Theorem root1eq1 23707

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 10686	. . . . . . . 8 |- 2 e. RR
2		simpl 459	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> N e. NN )
3		nndivre 10652	. . . . . . . 8 |- ( ( 2 e. RR /\ N e. NN ) -> ( 2 / N ) e. RR )
4	1, 2, 3	sylancr 670	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> ( 2 / N ) e. RR )
5	4	recnd 9674	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( 2 / N ) e. CC )
6		ax-icn 9603	. . . . . . . 8 |- _i e. CC
7		picn 23426	. . . . . . . 8 |- pi e. CC
8	6, 7	mulcli 9653	. . . . . . 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 9668	. . . . 5 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( 2 / N ) x. ( _i x. pi ) ) e. CC )
11		efexp 14167	. . . . 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 674	. . . 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 10949	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
14	13	adantl 468	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> K e. CC )
15		nncn 10624	. . . . . . . . 9 |- ( N e. NN -> N e. CC )
16	15	adantr 467	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> N e. CC )
17		2cn 10687	. . . . . . . . 9 |- 2 e. CC
18	17	a1i 11	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> 2 e. CC )
19		nnne0 10649	. . . . . . . . 9 |- ( N e. NN -> N =/= 0 )
20	19	adantr 467	. . . . . . . 8 |- ( ( N e. NN /\ K e. ZZ ) -> N =/= 0 )
21	14, 16, 18, 20	div32d 10413	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( K / N ) x. 2 ) = ( K x. ( 2 / N ) ) )
22	21	oveq1d 6310	. . . . . 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 10390	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> ( K / N ) e. CC )
24	23, 18, 9	mulassd 9671	. . . . . 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 9671	. . . . . 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 2495	. . . . 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 5874	. . . 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 10720	. . . . . . . 8 |- -u 1 e. CC
29	28	a1i 11	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> -u 1 e. CC )
30		neg1ne0 10722	. . . . . . . 8 |- -u 1 =/= 0
31	30	a1i 11	. . . . . . 7 |- ( ( N e. NN /\ K e. ZZ ) -> -u 1 =/= 0 )
32	29, 31, 5	cxpefd 23669	. . . . . 6 |- ( ( N e. NN /\ K e. ZZ ) -> ( -u 1 ^c ( 2 / N ) ) = ( exp ` ( ( 2 / N ) x. ( log ` -u 1 ) ) ) )
33		logm1 23550	. . . . . . . 8 |- ( log ` -u 1 ) = ( _i x. pi )
34	33	oveq2i 6306	. . . . . . 7 |- ( ( 2 / N ) x. ( log ` -u 1 ) ) = ( ( 2 / N ) x. ( _i x. pi ) )
35	34	fveq2i 5873	. . . . . 6 |- ( exp ` ( ( 2 / N ) x. ( log ` -u 1 ) ) ) = ( exp ` ( ( 2 / N ) x. ( _i x. pi ) ) )
36	32, 35	syl6eq 2503	. . . . 5 |- ( ( N e. NN /\ K e. ZZ ) -> ( -u 1 ^c ( 2 / N ) ) = ( exp ` ( ( 2 / N ) x. ( _i x. pi ) ) ) )
37	36	oveq1d 6310	. . . 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 2498	. . 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 2455	. 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 9653	. . . 4 |- ( 2 x. ( _i x. pi ) ) e. CC
41		mulcl 9628	. . . 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 669	. . 3 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( K / N ) x. ( 2 x. ( _i x. pi ) ) ) e. CC )
43		efeq1 23490	. . 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 17	. 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 9833	. . . . . 6 |- ( _i x. ( 2 x. pi ) ) = ( 2 x. ( _i x. pi ) )
46	45	oveq2i 6306	. . . . 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 10709	. . . . . . . 8 |- 2 =/= 0
49		ine0 10061	. . . . . . . . 9 |- _i =/= 0
50		pire 23425	. . . . . . . . . 10 |- pi e. RR
51		pipos 23427	. . . . . . . . . 10 |- 0 < pi
52	50, 51	gt0ne0ii 10157	. . . . . . . . 9 |- pi =/= 0
53	6, 7, 49, 52	mulne0i 10262	. . . . . . . 8 |- ( _i x. pi ) =/= 0
54	17, 8, 48, 53	mulne0i 10262	. . . . . . 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 10396	. . . . 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 2499	. . . 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 2515	. . 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 10966	. . . . 5 |- ( N e. NN -> N e. ZZ )
60	59	adantr 467	. . . 4 |- ( ( N e. NN /\ K e. ZZ ) -> N e. ZZ )
61		simpr 463	. . . 4 |- ( ( N e. NN /\ K e. ZZ ) -> K e. ZZ )
62		dvdsval2 14320	. . . 4 |- ( ( N e. ZZ /\ N =/= 0 /\ K e. ZZ ) -> ( N || K <-> ( K / N ) e. ZZ ) )
63	60, 20, 61, 62	syl3anc 1269	. . 3 |- ( ( N e. NN /\ K e. ZZ ) -> ( N || K <-> ( K / N ) e. ZZ ) )
64	58, 63	bitr4d 260	. 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 283	1 |- ( ( N e. NN /\ K e. ZZ ) -> ( ( ( -u 1 ^c ( 2 / N ) ) ^ K ) = 1 <-> N || K ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1446   e. wcel 1889   =/= wne 2624   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   CCcc 9542   RRcr 9543   0cc0 9544   1c1 9545   _ici 9546   x. cmul 9549   -ucneg 9866   / cdiv 10276   NNcn 10616   2c2 10666   ZZcz 10944   ^cexp 12279   expce 14126   picpi 14131   || cdvds 14317   logclog 23516   ^c ccxp 23517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622  ax-addf 9623  ax-mulf 9624

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  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-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-fi 7930  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ioc 11647  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12035  df-mod 12104  df-seq 12221  df-exp 12280  df-fac 12467  df-bc 12495  df-hash 12523  df-shft 13142  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-limsup 13538  df-clim 13564  df-rlim 13565  df-sum 13765  df-ef 14133  df-sin 14135  df-cos 14136  df-pi 14138  df-dvds 14318  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-starv 15217  df-sca 15218  df-vsca 15219  df-ip 15220  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-hom 15226  df-cco 15227  df-rest 15333  df-topn 15334  df-0g 15352  df-gsum 15353  df-topgen 15354  df-pt 15355  df-prds 15358  df-xrs 15412  df-qtop 15418  df-imas 15419  df-xps 15422  df-mre 15504  df-mrc 15505  df-acs 15507  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-submnd 16595  df-mulg 16688  df-cntz 16983  df-cmn 17444  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-fbas 18979  df-fg 18980  df-cnfld 18983  df-top 19933  df-bases 19934  df-topon 19935  df-topsp 19936  df-cld 20046  df-ntr 20047  df-cls 20048  df-nei 20126  df-lp 20164  df-perf 20165  df-cn 20255  df-cnp 20256  df-haus 20343  df-tx 20589  df-hmeo 20782  df-fil 20873  df-fm 20965  df-flim 20966  df-flf 20967  df-xms 21347  df-ms 21348  df-tms 21349  df-cncf 21922  df-limc 22833  df-dv 22834  df-log 23518  df-cxp 23519

This theorem is referenced by:  dchrptlem1  24204  dchrptlem2  24205