Theorem dchrfi 24980

Description: The group of Dirichlet characters is a finite group. (Contributed by Mario Carneiro, 19-Apr-2016.)

Hypotheses

Ref	Expression
dchrabl.g	|- G = (DChr ` N )
dchrfi.b	|- D = ( Base ` G )

Assertion

Ref	Expression
dchrfi	|- ( N e. NN -> D e. Fin )

Proof of Theorem dchrfi

Dummy variables x f z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snfi 8038	. . . 4 |- { 0 } e. Fin
2		cnex 10017	. . . . . . . . 9 |- CC e. _V
3	2	a1i 11	. . . . . . . 8 |- ( N e. NN -> CC e. _V )
4		ovexd 6680	. . . . . . . 8 |- ( ( N e. NN /\ z e. CC ) -> ( z ^ ( phi ` N ) ) e. _V )
5		1cnd 10056	. . . . . . . 8 |- ( ( N e. NN /\ z e. CC ) -> 1 e. CC )
6		eqidd 2623	. . . . . . . 8 |- ( N e. NN -> ( z e. CC |-> ( z ^ ( phi ` N ) ) ) = ( z e. CC |-> ( z ^ ( phi ` N ) ) ) )
7		fconstmpt 5163	. . . . . . . . 9 |- ( CC X. { 1 } ) = ( z e. CC |-> 1 )
8	7	a1i 11	. . . . . . . 8 |- ( N e. NN -> ( CC X. { 1 } ) = ( z e. CC |-> 1 ) )
9	3, 4, 5, 6, 8	offval2 6914	. . . . . . 7 |- ( N e. NN -> ( ( z e. CC |-> ( z ^ ( phi ` N ) ) ) oF - ( CC X. { 1 } ) ) = ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) )
10		ssid 3624	. . . . . . . . . 10 |- CC C_ CC
11	10	a1i 11	. . . . . . . . 9 |- ( N e. NN -> CC C_ CC )
12		1cnd 10056	. . . . . . . . 9 |- ( N e. NN -> 1 e. CC )
13		phicl 15474	. . . . . . . . . 10 |- ( N e. NN -> ( phi ` N ) e. NN )
14	13	nnnn0d 11351	. . . . . . . . 9 |- ( N e. NN -> ( phi ` N ) e. NN0 )
15		plypow 23961	. . . . . . . . 9 |- ( ( CC C_ CC /\ 1 e. CC /\ ( phi ` N ) e. NN0 ) -> ( z e. CC |-> ( z ^ ( phi ` N ) ) ) e. (Poly ` CC ) )
16	11, 12, 14, 15	syl3anc 1326	. . . . . . . 8 |- ( N e. NN -> ( z e. CC |-> ( z ^ ( phi ` N ) ) ) e. (Poly ` CC ) )
17		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
18		plyconst 23962	. . . . . . . . 9 |- ( ( CC C_ CC /\ 1 e. CC ) -> ( CC X. { 1 } ) e. (Poly ` CC ) )
19	10, 17, 18	mp2an 708	. . . . . . . 8 |- ( CC X. { 1 } ) e. (Poly ` CC )
20		plysubcl 23978	. . . . . . . 8 |- ( ( ( z e. CC |-> ( z ^ ( phi ` N ) ) ) e. (Poly ` CC ) /\ ( CC X. { 1 } ) e. (Poly ` CC ) ) -> ( ( z e. CC |-> ( z ^ ( phi ` N ) ) ) oF - ( CC X. { 1 } ) ) e. (Poly ` CC ) )
21	16, 19, 20	sylancl 694	. . . . . . 7 |- ( N e. NN -> ( ( z e. CC |-> ( z ^ ( phi ` N ) ) ) oF - ( CC X. { 1 } ) ) e. (Poly ` CC ) )
22	9, 21	eqeltrrd 2702	. . . . . 6 |- ( N e. NN -> ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) e. (Poly ` CC ) )
23		0cn 10032	. . . . . . 7 |- 0 e. CC
24		neg1ne0 11126	. . . . . . . 8 |- -u 1 =/= 0
25	13	0expd 13024	. . . . . . . . . . 11 |- ( N e. NN -> ( 0 ^ ( phi ` N ) ) = 0 )
26	25	oveq1d 6665	. . . . . . . . . 10 |- ( N e. NN -> ( ( 0 ^ ( phi ` N ) ) - 1 ) = ( 0 - 1 ) )
27		oveq1 6657	. . . . . . . . . . . . 13 |- ( z = 0 -> ( z ^ ( phi ` N ) ) = ( 0 ^ ( phi ` N ) ) )
28	27	oveq1d 6665	. . . . . . . . . . . 12 |- ( z = 0 -> ( ( z ^ ( phi ` N ) ) - 1 ) = ( ( 0 ^ ( phi ` N ) ) - 1 ) )
29		eqid 2622	. . . . . . . . . . . 12 |- ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) = ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) )
30		ovex 6678	. . . . . . . . . . . 12 |- ( ( 0 ^ ( phi ` N ) ) - 1 ) e. _V
31	28, 29, 30	fvmpt 6282	. . . . . . . . . . 11 |- ( 0 e. CC -> ( ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) ` 0 ) = ( ( 0 ^ ( phi ` N ) ) - 1 ) )
32	23, 31	ax-mp 5	. . . . . . . . . 10 |- ( ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) ` 0 ) = ( ( 0 ^ ( phi ` N ) ) - 1 )
33		df-neg 10269	. . . . . . . . . 10 |- -u 1 = ( 0 - 1 )
34	26, 32, 33	3eqtr4g 2681	. . . . . . . . 9 |- ( N e. NN -> ( ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) ` 0 ) = -u 1 )
