Theorem root1eq1 22192

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 complex numbers.) (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 10390	. . . . . . . 8 |- 2 e. RR
2		simpl 457	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> N e. NN )
3		nndivre 10356	. . . . . . . 8 |- ( ( 2 e. RR /\ N e. NN ) -> ( 2 / N ) e. RR )
4	1, 2, 3	sylancr 663	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> ( 2 / N ) e. RR )
5	4	recnd 9411	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( 2 / N ) e. CC )
6		ax-icn 9340	. . . . . . . 8 |- _i e. CC
7		picn 21921	. . . . . . . 8 |- pi e. CC
8	6, 7	mulcli 9390	. . . . . . 7 |- ( _i x. pi ) e. CC
9	8	a1i 11	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( _i x. pi ) e. CC )
10	5, 9	mulcld 9405	. . . . 5 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( 2 / N ) x. ( _i x. pi ) ) e. CC )
11		efexp 13384	. . . . 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 ) )
12	10, 11	sylancom 667	. . . 4 |- ( ( N e. NN /\ K e. ZZ ) -> ( exp ` ( K x. ( ( 2 / N ) x. ( _i x. pi ) ) ) ) = ( ( exp ` ( ( 2 / N ) x. ( _i x. pi ) ) ) ^ K ) )
13		zcn 10650	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
14	13	adantl 466	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> K e. CC )
15		nncn 10329	. . . . . . . . 9 |- ( N e. NN -> N e. CC )
16	15	adantr 465	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> N e. CC )
17		2cn 10391	. . . . . . . . 9 |- 2 e. CC
18	17	a1i 11	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> 2 e. CC )
19		nnne0 10353	. . . . . . . . 9 |- ( N e. NN -> N =/= 0 )
20	19	adantr 465	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> N =/= 0 )
21	14, 16, 18, 20	div32d 10129	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( K / N ) x. 2 ) = ( K x. ( 2 / N ) ) )
22	21	oveq1d 6105	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( K / N ) x. 2 ) x. ( _i x. pi ) ) = ( ( K x. ( 2 / N ) ) x. ( _i x. pi ) ) )
23	14, 16, 20	divcld 10106	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> ( K / N ) e. CC )
24	23, 18, 9	mulassd 9408	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( K / N ) x. 2 ) x. ( _i x. pi ) ) = ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) )
25	14, 5, 9	mulassd 9408	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( K x. ( 2 / N ) ) x. ( _i x. pi ) ) = ( K x. ( ( 2 / N ) x. ( _i x. pi ) ) ) )
26	22, 24, 25	3eqtr3d 2482	. . . . 5 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) = ( K x. ( ( 2 / N ) x. ( _i x. pi ) ) ) )
27	26	fveq2d 5694	. . . 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 ) ) ) ) )
28		neg1cn 10424	. . . . . . . 8 |- -u 1 e. CC
29	28	a1i 11	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> -u 1 e. CC )
30		neg1ne0 10426	. . . . . . . 8 |- -u 1 =/= 0
31	30	a1i 11	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> -u 1 =/= 0 )
32	29, 31, 5	cxpefd 22156	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( -u 1 ^c ( 2 / N ) ) = ( exp ` ( ( 2 / N ) x. ( log ` -u 1 ) ) ) )
33		logm1 22036	. . . . . . . 8 |- ( log ` -u 1 ) = ( _i x. pi )
34	33	oveq2i 6101	. . . . . . 7 |- ( ( 2 / N ) x. ( log ` -u 1 ) ) = ( ( 2 / N ) x. ( _i x. pi ) )
35	34	fveq2i 5693	. . . . . 6 |- ( exp ` ( ( 2 / N ) x. ( log ` -u 1 ) ) ) = ( exp ` ( ( 2 / N ) x. ( _i x. pi ) ) )
36	32, 35	syl6eq 2490	. . . . 5 |- ( ( N e. NN /\ K e. ZZ ) -> ( -u 1 ^c ( 2 / N ) ) = ( exp ` ( ( 2 / N ) x. ( _i x. pi ) ) ) )
37	36	oveq1d 6105	. . . 4 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ K ) = ( ( exp ` ( ( 2 / N ) x. ( _i x. pi ) ) ) ^ K ) )
38	12, 27, 37	3eqtr4rd 2485	. . 3 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( -u 1 ^c ( 2 / N ) ) ^ K ) = ( exp ` ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) ) )
39	38	eqeq1d 2450	. 2 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( -u 1 ^c ( 2 / N ) ) ^ K ) = 1 <-> ( exp ` ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) ) = 1 ) )
