Theorem dchrinv 21050

Description: The inverse of a Dirichlet character is the conjugate (which is also the multiplicative inverse, because the values of X are unimodular). (Contributed by Mario Carneiro, 28-Apr-2016.)

Hypotheses

Ref	Expression
dchrabs.g	|- G = (DChr ` N )
dchrabs.d	|- D = ( Base ` G )
dchrabs.x	|- ( ph -> X e. D )
dchrinv.i	|- I = ( inv g ` G )

Assertion

Ref	Expression
dchrinv	|- ( ph -> ( I ` X ) = ( * o. X ) )

Proof of Theorem dchrinv

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dchrabs.g	. . . . . . . 8 |- G = (DChr ` N )
2		eqid 2438	. . . . . . . 8 |- (ℤ/nℤ ` N ) = (ℤ/nℤ ` N )
3		dchrabs.d	. . . . . . . 8 |- D = ( Base ` G )
4		eqid 2438	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
5		dchrabs.x	. . . . . . . 8 |- ( ph -> X e. D )
6		cjf 11914	. . . . . . . . . 10 |- * : CC --> CC
7		eqid 2438	. . . . . . . . . . 11 |- ( Base ` (ℤ/nℤ ` N ) ) = ( Base ` (ℤ/nℤ ` N ) )
8	1, 2, 3, 7, 5	dchrf 21031	. . . . . . . . . 10 |- ( ph -> X : ( Base ` (ℤ/nℤ ` N ) ) --> CC )
9		fco 5603	. . . . . . . . . 10 |- ( ( * : CC --> CC /\ X : ( Base ` (ℤ/nℤ ` N ) ) --> CC ) -> ( * o. X ) : ( Base ` (ℤ/nℤ ` N ) ) --> CC )
10	6, 8, 9	sylancr 646	. . . . . . . . 9 |- ( ph -> ( * o. X ) : ( Base ` (ℤ/nℤ ` N ) ) --> CC )
11		eqid 2438	. . . . . . . . . . . . . . . . . . . . 21 |- (Unit ` (ℤ/nℤ ` N ) ) = (Unit ` (ℤ/nℤ ` N ) )
12	1, 3	dchrrcl 21029	. . . . . . . . . . . . . . . . . . . . . 22 |- ( X e. D -> N e. NN )
13	5, 12	syl 16	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> N e. NN )
14	1, 2, 7, 11, 13, 3	dchrelbas3 21027	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( X e. D <-> ( X : ( Base ` (ℤ/nℤ ` N ) ) --> CC /\ ( A. x e. (Unit ` (ℤ/nℤ ` N ) ) A. y e. (Unit ` (ℤ/nℤ ` N ) ) ( X ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) /\ ( X ` ( 1r ` (ℤ/nℤ ` N ) ) ) = 1 /\ A. x e. ( Base ` (ℤ/nℤ ` N ) ) ( ( X ` x ) =/= 0 -> x e. (Unit ` (ℤ/nℤ ` N ) ) ) ) ) ) )
15	5, 14	mpbid 203	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( X : ( Base ` (ℤ/nℤ ` N ) ) --> CC /\ ( A. x e. (Unit ` (ℤ/nℤ ` N ) ) A. y e. (Unit ` (ℤ/nℤ ` N ) ) ( X ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) /\ ( X ` ( 1r ` (ℤ/nℤ ` N ) ) ) = 1 /\ A. x e. ( Base ` (ℤ/nℤ ` N ) ) ( ( X ` x ) =/= 0 -> x e. (Unit ` (ℤ/nℤ ` N ) ) ) ) ) )
16	15	simprd 451	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( A. x e. (Unit ` (ℤ/nℤ ` N ) ) A. y e. (Unit ` (ℤ/nℤ ` N ) ) ( X ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) /\ ( X ` ( 1r ` (ℤ/nℤ ` N ) ) ) = 1 /\ A. x e. ( Base ` (ℤ/nℤ ` N ) ) ( ( X ` x ) =/= 0 -> x e. (Unit ` (ℤ/nℤ ` N ) ) ) ) )
17	16	simp1d 970	. . . . . . . . . . . . . . . . 17 |- ( ph -> A. x e. (Unit ` (ℤ/nℤ ` N ) ) A. y e. (Unit ` (ℤ/nℤ ` N ) ) ( X ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) )
18	17	r19.21bi 2806	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> A. y e. (Unit ` (ℤ/nℤ ` N ) ) ( X ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) )