35	34	neeq1d 2853	. . . . . . . 8 |- ( N e. NN -> ( ( ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) ` 0 ) =/= 0 <-> -u 1 =/= 0 ) )
36	24, 35	mpbiri 248	. . . . . . 7 |- ( N e. NN -> ( ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) ` 0 ) =/= 0 )
37		ne0p 23963	. . . . . . 7 |- ( ( 0 e. CC /\ ( ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) ` 0 ) =/= 0 ) -> ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) =/= 0p )
38	23, 36, 37	sylancr 695	. . . . . 6 |- ( N e. NN -> ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) =/= 0p )
39	29	mptiniseg 5629	. . . . . . . . 9 |- ( 0 e. CC -> ( `' ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) " { 0 } ) = { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } )
40	23, 39	ax-mp 5	. . . . . . . 8 |- ( `' ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) " { 0 } ) = { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 }
41	40	eqcomi 2631	. . . . . . 7 |- { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } = ( `' ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) " { 0 } )
42	41	fta1 24063	. . . . . 6 |- ( ( ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) e. (Poly ` CC ) /\ ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) =/= 0p ) -> ( { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } e. Fin /\ ( # ` { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) <_ (deg ` ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) ) ) )
43	22, 38, 42	syl2anc 693	. . . . 5 |- ( N e. NN -> ( { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } e. Fin /\ ( # ` { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) <_ (deg ` ( z e. CC |-> ( ( z ^ ( phi ` N ) ) - 1 ) ) ) ) )
44	43	simpld 475	. . . 4 |- ( N e. NN -> { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } e. Fin )
45		unfi 8227	. . . 4 |- ( ( { 0 } e. Fin /\ { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } e. Fin ) -> ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) e. Fin )
46	1, 44, 45	sylancr 695	. . 3 |- ( N e. NN -> ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) e. Fin )
47		eqid 2622	. . . 4 |- (ℤ/nℤ ` N ) = (ℤ/nℤ ` N )
48		eqid 2622	. . . 4 |- ( Base ` (ℤ/nℤ ` N ) ) = ( Base ` (ℤ/nℤ ` N ) )
49	47, 48	znfi 19908	. . 3 |- ( N e. NN -> ( Base ` (ℤ/nℤ ` N ) ) e. Fin )
50		mapfi 8262	. . 3 |- ( ( ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) e. Fin /\ ( Base ` (ℤ/nℤ ` N ) ) e. Fin ) -> ( ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) ^m ( Base ` (ℤ/nℤ ` N ) ) ) e. Fin )
51	46, 49, 50	syl2anc 693	. 2 |- ( N e. NN -> ( ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) ^m ( Base ` (ℤ/nℤ ` N ) ) ) e. Fin )
52		dchrabl.g	. . . . . . . 8 |- G = (DChr ` N )
53		dchrfi.b	. . . . . . . 8 |- D = ( Base ` G )
54		simpr 477	. . . . . . . 8 |- ( ( N e. NN /\ f e. D ) -> f e. D )
55	52, 47, 53, 48, 54	dchrf 24967	. . . . . . 7 |- ( ( N e. NN /\ f e. D ) -> f : ( Base ` (ℤ/nℤ ` N ) ) --> CC )
56		ffn 6045	. . . . . . 7 |- ( f : ( Base ` (ℤ/nℤ ` N ) ) --> CC -> f Fn ( Base ` (ℤ/nℤ ` N ) ) )
57	55, 56	syl 17	. . . . . 6 |- ( ( N e. NN /\ f e. D ) -> f Fn ( Base ` (ℤ/nℤ ` N ) ) )
58		df-ne 2795	. . . . . . . . . . 11 |- ( ( f ` x ) =/= 0 <-> -. ( f ` x ) = 0 )
59		fvex 6201	. . . . . . . . . . . 12 |- ( f ` x ) e. _V
60	59	elsn 4192	. . . . . . . . . . 11 |- ( ( f ` x ) e. { 0 } <-> ( f ` x ) = 0 )
61	58, 60	xchbinxr 325	. . . . . . . . . 10 |- ( ( f ` x ) =/= 0 <-> -. ( f ` x ) e. { 0 } )
62		simpl 473	. . . . . . . . . . . . 13 |- ( ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) -> x e. ( Base ` (ℤ/nℤ ` N ) ) )
