Theorem efnthr 16142

Description: An equation involving an N-th power. The expression exp ( 2 pi _i / N ) is a way to write the primitive N-th root of unity with the smallest positive argument. We could write it as -u 1 ^c ( 2 / N ) except that we only support positive real numbers as the base for ^c. The converse is presumably also provable but we do not have a proof yet. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by Jim Kingdon, 18-Sep-2026.)

Assertion

Ref	Expression
efnthr	|- ( ( A e. CC /\ N e. NN /\ B e. RR+ ) -> ( E. n e. ( 0 ... ( N - 1 ) ) A = ( ( B ^c ( 1 / N ) ) x. ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ) -> ( A ^ N ) = B ) )

Distinct variable groups:   A, n   B, n   n, N

Proof of Theorem efnthr

Step	Hyp	Ref	Expression
1		simp3 1030	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN /\ B e. RR+ ) -> B e. RR+ )
2		nnrecre 9344	. . . . . . . . 9 |- ( N e. NN -> ( 1 / N ) e. RR )
3	2	3ad2ant2 1050	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN /\ B e. RR+ ) -> ( 1 / N ) e. RR )
4	1, 3	rpcxpcld 16134	. . . . . . 7 |- ( ( A e. CC /\ N e. NN /\ B e. RR+ ) -> ( B ^c ( 1 / N ) ) e. RR+ )
5	4	adantr 276	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( B ^c ( 1 / N ) ) e. RR+ )
6	5	rpcnd 10110	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( B ^c ( 1 / N ) ) e. CC )
7		ax-icn 8275	. . . . . . . . . 10 |- _i e. CC
8		2cn 9378	. . . . . . . . . . 11 |- 2 e. CC
9		picn 15980	. . . . . . . . . . 11 |- pi e. CC
10	8, 9	mulcli 8332	. . . . . . . . . 10 |- ( 2 x. pi ) e. CC
11	7, 10	mulcli 8332	. . . . . . . . 9 |- ( _i x. ( 2 x. pi ) ) e. CC
12	11	a1i 9	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN /\ B e. RR+ ) -> ( _i x. ( 2 x. pi ) ) e. CC )
13		nncn 9315	. . . . . . . . 9 |- ( N e. NN -> N e. CC )
14	13	3ad2ant2 1050	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN /\ B e. RR+ ) -> N e. CC )
15		nnap0 9336	. . . . . . . . 9 |- ( N e. NN -> N # 0 )
16	15	3ad2ant2 1050	. . . . . . . 8 |- ( ( A e. CC /\ N e. NN /\ B e. RR+ ) -> N # 0 )
17	12, 14, 16	divclapd 9123	. . . . . . 7 |- ( ( A e. CC /\ N e. NN /\ B e. RR+ ) -> ( ( _i x. ( 2 x. pi ) ) / N ) e. CC )
18		efcl 12450	. . . . . . 7 |- ( ( ( _i x. ( 2 x. pi ) ) / N ) e. CC -> ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) e. CC )
19	17, 18	syl 14	. . . . . 6 |- ( ( A e. CC /\ N e. NN /\ B e. RR+ ) -> ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) e. CC )
20		elfznn0 10532	. . . . . 6 |- ( n e. ( 0 ... ( N - 1 ) ) -> n e. NN0 )
21		expcl 11009	. . . . . 6 |- ( ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) e. CC /\ n e. NN0 ) -> ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) e. CC )
22	19, 20, 21	syl2an 289	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) e. CC )
23		simpl2 1032	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> N e. NN )
24	23	nnnn0d 9625	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> N e. NN0 )
25	6, 22, 24	mulexpd 11141	. . . 4 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( B ^c ( 1 / N ) ) x. ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ) ^ N ) = ( ( ( B ^c ( 1 / N ) ) ^ N ) x. ( ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ^ N ) ) )
26	1	adantr 276	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> B e. RR+ )
27		rpcxproot 16113	. . . . . 6 |- ( ( B e. RR+ /\ N e. NN ) -> ( ( B ^c ( 1 / N ) ) ^ N ) = B )
28	26, 23, 27	syl2anc 415	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( B ^c ( 1 / N ) ) ^ N ) = B )
29	20	adantl 277	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> n e. NN0 )
30	29	nn0cnd 9627	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> n e. CC )
31	23	nncnd 9321	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> N e. CC )
32	30, 31	mulcomd 8348	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( n x. N ) = ( N x. n ) )
33	32	oveq2d 6101	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ ( n x. N ) ) = ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ ( N x. n ) ) )
34	19	adantr 276	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) e. CC )
35	34, 24, 29	expmuld 11129	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ ( n x. N ) ) = ( ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ^ N ) )
36	34, 29, 24	expmuld 11129	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ ( N x. n ) ) = ( ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ N ) ^ n ) )
37	33, 35, 36	3eqtr3d 2279	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ^ N ) = ( ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ N ) ^ n ) )
38		root1idef 16050	. . . . . . . 8 |- ( N e. NN -> ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ N ) = 1 )
39	23, 38	syl 14	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ N ) = 1 )
40	39	oveq1d 6100	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ N ) ^ n ) = ( 1 ^ n ) )
41		elfzelz 10439	. . . . . . . 8 |- ( n e. ( 0 ... ( N - 1 ) ) -> n e. ZZ )
42	41	adantl 277	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> n e. ZZ )
43		1exp 11020	. . . . . . 7 |- ( n e. ZZ -> ( 1 ^ n ) = 1 )
44	42, 43	syl 14	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( 1 ^ n ) = 1 )
45	37, 40, 44	3eqtrd 2275	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ^ N ) = 1 )
46	28, 45	oveq12d 6103	. . . 4 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( B ^c ( 1 / N ) ) ^ N ) x. ( ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ^ N ) ) = ( B x. 1 ) )
47	26	rpcnd 10110	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> B e. CC )
48	47	mulridd 8344	. . . 4 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( B x. 1 ) = B )
49	25, 46, 48	3eqtrd 2275	. . 3 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( B ^c ( 1 / N ) ) x. ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ) ^ N ) = B )
50		oveq1 6092	. . . 4 |- ( A = ( ( B ^c ( 1 / N ) ) x. ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ) -> ( A ^ N ) = ( ( ( B ^c ( 1 / N ) ) x. ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ) ^ N ) )
51	50	eqeq1d 2247	. . 3 |- ( A = ( ( B ^c ( 1 / N ) ) x. ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ) -> ( ( A ^ N ) = B <-> ( ( ( B ^c ( 1 / N ) ) x. ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ) ^ N ) = B ) )
52	49, 51	syl5ibrcom 157	. 2 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( A = ( ( B ^c ( 1 / N ) ) x. ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ) -> ( A ^ N ) = B ) )
53	52	rexlimdva 2668	1 |- ( ( A e. CC /\ N e. NN /\ B e. RR+ ) -> ( E. n e. ( 0 ... ( N - 1 ) ) A = ( ( B ^c ( 1 / N ) ) x. ( ( exp ` ( ( _i x. ( 2 x. pi ) ) / N ) ) ^ n ) ) -> ( A ^ N ) = B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8178   RRcr 8179   0cc0 8180   1c1 8181   _ici 8182   x. cmul 8185   - cmin 8499   # cap 8912   / cdiv 9005   NNcn 9307   2c2 9358   NN0cn0 9568   ZZcz 9649   RR+crp 10065   ...cfz 10422   ^cexp 10990   expce 12428   picpi 12433   ^c ccxp 16052

