Theorem dchrinv 24986

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 = ( invg ` 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 2622	. . . . . . . 8 |- (ℤ/nℤ ` N ) = (ℤ/nℤ ` N )
3		dchrabs.d	. . . . . . . 8 |- D = ( Base ` G )
4		eqid 2622	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
5		dchrabs.x	. . . . . . . 8 |- ( ph -> X e. D )
6		cjf 13844	. . . . . . . . . 10 |- * : CC --> CC
7		eqid 2622	. . . . . . . . . . 11 |- ( Base ` (ℤ/nℤ ` N ) ) = ( Base ` (ℤ/nℤ ` N ) )
8	1, 2, 3, 7, 5	dchrf 24967	. . . . . . . . . 10 |- ( ph -> X : ( Base ` (ℤ/nℤ ` N ) ) --> CC )
9		fco 6058	. . . . . . . . . 10 |- ( ( * : CC --> CC /\ X : ( Base ` (ℤ/nℤ ` N ) ) --> CC ) -> ( * o. X ) : ( Base ` (ℤ/nℤ ` N ) ) --> CC )
10	6, 8, 9	sylancr 695	. . . . . . . . 9 |- ( ph -> ( * o. X ) : ( Base ` (ℤ/nℤ ` N ) ) --> CC )
11		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 |- (Unit ` (ℤ/nℤ ` N ) ) = (Unit ` (ℤ/nℤ ` N ) )
12	1, 3	dchrrcl 24965	. . . . . . . . . . . . . . . . . . . . . 22 |- ( X e. D -> N e. NN )
13	5, 12	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> N e. NN )
14	1, 2, 7, 11, 13, 3	dchrelbas3 24963	. . . . . . . . . . . . . . . . . . . 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 222	. . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 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 1073	. . . . . . . . . . . . . . . . 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 2932	. . . . . . . . . . . . . . . 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 2932	. . . . . . . . . . . . . . 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 679	. . . . . . . . . . . . . 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 6195	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> X : ( Base ` (ℤ/nℤ ` N ) ) --> CC )
23	7, 11	unitss 18660	. . . . . . . . . . . . . . . 16 |- (Unit ` (ℤ/nℤ ` N ) ) C_ ( Base ` (ℤ/nℤ ` N ) )
24		simprl 794	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> x e. (Unit ` (ℤ/nℤ ` N ) ) )
25	23, 24	sseldi 3601	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> x e. ( Base ` (ℤ/nℤ ` N ) ) )
26	22, 25	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( X ` x ) e. CC )
27		simprr 796	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> y e. (Unit ` (ℤ/nℤ ` N ) ) )
28	23, 27	sseldi 3601	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> y e. ( Base ` (ℤ/nℤ ` N ) ) )
29	22, 28	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( X ` y ) e. CC )
30	26, 29	cjmuld 13961	. . . . . . . . . . . . 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 2656	. . . . . . . . . . . 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 11351	. . . . . . . . . . . . . . . 16 |- ( ph -> N e. NN0 )
33	2	zncrng 19893	. . . . . . . . . . . . . . . 16 |- ( N e. NN0 -> (ℤ/nℤ ` N ) e. CRing )
34		crngring 18558	. . . . . . . . . . . . . . . 16 |- ( (ℤ/nℤ ` N ) e. CRing -> (ℤ/nℤ ` N ) e. Ring )
35	32, 33, 34	3syl 18	. . . . . . . . . . . . . . 15 |- ( ph -> (ℤ/nℤ ` N ) e. Ring )
36	35	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> (ℤ/nℤ ` N ) e. Ring )
37		eqid 2622	. . . . . . . . . . . . . . 15 |- ( .r ` (ℤ/nℤ ` N ) ) = ( .r ` (ℤ/nℤ ` N ) )
38	7, 37	ringcl 18561	. . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . 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 6275	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 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 6275	. . . . . . . . . . . . . 14 |- ( ( X : ( Base ` (ℤ/nℤ ` N ) ) --> CC /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( * o. X ) ` x ) = ( * ` ( X ` x ) ) )
43	22, 25, 42	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( ( * o. X ) ` x ) = ( * ` ( X ` x ) ) )