63		ffvelrn 6357	. . . . . . . . . . . . 13 |- ( ( f : ( Base ` (ℤ/nℤ ` N ) ) --> CC /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( f ` x ) e. CC )
64	55, 62, 63	syl2an 494	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( f ` x ) e. CC )
65	52, 47, 53	dchrmhm 24966	. . . . . . . . . . . . . . . . . 18 |- D C_ ( (mulGrp ` (ℤ/nℤ ` N ) ) MndHom (mulGrp ` ℂfld ) )
66		simplr 792	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> f e. D )
67	65, 66	sseldi 3601	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> f e. ( (mulGrp ` (ℤ/nℤ ` N ) ) MndHom (mulGrp ` ℂfld ) ) )
68	14	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( phi ` N ) e. NN0 )
69		simprl 794	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> x e. ( Base ` (ℤ/nℤ ` N ) ) )
70		eqid 2622	. . . . . . . . . . . . . . . . . . 19 |- (mulGrp ` (ℤ/nℤ ` N ) ) = (mulGrp ` (ℤ/nℤ ` N ) )
71	70, 48	mgpbas 18495	. . . . . . . . . . . . . . . . . 18 |- ( Base ` (ℤ/nℤ ` N ) ) = ( Base ` (mulGrp ` (ℤ/nℤ ` N ) ) )
72		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- (.g ` (mulGrp ` (ℤ/nℤ ` N ) ) ) = (.g ` (mulGrp ` (ℤ/nℤ ` N ) ) )
73		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- (.g ` (mulGrp ` ℂfld ) ) = (.g ` (mulGrp ` ℂfld ) )
74	71, 72, 73	mhmmulg 17583	. . . . . . . . . . . . . . . . 17 |- ( ( f e. ( (mulGrp ` (ℤ/nℤ ` N ) ) MndHom (mulGrp ` ℂfld ) ) /\ ( phi ` N ) e. NN0 /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( f ` ( ( phi ` N ) (.g ` (mulGrp ` (ℤ/nℤ ` N ) ) ) x ) ) = ( ( phi ` N ) (.g ` (mulGrp ` ℂfld ) ) ( f ` x ) ) )
75	67, 68, 69, 74	syl3anc 1326	. . . . . . . . . . . . . . . 16 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( f ` ( ( phi ` N ) (.g ` (mulGrp ` (ℤ/nℤ ` N ) ) ) x ) ) = ( ( phi ` N ) (.g ` (mulGrp ` ℂfld ) ) ( f ` x ) ) )
76		nnnn0 11299	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( N e. NN -> N e. NN0 )
77	47	zncrng 19893	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( N e. NN0 -> (ℤ/nℤ ` N ) e. CRing )
78	76, 77	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( N e. NN -> (ℤ/nℤ ` N ) e. CRing )
79		crngring 18558	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( (ℤ/nℤ ` N ) e. CRing -> (ℤ/nℤ ` N ) e. Ring )
80	78, 79	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N e. NN -> (ℤ/nℤ ` N ) e. Ring )
81	80	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> (ℤ/nℤ ` N ) e. Ring )
82		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 |- (Unit ` (ℤ/nℤ ` N ) ) = (Unit ` (ℤ/nℤ ` N ) )
83		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) = ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) )
84	82, 83	unitgrp 18667	. . . . . . . . . . . . . . . . . . . . . 22 |- ( (ℤ/nℤ ` N ) e. Ring -> ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) e. Grp )
85	81, 84	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) e. Grp )
86	47, 82	znunithash 19913	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( N e. NN -> ( # ` (Unit ` (ℤ/nℤ ` N ) ) ) = ( phi ` N ) )
87	86, 14	eqeltrd 2701	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N e. NN -> ( # ` (Unit ` (ℤ/nℤ ` N ) ) ) e. NN0 )
88		fvex 6201	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (Unit ` (ℤ/nℤ ` N ) ) e. _V
89		hashclb 13149	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( (Unit ` (ℤ/nℤ ` N ) ) e. _V -> ( (Unit ` (ℤ/nℤ ` N ) ) e. Fin <-> ( # ` (Unit ` (ℤ/nℤ ` N ) ) ) e. NN0 ) )
90	88, 89	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( (Unit ` (ℤ/nℤ ` N ) ) e. Fin <-> ( # ` (Unit ` (ℤ/nℤ ` N ) ) ) e. NN0 )
91	87, 90	sylibr 224	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. NN -> (Unit ` (ℤ/nℤ ` N ) ) e. Fin )