19	18	r19.21bi 2806	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( X ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) )
20	19	anasss 630	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( X ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) )
21	20	fveq2d 5735	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( * ` ( X ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) ) = ( * ` ( ( X ` x ) x. ( X ` y ) ) ) )
22	8	adantr 453	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> X : ( Base ` (ℤ/nℤ ` N ) ) --> CC )
23	7, 11	unitss 15770	. . . . . . . . . . . . . . . 16 |- (Unit ` (ℤ/nℤ ` N ) ) C_ ( Base ` (ℤ/nℤ ` N ) )
24		simprl 734	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> x e. (Unit ` (ℤ/nℤ ` N ) ) )
25	23, 24	sseldi 3348	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> x e. ( Base ` (ℤ/nℤ ` N ) ) )
26	22, 25	ffvelrnd 5874	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( X ` x ) e. CC )
27		simprr 735	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> y e. (Unit ` (ℤ/nℤ ` N ) ) )
28	23, 27	sseldi 3348	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> y e. ( Base ` (ℤ/nℤ ` N ) ) )
29	22, 28	ffvelrnd 5874	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( X ` y ) e. CC )
30	26, 29	cjmuld 12031	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( * ` ( ( X ` x ) x. ( X ` y ) ) ) = ( ( * ` ( X ` x ) ) x. ( * ` ( X ` y ) ) ) )
31	21, 30	eqtrd 2470	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( * ` ( X ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) ) = ( ( * ` ( X ` x ) ) x. ( * ` ( X ` y ) ) ) )
32	13	nnnn0d 10279	. . . . . . . . . . . . . . . 16 |- ( ph -> N e. NN0 )
33	2	zncrng 16830	. . . . . . . . . . . . . . . 16 |- ( N e. NN0 -> (ℤ/nℤ ` N ) e. CRing )
34		crngrng 15679	. . . . . . . . . . . . . . . 16 |- ( (ℤ/nℤ ` N ) e. CRing -> (ℤ/nℤ ` N ) e. Ring )
35	32, 33, 34	3syl 19	. . . . . . . . . . . . . . 15 |- ( ph -> (ℤ/nℤ ` N ) e. Ring )
36	35	adantr 453	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> (ℤ/nℤ ` N ) e. Ring )
37		eqid 2438	. . . . . . . . . . . . . . 15 |- ( .r ` (ℤ/nℤ ` N ) ) = ( .r ` (ℤ/nℤ ` N ) )
38	7, 37	rngcl 15682	. . . . . . . . . . . . . 14 |- ( ( (ℤ/nℤ ` N ) e. Ring /\ x e. ( Base ` (ℤ/nℤ ` N ) ) /\ y e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( x ( .r ` (ℤ/nℤ ` N ) ) y ) e. ( Base ` (ℤ/nℤ ` N ) ) )
39	36, 25, 28, 38	syl3anc 1185	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( x ( .r ` (ℤ/nℤ ` N ) ) y ) e. ( Base ` (ℤ/nℤ ` N ) ) )
40		fvco3 5803	. . . . . . . . . . . . 13 |- ( ( X : ( Base ` (ℤ/nℤ ` N ) ) --> CC /\ ( x ( .r ` (ℤ/nℤ ` N ) ) y ) e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( * o. X ) ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) = ( * ` ( X ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) ) )
41	22, 39, 40	syl2anc 644	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( ( * o. X ) ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) = ( * ` ( X ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) ) )