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299  ax-caucvg 8300  ax-pre-suploc 8301  ax-addf 8302  ax-mulf 8303

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7325  df-inf 7326  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-5 9369  df-6 9370  df-7 9371  df-8 9372  df-9 9373  df-n0 9569  df-z 9650  df-uz 9932  df-q 10030  df-rp 10066  df-xneg 10185  df-xadd 10186  df-ioo 10305  df-ioc 10306  df-ico 10307  df-icc 10308  df-fz 10423  df-fzo 10561  df-seqfrec 10900  df-exp 10991  df-fac 11180  df-bc 11202  df-ihash 11231  df-shft 11596  df-cj 11623  df-re 11624  df-im 11625  df-rsqrt 11780  df-abs 11781  df-clim 12064  df-sumdc 12139  df-ef 12434  df-e 12435  df-sin 12436  df-cos 12437  df-pi 12439  df-rest 13648  df-topgen 13667  df-psmet 14964  df-xmet 14965  df-met 14966  df-bl 14967  df-mopn 14968  df-top 15190  df-topon 15203  df-bases 15235  df-ntr 15288  df-cn 15380  df-cnp 15381  df-tx 15445  df-cncf 15763  df-limced 15848  df-dvap 15849  df-relog 16053  df-rpcxp 16054

This theorem is used by: (None)