92	91	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> (Unit ` (ℤ/nℤ ` N ) ) e. Fin )
93		simprr 796	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( f ` x ) =/= 0 )
94	52, 47, 53, 48, 82, 66, 69	dchrn0 24975	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( ( f ` x ) =/= 0 <-> x e. (Unit ` (ℤ/nℤ ` N ) ) ) )
95	93, 94	mpbid 222	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> x e. (Unit ` (ℤ/nℤ ` N ) ) )
96	82, 83	unitgrpbas 18666	. . . . . . . . . . . . . . . . . . . . . 22 |- (Unit ` (ℤ/nℤ ` N ) ) = ( Base ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) )
97		eqid 2622	. . . . . . . . . . . . . . . . . . . . . 22 |- ( od ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) = ( od ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) )
98	96, 97	oddvds2 17983	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) e. Grp /\ (Unit ` (ℤ/nℤ ` N ) ) e. Fin /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( od ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) ` x ) || ( # ` (Unit ` (ℤ/nℤ ` N ) ) ) )
99	85, 92, 95, 98	syl3anc 1326	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( ( od ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) ` x ) || ( # ` (Unit ` (ℤ/nℤ ` N ) ) ) )
100	86	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( # ` (Unit ` (ℤ/nℤ ` N ) ) ) = ( phi ` N ) )
101	99, 100	breqtrd 4679	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( ( od ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) ` x ) || ( phi ` N ) )
102	13	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( phi ` N ) e. NN )
103	102	nnzd 11481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( phi ` N ) e. ZZ )
104		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 |- (.g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) = (.g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) )
105		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) = ( 0g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) )
106	96, 97, 104, 105	oddvds 17966	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) e. Grp /\ x e. (Unit ` (ℤ/nℤ ` N ) ) /\ ( phi ` N ) e. ZZ ) -> ( ( ( od ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) ` x ) || ( phi ` N ) <-> ( ( phi ` N ) (.g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) x ) = ( 0g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) ) )
107	85, 95, 103, 106	syl3anc 1326	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( ( ( od ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) ` x ) || ( phi ` N ) <-> ( ( phi ` N ) (.g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) x ) = ( 0g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) ) )
108	101, 107	mpbid 222	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( ( phi ` N ) (.g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) x ) = ( 0g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) )
109	82, 70	unitsubm 18670	. . . . . . . . . . . . . . . . . . . 20 |- ( (ℤ/nℤ ` N ) e. Ring -> (Unit ` (ℤ/nℤ ` N ) ) e. (SubMnd ` (mulGrp ` (ℤ/nℤ ` N ) ) ) )
110	81, 109	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> (Unit ` (ℤ/nℤ ` N ) ) e. (SubMnd ` (mulGrp ` (ℤ/nℤ ` N ) ) ) )
111	72, 83, 104	submmulg 17586	. . . . . . . . . . . . . . . . . . 19 |- ( ( (Unit ` (ℤ/nℤ ` N ) ) e. (SubMnd ` (mulGrp ` (ℤ/nℤ ` N ) ) ) /\ ( phi ` N ) e. NN0 /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( phi ` N ) (.g ` (mulGrp ` (ℤ/nℤ ` N ) ) ) x ) = ( ( phi ` N ) (.g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) x ) )