42		fvco3 5803	. . . . . . . . . . . . . 14 |- ( ( X : ( Base ` (ℤ/nℤ ` N ) ) --> CC /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( * o. X ) ` x ) = ( * ` ( X ` x ) ) )
43	22, 25, 42	syl2anc 644	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( ( * o. X ) ` x ) = ( * ` ( X ` x ) ) )
44		fvco3 5803	. . . . . . . . . . . . . 14 |- ( ( X : ( Base ` (ℤ/nℤ ` N ) ) --> CC /\ y e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( * o. X ) ` y ) = ( * ` ( X ` y ) ) )
45	22, 28, 44	syl2anc 644	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( ( * o. X ) ` y ) = ( * ` ( X ` y ) ) )
46	43, 45	oveq12d 6102	. . . . . . . . . . . 12 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( ( ( * o. X ) ` x ) x. ( ( * o. X ) ` y ) ) = ( ( * ` ( X ` x ) ) x. ( * ` ( X ` y ) ) ) )
47	31, 41, 46	3eqtr4d 2480	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( ( * o. X ) ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) = ( ( ( * o. X ) ` x ) x. ( ( * o. X ) ` y ) ) )
48	47	ralrimivva 2800	. . . . . . . . . 10 |- ( ph -> A. x e. (Unit ` (ℤ/nℤ ` N ) ) A. y e. (Unit ` (ℤ/nℤ ` N ) ) ( ( * o. X ) ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) = ( ( ( * o. X ) ` x ) x. ( ( * o. X ) ` y ) ) )
49		eqid 2438	. . . . . . . . . . . . . 14 |- ( 1r ` (ℤ/nℤ ` N ) ) = ( 1r ` (ℤ/nℤ ` N ) )
50	7, 49	rngidcl 15689	. . . . . . . . . . . . 13 |- ( (ℤ/nℤ ` N ) e. Ring -> ( 1r ` (ℤ/nℤ ` N ) ) e. ( Base ` (ℤ/nℤ ` N ) ) )
51	35, 50	syl 16	. . . . . . . . . . . 12 |- ( ph -> ( 1r ` (ℤ/nℤ ` N ) ) e. ( Base ` (ℤ/nℤ ` N ) ) )
52		fvco3 5803	. . . . . . . . . . . 12 |- ( ( X : ( Base ` (ℤ/nℤ ` N ) ) --> CC /\ ( 1r ` (ℤ/nℤ ` N ) ) e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( * o. X ) ` ( 1r ` (ℤ/nℤ ` N ) ) ) = ( * ` ( X ` ( 1r ` (ℤ/nℤ ` N ) ) ) ) )
53	8, 51, 52	syl2anc 644	. . . . . . . . . . 11 |- ( ph -> ( ( * o. X ) ` ( 1r ` (ℤ/nℤ ` N ) ) ) = ( * ` ( X ` ( 1r ` (ℤ/nℤ ` N ) ) ) ) )
54	16	simp2d 971	. . . . . . . . . . . . 13 |- ( ph -> ( X ` ( 1r ` (ℤ/nℤ ` N ) ) ) = 1 )
55	54	fveq2d 5735	. . . . . . . . . . . 12 |- ( ph -> ( * ` ( X ` ( 1r ` (ℤ/nℤ ` N ) ) ) ) = ( * ` 1 ) )
56		1re 9095	. . . . . . . . . . . . 13 |- 1 e. RR
57		cjre 11949	. . . . . . . . . . . . 13 |- ( 1 e. RR -> ( * ` 1 ) = 1 )
58	56, 57	ax-mp 5	. . . . . . . . . . . 12 |- ( * ` 1 ) = 1
59	55, 58	syl6eq 2486	. . . . . . . . . . 11 |- ( ph -> ( * ` ( X ` ( 1r ` (ℤ/nℤ ` N ) ) ) ) = 1 )
60	53, 59	eqtrd 2470	. . . . . . . . . 10 |- ( ph -> ( ( * o. X ) ` ( 1r ` (ℤ/nℤ ` N ) ) ) = 1 )
61	16	simp3d 972	. . . . . . . . . . 11 |- ( ph -> A. x e. ( Base ` (ℤ/nℤ ` N ) ) ( ( X ` x ) =/= 0 -> x e. (Unit ` (ℤ/nℤ ` N ) ) ) )
62	8, 42	sylan 459	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( * o. X ) ` x ) = ( * ` ( X ` x ) ) )
63		cj0 11968	. . . . . . . . . . . . . . . . . 18 |- ( * ` 0 ) = 0
64	63	eqcomi 2442	. . . . . . . . . . . . . . . . 17 |- 0 = ( * ` 0 )
65	64	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> 0 = ( * ` 0 ) )
66	62, 65	eqeq12d 2452	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( ( * o. X ) ` x ) = 0 <-> ( * ` ( X ` x ) ) = ( * ` 0 ) ) )
