i 11	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( 2 x. ( _i x. pi ) ) =/= 0 )
58	24, 50, 57	divcan4d 9752	. . . . 5 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) / ( 2 x. ( _i x. pi ) ) ) = ( K / N ) )
59	49, 58	syl5eq 2448	. . . 4 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) / ( _i x. ( 2 x. pi ) ) ) = ( K / N ) )
60	59	eleq1d 2470	. . 3 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) / ( _i x. ( 2 x. pi ) ) ) e. ZZ <-> ( K / N ) e. ZZ ) )
61		nnz 10259	. . . . 5 |- ( N e. NN -> N e. ZZ )
62	61	adantr 452	. . . 4 |- ( ( N e. NN /\ K e. ZZ ) -> N e. ZZ )
63		simpr 448	. . . 4 |- ( ( N e. NN /\ K e. ZZ ) -> K e. ZZ )
64		dvdsval2 12810	. . . 4 |- ( ( N e. ZZ /\ N =/= 0 /\ K e. ZZ ) -> ( N || K <-> ( K / N ) e. ZZ ) )
65	62, 21, 63, 64	syl3anc 1184	. . 3 |- ( ( N e. NN /\ K e. ZZ ) -> ( N || K <-> ( K / N ) e. ZZ ) )
66	60, 65	bitr4d 248	. 2 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) / ( _i x. ( 2 x. pi ) ) ) e. ZZ <-> N || K ) )
67	42, 47, 66	3bitrd 271	1 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( -u 1 ^ c ( 2 / N ) ) ^ K ) = 1 <-> N || K ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   _ici 8948   x. cmul 8951   -ucneg 9248   / cdiv 9633   NNcn 9956   2c2 10005   ZZcz 10238   ^cexp 11337   expce 12619   picpi 12624   || cdivides 12807   logclog 20405   ^ c ccxp 20406

This theorem is referenced by:  dchrptlem1  21001  dchrptlem2  21002

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408