112	110, 68, 95, 111	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( ( phi ` N ) (.g ` (mulGrp ` (ℤ/nℤ ` N ) ) ) x ) = ( ( phi ` N ) (.g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) x ) )
113		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 |- ( 1r ` (ℤ/nℤ ` N ) ) = ( 1r ` (ℤ/nℤ ` N ) )
114	70, 113	ringidval 18503	. . . . . . . . . . . . . . . . . . . 20 |- ( 1r ` (ℤ/nℤ ` N ) ) = ( 0g ` (mulGrp ` (ℤ/nℤ ` N ) ) )
115	83, 114	subm0 17356	. . . . . . . . . . . . . . . . . . 19 |- ( (Unit ` (ℤ/nℤ ` N ) ) e. (SubMnd ` (mulGrp ` (ℤ/nℤ ` N ) ) ) -> ( 1r ` (ℤ/nℤ ` N ) ) = ( 0g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) )
116	110, 115	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( 1r ` (ℤ/nℤ ` N ) ) = ( 0g ` ( (mulGrp ` (ℤ/nℤ ` N ) ) ↾s (Unit ` (ℤ/nℤ ` N ) ) ) ) )
117	108, 112, 116	3eqtr4d 2666	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( ( phi ` N ) (.g ` (mulGrp ` (ℤ/nℤ ` N ) ) ) x ) = ( 1r ` (ℤ/nℤ ` N ) ) )
118	117	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( f ` ( ( phi ` N ) (.g ` (mulGrp ` (ℤ/nℤ ` N ) ) ) x ) ) = ( f ` ( 1r ` (ℤ/nℤ ` N ) ) ) )
119	75, 118	eqtr3d 2658	. . . . . . . . . . . . . . 15 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( ( phi ` N ) (.g ` (mulGrp ` ℂfld ) ) ( f ` x ) ) = ( f ` ( 1r ` (ℤ/nℤ ` N ) ) ) )
120		cnfldexp 19779	. . . . . . . . . . . . . . . 16 |- ( ( ( f ` x ) e. CC /\ ( phi ` N ) e. NN0 ) -> ( ( phi ` N ) (.g ` (mulGrp ` ℂfld ) ) ( f ` x ) ) = ( ( f ` x ) ^ ( phi ` N ) ) )
121	64, 68, 120	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( ( phi ` N ) (.g ` (mulGrp ` ℂfld ) ) ( f ` x ) ) = ( ( f ` x ) ^ ( phi ` N ) ) )
122		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
123		cnfld1 19771	. . . . . . . . . . . . . . . . . 18 |- 1 = ( 1r ` ℂfld )
124	122, 123	ringidval 18503	. . . . . . . . . . . . . . . . 17 |- 1 = ( 0g ` (mulGrp ` ℂfld ) )
125	114, 124	mhm0 17343	. . . . . . . . . . . . . . . 16 |- ( f e. ( (mulGrp ` (ℤ/nℤ ` N ) ) MndHom (mulGrp ` ℂfld ) ) -> ( f ` ( 1r ` (ℤ/nℤ ` N ) ) ) = 1 )
126	67, 125	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( f ` ( 1r ` (ℤ/nℤ ` N ) ) ) = 1 )
127	119, 121, 126	3eqtr3d 2664	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( ( f ` x ) ^ ( phi ` N ) ) = 1 )
128	127	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( ( ( f ` x ) ^ ( phi ` N ) ) - 1 ) = ( 1 - 1 ) )
129		1m1e0 11089	. . . . . . . . . . . . 13 |- ( 1 - 1 ) = 0
130	128, 129	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( ( ( f ` x ) ^ ( phi ` N ) ) - 1 ) = 0 )
131		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( z = ( f ` x ) -> ( z ^ ( phi ` N ) ) = ( ( f ` x ) ^ ( phi ` N ) ) )
132	131	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( z = ( f ` x ) -> ( ( z ^ ( phi ` N ) ) - 1 ) = ( ( ( f ` x ) ^ ( phi ` N ) ) - 1 ) )
133	132	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( z = ( f ` x ) -> ( ( ( z ^ ( phi ` N ) ) - 1 ) = 0 <-> ( ( ( f ` x ) ^ ( phi ` N ) ) - 1 ) = 0 ) )
134	133	elrab 3363	. . . . . . . . . . . 12 |- ( ( f ` x ) e. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } <-> ( ( f ` x ) e. CC /\ ( ( ( f ` x ) ^ ( phi ` N ) ) - 1 ) = 0 ) )
135	64, 130, 134	sylanbrc 698	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ f e. D ) /\ ( x e. ( Base ` (ℤ/nℤ ` N ) ) /\ ( f ` x ) =/= 0 ) ) -> ( f ` x ) e. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } )
136	135	expr 643	. . . . . . . . . 10 |- ( ( ( N e. NN /\ f e. D ) /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( f ` x ) =/= 0 -> ( f ` x ) e. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) )