67	8	ffvelrnda 5873	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( X ` x ) e. CC )
68		0cn 9089	. . . . . . . . . . . . . . . 16 |- 0 e. CC
69		cj11 11972	. . . . . . . . . . . . . . . 16 |- ( ( ( X ` x ) e. CC /\ 0 e. CC ) -> ( ( * ` ( X ` x ) ) = ( * ` 0 ) <-> ( X ` x ) = 0 ) )
70	67, 68, 69	sylancl 645	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( * ` ( X ` x ) ) = ( * ` 0 ) <-> ( X ` x ) = 0 ) )
71	66, 70	bitrd 246	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( ( * o. X ) ` x ) = 0 <-> ( X ` x ) = 0 ) )
72	71	necon3bid 2638	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( ( * o. X ) ` x ) =/= 0 <-> ( X ` x ) =/= 0 ) )
73	72	imbi1d 310	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( ( ( * o. X ) ` x ) =/= 0 -> x e. (Unit ` (ℤ/nℤ ` N ) ) ) <-> ( ( X ` x ) =/= 0 -> x e. (Unit ` (ℤ/nℤ ` N ) ) ) ) )
74	73	ralbidva 2723	. . . . . . . . . . 11 |- ( ph -> ( A. x e. ( Base ` (ℤ/nℤ ` N ) ) ( ( ( * o. X ) ` x ) =/= 0 -> x e. (Unit ` (ℤ/nℤ ` N ) ) ) <-> A. x e. ( Base ` (ℤ/nℤ ` N ) ) ( ( X ` x ) =/= 0 -> x e. (Unit ` (ℤ/nℤ ` N ) ) ) ) )
75	61, 74	mpbird 225	. . . . . . . . . 10 |- ( ph -> A. x e. ( Base ` (ℤ/nℤ ` N ) ) ( ( ( * o. X ) ` x ) =/= 0 -> x e. (Unit ` (ℤ/nℤ ` N ) ) ) )
76	48, 60, 75	3jca 1135	. . . . . . . . 9 |- ( ph -> ( A. x e. (Unit ` (ℤ/nℤ ` N ) ) A. y e. (Unit ` (ℤ/nℤ ` N ) ) ( ( * o. X ) ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) = ( ( ( * o. X ) ` x ) x. ( ( * o. X ) ` y ) ) /\ ( ( * o. X ) ` ( 1r ` (ℤ/nℤ ` N ) ) ) = 1 /\ A. x e. ( Base ` (ℤ/nℤ ` N ) ) ( ( ( * o. X ) ` x ) =/= 0 -> x e. (Unit ` (ℤ/nℤ ` N ) ) ) ) )
77	1, 2, 7, 11, 13, 3	dchrelbas3 21027	. . . . . . . . 9 |- ( ph -> ( ( * o. X ) e. D <-> ( ( * o. X ) : ( Base ` (ℤ/nℤ ` N ) ) --> CC /\ ( A. x e. (Unit ` (ℤ/nℤ ` N ) ) A. y e. (Unit ` (ℤ/nℤ ` N ) ) ( ( * o. X ) ` ( x ( .r ` (ℤ/nℤ ` N ) ) y ) ) = ( ( ( * o. X ) ` x ) x. ( ( * o. X ) ` y ) ) /\ ( ( * o. X ) ` ( 1r ` (ℤ/nℤ ` N ) ) ) = 1 /\ A. x e. ( Base ` (ℤ/nℤ ` N ) ) ( ( ( * o. X ) ` x ) =/= 0 -> x e. (Unit ` (ℤ/nℤ ` N ) ) ) ) ) ) )
78	10, 76, 77	mpbir2and 890	. . . . . . . 8 |- ( ph -> ( * o. X ) e. D )
79	1, 2, 3, 4, 5, 78	dchrmul 21037	. . . . . . 7 |- ( ph -> ( X ( +g ` G ) ( * o. X ) ) = ( X o F x. ( * o. X ) ) )
80	79	adantr 453	. . . . . 6 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( X ( +g ` G ) ( * o. X ) ) = ( X o F x. ( * o. X ) ) )
81	80	fveq1d 5733	. . . . 5 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( X ( +g ` G ) ( * o. X ) ) ` x ) = ( ( X o F x. ( * o. X ) ) ` x ) )
82	23	sseli 3346	. . . . . . . . 9 |- ( x e. (Unit ` (ℤ/nℤ ` N ) ) -> x e. ( Base ` (ℤ/nℤ ` N ) ) )
83	82, 62	sylan2 462	. . . . . . . 8 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( * o. X ) ` x ) = ( * ` ( X ` x ) ) )
84	83	oveq2d 6100	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( X ` x ) x. ( ( * o. X ) ` x ) ) = ( ( X ` x ) x. ( * ` ( X ` x ) ) ) )
85	82, 67	sylan2 462	. . . . . . . 8 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( X ` x ) e. CC )
