Theorem efnthr 16188

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 9351	. . . . . . . . 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 16180	. . . . . . 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 10117	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( B ^c ( 1 / N ) ) e. CC )
7		ax-icn 8281	. . . . . . . . . 10 |- _i e. CC
8		2cn 9385	. . . . . . . . . . 11 |- 2 e. CC
9		picn 16026	. . . . . . . . . . 11 |- pi e. CC
10	8, 9	mulcli 8338	. . . . . . . . . 10 |- ( 2 x. pi ) e. CC
11	7, 10	mulcli 8338	. . . . . . . . 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 9322	. . . . . . . . 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 9343	. . . . . . . . 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 9129	. . . . . . 7 |- ( ( A e. CC /\ N e. NN /\ B e. RR+ ) -> ( ( _i x. ( 2 x. pi ) ) / N ) e. CC )
18		efcl 12457	. . . . . . 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 10539	. . . . . 6 |- ( n e. ( 0 ... ( N - 1 ) ) -> n e. NN0 )
21		expcl 11016	. . . . . 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 9632	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> N e. NN0 )
25	6, 22, 24	mulexpd 11148	. . . 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 16159	. . . . . 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 9634	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> n e. CC )
31	23	nncnd 9328	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> N e. CC )
32	30, 31	mulcomd 8354	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> ( n x. N ) = ( N x. n ) )
33	32	oveq2d 6104	. . . . . . 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 11136	. . . . . . 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 11136	. . . . . . 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 16096	. . . . . . . 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 6103	. . . . . 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 10446	. . . . . . . 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 11027	. . . . . . 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 6106	. . . 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 10117	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ B e. RR+ ) /\ n e. ( 0 ... ( N - 1 ) ) ) -> B e. CC )
48	47	mulridd 8350	. . . 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 6095	. . . 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 4131   ` cfv 5380  (class class class)co 6088   CCcc 8184   RRcr 8185   0cc0 8186   1c1 8187   _ici 8188   x. cmul 8191   - cmin 8505   # cap 8918   / cdiv 9011   NNcn 9314   2c2 9365   NN0cn0 9575   ZZcz 9656   RR+crp 10072   ...cfz 10429   ^cexp 10997   expce 12435   picpi 12440   ^c ccxp 16098

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306  ax-pre-suploc 8307  ax-addf 8308  ax-mulf 8309

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-map 6928  df-pm 6929  df-en 7028  df-dom 7029  df-fin 7030  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-5 9376  df-6 9377  df-7 9378  df-8 9379  df-9 9380  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-xneg 10192  df-xadd 10193  df-ioo 10312  df-ioc 10313  df-ico 10314  df-icc 10315  df-fz 10430  df-fzo 10568  df-seqfrec 10907  df-exp 10998  df-fac 11187  df-bc 11209  df-ihash 11238  df-shft 11603  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788  df-clim 12071  df-sumdc 12146  df-ef 12441  df-e 12442  df-sin 12443  df-cos 12444  df-pi 12446  df-rest 13655  df-topgen 13674  df-psmet 14990  df-xmet 14991  df-met 14992  df-bl 14993  df-mopn 14994  df-top 15236  df-topon 15249  df-bases 15281  df-ntr 15334  df-cn 15426  df-cnp 15427  df-tx 15491  df-cncf 15809  df-limced 15894  df-dvap 15895  df-relog 16099  df-rpcxp 16100

This theorem is used by: (None)