Theorem root1eq1 22852

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 10596	. . . . . . . 8 |- 2 e. RR
2		simpl 457	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> N e. NN )
3		nndivre 10562	. . . . . . . 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 9613	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( 2 / N ) e. CC )
6		ax-icn 9542	. . . . . . . 8 |- _i e. CC
7		picn 22581	. . . . . . . 8 |- pi e. CC
8	6, 7	mulcli 9592	. . . . . . 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 9607	. . . . 5 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( 2 / N ) x. ( _i x. pi ) ) e. CC )
11		efexp 13688	. . . . 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 10860	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
14	13	adantl 466	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> K e. CC )
15		nncn 10535	. . . . . . . . 9 |- ( N e. NN -> N e. CC )
16	15	adantr 465	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> N e. CC )
17		2cn 10597	. . . . . . . . 9 |- 2 e. CC
18	17	a1i 11	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> 2 e. CC )
19		nnne0 10559	. . . . . . . . 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 10334	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( K / N ) x. 2 ) = ( K x. ( 2 / N ) ) )
22	21	oveq1d 6292	. . . . . 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 10311	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> ( K / N ) e. CC )
24	23, 18, 9	mulassd 9610	. . . . . 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 9610	. . . . . 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 2511	. . . . 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 5863	. . . 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 10630	. . . . . . . 8 |- -u 1 e. CC
29	28	a1i 11	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> -u 1 e. CC )
30		neg1ne0 10632	. . . . . . . 8 |- -u 1 =/= 0
31	30	a1i 11	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> -u 1 =/= 0 )
32	29, 31, 5	cxpefd 22816	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( -u 1 ^c ( 2 / N ) ) = ( exp ` ( ( 2 / N ) x. ( log ` -u 1 ) ) ) )
33		logm1 22696	. . . . . . . 8 |- ( log ` -u 1 ) = ( _i x. pi )
34	33	oveq2i 6288	. . . . . . 7 |- ( ( 2 / N ) x. ( log ` -u 1 ) ) = ( ( 2 / N ) x. ( _i x. pi ) )
35	34	fveq2i 5862	. . . . . 6 |- ( exp ` ( ( 2 / N ) x. ( log ` -u 1 ) ) ) = ( exp ` ( ( 2 / N ) x. ( _i x. pi ) ) )
36	32, 35	syl6eq 2519	. . . . 5 |- ( ( N e. NN /\ K e. ZZ ) -> ( -u 1 ^c ( 2 / N ) ) = ( exp ` ( ( 2 / N ) x. ( _i x. pi ) ) ) )
37	36	oveq1d 6292	. . . 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 2514	. . 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 2464	. 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 9592	. . . 4 |- ( 2 x. ( _i x. pi ) ) e. CC
41		mulcl 9567	. . . 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 22644	. . 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 9765	. . . . . 6 |- ( _i x. ( 2 x. pi ) ) = ( 2 x. ( _i x. pi ) )
46	45	oveq2i 6288	. . . . 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 10619	. . . . . . . 8 |- 2 =/= 0
49		ine0 9983	. . . . . . . . 9 |- _i =/= 0
50		pire 22580	. . . . . . . . . 10 |- pi e. RR
51		pipos 22582	. . . . . . . . . 10 |- 0 < pi
52	50, 51	gt0ne0ii 10080	. . . . . . . . 9 |- pi =/= 0
53	6, 7, 49, 52	mulne0i 10183	. . . . . . . 8 |- ( _i x. pi ) =/= 0
54	17, 8, 48, 53	mulne0i 10183	. . . . . . 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 10317	. . . . 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 2515	. . . 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 2531	. . 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 10877	. . . . 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 13841	. . . 4 |- ( ( N e. ZZ /\ N =/= 0 /\ K e. ZZ ) -> ( N || K <-> ( K / N ) e. ZZ ) )
63	60, 20, 61, 62	syl3anc 1223	. . 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 1374   e. wcel 1762   =/= wne 2657   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484   _ici 9485   x. cmul 9488   -ucneg 9797   / cdiv 10197   NNcn 10527   2c2 10576   ZZcz 10855   ^cexp 12124   expce 13650   picpi 13655   || cdivides 13838   logclog 22665   ^c ccxp 22666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-ioc 11525  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-fac 12311  df-bc 12338  df-hash 12363  df-shft 12852  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-sum 13460  df-ef 13656  df-sin 13658  df-cos 13659  df-pi 13661  df-dvds 13839  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-fbas 18182  df-fg 18183  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lp 19398  df-perf 19399  df-cn 19489  df-cnp 19490  df-haus 19577  df-tx 19793  df-hmeo 19986  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-xms 20553  df-ms 20554  df-tms 20555  df-cncf 21112  df-limc 22000  df-dv 22001  df-log 22667  df-cxp 22668

This theorem is referenced by:  dchrptlem1  23262  dchrptlem2  23263