40	17, 8	mulcli 9390	. . . 4 |- ( 2 x. ( _i x. pi ) ) e. CC
41		mulcl 9365	. . . 4 |- ( ( ( K / N ) e. CC /\ ( 2 x. ( _i x. pi ) ) e. CC ) -> ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) e. CC )
42	23, 40, 41	sylancl 662	. . 3 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) e. CC )
43		efeq1 21984	. . 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 ) )
44	42, 43	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 ) )
45	6, 17, 7	mul12i 9563	. . . . . 6 |- ( _i x. ( 2 x. pi ) ) = ( 2 x. ( _i x. pi ) )
46	45	oveq2i 6101	. . . . 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 ) ) )
47	40	a1i 11	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( 2 x. ( _i x. pi ) ) e. CC )
48		2ne0 10413	. . . . . . . 8 |- 2 =/= 0
49		ine0 9779	. . . . . . . . 9 |- _i =/= 0
50		pire 21920	. . . . . . . . . 10 |- pi e. RR
51		pipos 21922	. . . . . . . . . 10 |- 0 < pi
52	50, 51	gt0ne0ii 9875	. . . . . . . . 9 |- pi =/= 0
53	6, 7, 49, 52	mulne0i 9978	. . . . . . . 8 |- ( _i x. pi ) =/= 0
54	17, 8, 48, 53	mulne0i 9978	. . . . . . 7 |- ( 2 x. ( _i x. pi ) ) =/= 0
55	54	a1i 11	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( 2 x. ( _i x. pi ) ) =/= 0 )
56	23, 47, 55	divcan4d 10112	. . . . 5 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) / ( 2 x. ( _i x. pi ) ) ) = ( K / N ) )
57	46, 56	syl5eq 2486	. . . 4 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) / ( _i x. ( 2 x. pi ) ) ) = ( K / N ) )
58	57	eleq1d 2508	. . 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 ) )
59		nnz 10667	. . . . 5 |- ( N e. NN -> N e. ZZ )
60	59	adantr 465	. . . 4 |- ( ( N e. NN /\ K e. ZZ ) -> N e. ZZ )
61		simpr 461	. . . 4 |- ( ( N e. NN /\ K e. ZZ ) -> K e. ZZ )
62		dvdsval2 13537	. . . 4 |- ( ( N e. ZZ /\ N =/= 0 /\ K e. ZZ ) -> ( N || K <-> ( K / N ) e. ZZ ) )
63	60, 20, 61, 62	syl3anc 1218	. . 3 |- ( ( N e. NN /\ K e. ZZ ) -> ( N || K <-> ( K / N ) e. ZZ ) )
64	58, 63	bitr4d 256	. 2 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) / ( _i x. ( 2 x. pi ) ) ) e. ZZ <-> N || K ) )
65	39, 44, 64	3bitrd 279	1 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( -u 1 ^c ( 2 / N ) ) ^ K ) = 1 <-> N || K ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2605   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282   _ici 9283   x. cmul 9286   -ucneg 9595   / cdiv 9992   NNcn 10321   2c2 10370   ZZcz 10645   ^cexp 11864   expce 13346   picpi 13351   || cdivides 13534   logclog 22005   ^c ccxp 22006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-fi 7660  df-sup 7690  df-oi 7723  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-q 10953  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-ioo 11303  df-ioc 11304  df-ico 11305  df-icc 11306  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-fac 12051  df-bc 12078  df-hash 12103  df-shft 12555  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-limsup 12948  df-clim 12965  df-rlim 12966  df-sum 13163  df-ef 13352  df-sin 13354  df-cos 13355  df-pi 13357  df-dvds 13535  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-hom 14261  df-cco 14262  df-rest 14360  df-topn 14361  df-0g 14379  df-gsum 14380  df-topgen 14381  df-pt 14382  df-prds 14385  df-xrs 14439  df-qtop 14444  df-imas 14445  df-xps 14447  df-mre 14523  df-mrc 14524  df-acs 14526  df-mnd 15414  df-submnd 15464  df-mulg 15547  df-cntz 15834  df-cmn 16278  df-psmet 17808  df-xmet 17809  df-met 17810  df-bl 17811  df-mopn 17812  df-fbas 17813  df-fg 17814  df-cnfld 17818  df-top 18502  df-bases 18504  df-topon 18505  df-topsp 18506  df-cld 18622  df-ntr 18623  df-cls 18624  df-nei 18701  df-lp 18739  df-perf 18740  df-cn 18830  df-cnp 18831  df-haus 18918  df-tx 19134  df-hmeo 19327  df-fil 19418  df-fm 19510  df-flim 19511  df-flf 19512  df-xms 19894  df-ms 19895  df-tms 19896  df-cncf 20453  df-limc 21340  df-dv 21341  df-log 22007  df-cxp 22008

This theorem is referenced by:  dchrptlem1  22602  dchrptlem2  22603