86	85	absvalsqd 12249	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( abs ` ( X ` x ) ) ^ 2 ) = ( ( X ` x ) x. ( * ` ( X ` x ) ) ) )
87	5	adantr 453	. . . . . . . . . 10 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> X e. D )
88		simpr 449	. . . . . . . . . 10 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> x e. (Unit ` (ℤ/nℤ ` N ) ) )
89	1, 3, 87, 2, 11, 88	dchrabs 21049	. . . . . . . . 9 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( abs ` ( X ` x ) ) = 1 )
90	89	oveq1d 6099	. . . . . . . 8 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( abs ` ( X ` x ) ) ^ 2 ) = ( 1 ^ 2 ) )
91		sq1 11481	. . . . . . . 8 |- ( 1 ^ 2 ) = 1
92	90, 91	syl6eq 2486	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( abs ` ( X ` x ) ) ^ 2 ) = 1 )
93	84, 86, 92	3eqtr2d 2476	. . . . . 6 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( X ` x ) x. ( ( * o. X ) ` x ) ) = 1 )
94	8	adantr 453	. . . . . . . 8 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> X : ( Base ` (ℤ/nℤ ` N ) ) --> CC )
95		ffn 5594	. . . . . . . 8 |- ( X : ( Base ` (ℤ/nℤ ` N ) ) --> CC -> X Fn ( Base ` (ℤ/nℤ ` N ) ) )
96	94, 95	syl 16	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> X Fn ( Base ` (ℤ/nℤ ` N ) ) )
97		ffn 5594	. . . . . . . . 9 |- ( ( * o. X ) : ( Base ` (ℤ/nℤ ` N ) ) --> CC -> ( * o. X ) Fn ( Base ` (ℤ/nℤ ` N ) ) )
98	10, 97	syl 16	. . . . . . . 8 |- ( ph -> ( * o. X ) Fn ( Base ` (ℤ/nℤ ` N ) ) )
99	98	adantr 453	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( * o. X ) Fn ( Base ` (ℤ/nℤ ` N ) ) )
100		fvex 5745	. . . . . . . 8 |- ( Base ` (ℤ/nℤ ` N ) ) e. _V
101	100	a1i 11	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( Base ` (ℤ/nℤ ` N ) ) e. _V )
102	82	adantl 454	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> x e. ( Base ` (ℤ/nℤ ` N ) ) )
103		fnfvof 6320	. . . . . . 7 |- ( ( ( X Fn ( Base ` (ℤ/nℤ ` N ) ) /\ ( * o. X ) Fn ( Base ` (ℤ/nℤ ` N ) ) ) /\ ( ( Base ` (ℤ/nℤ ` N ) ) e. _V /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) ) -> ( ( X o F x. ( * o. X ) ) ` x ) = ( ( X ` x ) x. ( ( * o. X ) ` x ) ) )
104	96, 99, 101, 102, 103	syl22anc 1186	. . . . . 6 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( X o F x. ( * o. X ) ) ` x ) = ( ( X ` x ) x. ( ( * o. X ) ` x ) ) )
105		eqid 2438	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
106	13	adantr 453	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> N e. NN )
107	1, 2, 105, 11, 106, 88	dchr1 21046	. . . . . 6 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( 0g ` G ) ` x ) = 1 )
108	93, 104, 107	3eqtr4d 2480	. . . . 5 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( X o F x. ( * o. X ) ) ` x ) = ( ( 0g ` G ) ` x ) )
109	81, 108	eqtrd 2470	. . . 4 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( X ( +g ` G ) ( * o. X ) ) ` x ) = ( ( 0g ` G ) ` x ) )
110	109	ralrimiva 2791	. . 3 |- ( ph -> A. x e. (Unit ` (ℤ/nℤ ` N ) ) ( ( X ( +g ` G ) ( * o. X ) ) ` x ) = ( ( 0g ` G ) ` x ) )