137	61, 136	syl5bir 233	. . . . . . . . 9 |- ( ( ( N e. NN /\ f e. D ) /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( -. ( f ` x ) e. { 0 } -> ( f ` x ) e. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) )
138	137	orrd 393	. . . . . . . 8 |- ( ( ( N e. NN /\ f e. D ) /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( f ` x ) e. { 0 } \/ ( f ` x ) e. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) )
139		elun 3753	. . . . . . . 8 |- ( ( f ` x ) e. ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) <-> ( ( f ` x ) e. { 0 } \/ ( f ` x ) e. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) )
140	138, 139	sylibr 224	. . . . . . 7 |- ( ( ( N e. NN /\ f e. D ) /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( f ` x ) e. ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) )
141	140	ralrimiva 2966	. . . . . 6 |- ( ( N e. NN /\ f e. D ) -> A. x e. ( Base ` (ℤ/nℤ ` N ) ) ( f ` x ) e. ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) )
142		ffnfv 6388	. . . . . 6 |- ( f : ( Base ` (ℤ/nℤ ` N ) ) --> ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) <-> ( f Fn ( Base ` (ℤ/nℤ ` N ) ) /\ A. x e. ( Base ` (ℤ/nℤ ` N ) ) ( f ` x ) e. ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) ) )
143	57, 141, 142	sylanbrc 698	. . . . 5 |- ( ( N e. NN /\ f e. D ) -> f : ( Base ` (ℤ/nℤ ` N ) ) --> ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) )
144	143	ex 450	. . . 4 |- ( N e. NN -> ( f e. D -> f : ( Base ` (ℤ/nℤ ` N ) ) --> ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) ) )
145	46, 49	elmapd 7871	. . . 4 |- ( N e. NN -> ( f e. ( ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) ^m ( Base ` (ℤ/nℤ ` N ) ) ) <-> f : ( Base ` (ℤ/nℤ ` N ) ) --> ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) ) )
146	144, 145	sylibrd 249	. . 3 |- ( N e. NN -> ( f e. D -> f e. ( ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) ^m ( Base ` (ℤ/nℤ ` N ) ) ) ) )
147	146	ssrdv 3609	. 2 |- ( N e. NN -> D C_ ( ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) ^m ( Base ` (ℤ/nℤ ` N ) ) ) )
148		ssfi 8180	. 2 |- ( ( ( ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) ^m ( Base ` (ℤ/nℤ ` N ) ) ) e. Fin /\ D C_ ( ( { 0 } u. { z e. CC | ( ( z ^ ( phi ` N ) ) - 1 ) = 0 } ) ^m ( Base ` (ℤ/nℤ ` N ) ) ) ) -> D e. Fin )
149	51, 147, 148	syl2anc 693	1 |- ( N e. NN -> D e. Fin )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   u. cun 3572   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   ^m cmap 7857   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   <_ cle 10075   - cmin 10266   -ucneg 10267   NNcn 11020   NN0cn0 11292   ZZcz 11377   ^cexp 12860   #chash 13117   || cdvds 14983   phicphi 15469   Basecbs 15857   ↾_s cress 15858   0gc0g 16100   MndHom cmhm 17333  SubMndcsubmnd 17334   Grpcgrp 17422  ._gcmg 17540   odcod 17944  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547   CRingccrg 18548  Unitcui 18639  ℂ_fldccnfld 19746  ℤ/nℤczn 19851   0pc0p 23436  Polycply 23940  degcdgr 23943  DChrcdchr 24957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-dvds 14984  df-gcd 15217  df-phi 15471  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-imas 16168  df-qus 16169  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-nsg 17592  df-eqg 17593  df-ghm 17658  df-od 17948  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rsp 19175  df-2idl 19232  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-zn 19855  df-0p 23437  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947  df-quot 24046  df-dchr 24958

This theorem is referenced by:  sumdchr2  24995  dchrhash  24996  rpvmasum2  25201  dchrisum0re  25202