111	1, 2, 3, 4, 5, 78	dchrmulcl 21038	. . . 4 |- ( ph -> ( X ( +g ` G ) ( * o. X ) ) e. D )
112	1	dchrabl 21043	. . . . . 6 |- ( N e. NN -> G e. Abel )
113		ablgrp 15422	. . . . . 6 |- ( G e. Abel -> G e. Grp )
114	13, 112, 113	3syl 19	. . . . 5 |- ( ph -> G e. Grp )
115	3, 105	grpidcl 14838	. . . . 5 |- ( G e. Grp -> ( 0g ` G ) e. D )
116	114, 115	syl 16	. . . 4 |- ( ph -> ( 0g ` G ) e. D )
117	1, 2, 3, 11, 111, 116	dchreq 21047	. . 3 |- ( ph -> ( ( X ( +g ` G ) ( * o. X ) ) = ( 0g ` G ) <-> A. x e. (Unit ` (ℤ/nℤ ` N ) ) ( ( X ( +g ` G ) ( * o. X ) ) ` x ) = ( ( 0g ` G ) ` x ) ) )
118	110, 117	mpbird 225	. 2 |- ( ph -> ( X ( +g ` G ) ( * o. X ) ) = ( 0g ` G ) )
119		dchrinv.i	. . . 4 |- I = ( inv g ` G )
120	3, 4, 105, 119	grpinvid1 14858	. . 3 |- ( ( G e. Grp /\ X e. D /\ ( * o. X ) e. D ) -> ( ( I ` X ) = ( * o. X ) <-> ( X ( +g ` G ) ( * o. X ) ) = ( 0g ` G ) ) )
121	114, 5, 78, 120	syl3anc 1185	. 2 |- ( ph -> ( ( I ` X ) = ( * o. X ) <-> ( X ( +g ` G ) ( * o. X ) ) = ( 0g ` G ) ) )
122	118, 121	mpbird 225	1 |- ( ph -> ( I ` X ) = ( * o. X ) )

Syntax hints:   -> wi 4   <-> wb 178   /\ wa 360   /\ w3a 937   = wceq 1653   e. wcel 1726   =/= wne 2601   A.wral 2707   _Vcvv 2958   o. ccom 4885   Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084   o Fcof 6306   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996   x. cmul 9000   NNcn 10005   2c2 10054   NN0cn0 10226   ^cexp 11387   *ccj 11906   abscabs 12044   Basecbs 13474   +g cplusg 13534   .rcmulr 13535   0gc0g 13728   Grpcgrp 14690   inv gcminusg 14691   Abelcabel 15418   Ringcrg 15665   CRingccrg 15666   1rcur 15667  Unitcui 15749  ℤ/nℤczn 16786  DChrcdchr 21021

This theorem is referenced by:  dchr2sum  21062  dchrisum0re  21212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-disj 4186  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-omul 6732  df-er 6908  df-ec 6910  df-qs 6914  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-acn 7834  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ioc 10926  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-sum 12485  df-ef 12675  df-sin 12677  df-cos 12678  df-pi 12680  df-dvds 12858  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-divs 13740  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-mhm 14743  df-submnd 14744  df-grp 14817  df-minusg 14818  df-sbg 14819  df-mulg 14820  df-subg 14946  df-nsg 14947  df-eqg 14948  df-ghm 15009  df-cntz 15121  df-od 15172  df-cmn 15419  df-abl 15420  df-mgp 15654  df-rng 15668  df-cring 15669  df-ur 15670  df-oppr 15733  df-dvdsr 15751  df-unit 15752  df-invr 15782  df-dvr 15793  df-rnghom 15824  df-drng 15842  df-subrg 15871  df-lmod 15957  df-lss 16014  df-lsp 16053  df-sra 16249  df-rgmod 16250  df-lidl 16251  df-rsp 16252  df-2idl 16308  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-zrh 16787  df-zn 16790  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-cn 17296  df-cnp 17297  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cncf 18913  df-limc 19758  df-dv 19759  df-log 20459  df-cxp 20460  df-dchr 21022