44		fvco3 6275	. . . . . . . . . . . . . 14 |- ( ( X : ( Base ` (ℤ/nℤ ` N ) ) --> CC /\ y e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( * o. X ) ` y ) = ( * ` ( X ` y ) ) )
45	22, 28, 44	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. (Unit ` (ℤ/nℤ ` N ) ) /\ y e. (Unit ` (ℤ/nℤ ` N ) ) ) ) -> ( ( * o. X ) ` y ) = ( * ` ( X ` y ) ) )
46	43, 45	oveq12d 6668	. . . . . . . . . . . 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 2666	. . . . . . . . . . 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 2971	. . . . . . . . . 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 2622	. . . . . . . . . . . . . 14 |- ( 1r ` (ℤ/nℤ ` N ) ) = ( 1r ` (ℤ/nℤ ` N ) )
50	7, 49	ringidcl 18568	. . . . . . . . . . . . 13 |- ( (ℤ/nℤ ` N ) e. Ring -> ( 1r ` (ℤ/nℤ ` N ) ) e. ( Base ` (ℤ/nℤ ` N ) ) )
51	35, 50	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( 1r ` (ℤ/nℤ ` N ) ) e. ( Base ` (ℤ/nℤ ` N ) ) )
52		fvco3 6275	. . . . . . . . . . . 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 693	. . . . . . . . . . 11 |- ( ph -> ( ( * o. X ) ` ( 1r ` (ℤ/nℤ ` N ) ) ) = ( * ` ( X ` ( 1r ` (ℤ/nℤ ` N ) ) ) ) )
54	16	simp2d 1074	. . . . . . . . . . . . 13 |- ( ph -> ( X ` ( 1r ` (ℤ/nℤ ` N ) ) ) = 1 )
55	54	fveq2d 6195	. . . . . . . . . . . 12 |- ( ph -> ( * ` ( X ` ( 1r ` (ℤ/nℤ ` N ) ) ) ) = ( * ` 1 ) )
56		1re 10039	. . . . . . . . . . . . 13 |- 1 e. RR
57		cjre 13879	. . . . . . . . . . . . 13 |- ( 1 e. RR -> ( * ` 1 ) = 1 )
58	56, 57	ax-mp 5	. . . . . . . . . . . 12 |- ( * ` 1 ) = 1
59	55, 58	syl6eq 2672	. . . . . . . . . . 11 |- ( ph -> ( * ` ( X ` ( 1r ` (ℤ/nℤ ` N ) ) ) ) = 1 )
60	53, 59	eqtrd 2656	. . . . . . . . . 10 |- ( ph -> ( ( * o. X ) ` ( 1r ` (ℤ/nℤ ` N ) ) ) = 1 )
61	16	simp3d 1075	. . . . . . . . . . 11 |- ( ph -> A. x e. ( Base ` (ℤ/nℤ ` N ) ) ( ( X ` x ) =/= 0 -> x e. (Unit ` (ℤ/nℤ ` N ) ) ) )
62	8, 42	sylan 488	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( * o. X ) ` x ) = ( * ` ( X ` x ) ) )
63		cj0 13898	. . . . . . . . . . . . . . . . . 18 |- ( * ` 0 ) = 0
64	63	eqcomi 2631	. . . . . . . . . . . . . . . . 17 |- 0 = ( * ` 0 )
65	64	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> 0 = ( * ` 0 ) )
66	62, 65	eqeq12d 2637	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( ( * o. X ) ` x ) = 0 <-> ( * ` ( X ` x ) ) = ( * ` 0 ) ) )
67	8	ffvelrnda 6359	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( X ` x ) e. CC )
68		0cn 10032	. . . . . . . . . . . . . . . 16 |- 0 e. CC
69		cj11 13902	. . . . . . . . . . . . . . . 16 |- ( ( ( X ` x ) e. CC /\ 0 e. CC ) -> ( ( * ` ( X ` x ) ) = ( * ` 0 ) <-> ( X ` x ) = 0 ) )
70	67, 68, 69	sylancl 694	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( * ` ( X ` x ) ) = ( * ` 0 ) <-> ( X ` x ) = 0 ) )
71	66, 70	bitrd 268	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( ( * o. X ) ` x ) = 0 <-> ( X ` x ) = 0 ) )
72	71	necon3bid 2838	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( Base ` (ℤ/nℤ ` N ) ) ) -> ( ( ( * o. X ) ` x ) =/= 0 <-> ( X ` x ) =/= 0 ) )
73	72	imbi1d 331	. . . . . . . . . . . 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 2985	. . . . . . . . . . 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 247	. . . . . . . . . 10 |- ( ph -> A. x e. ( Base ` (ℤ/nℤ ` N ) ) ( ( ( * o. X ) ` x ) =/= 0 -> x e. (Unit ` (ℤ/nℤ ` N ) ) ) )
76	48, 60, 75	3jca 1242	. . . . . . . . 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 24963	. . . . . . . . 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 957	. . . . . . . 8 |- ( ph -> ( * o. X ) e. D )
79	1, 2, 3, 4, 5, 78	dchrmul 24973	. . . . . . 7 |- ( ph -> ( X ( +g ` G ) ( * o. X ) ) = ( X oF x. ( * o. X ) ) )
80	79	adantr 481	. . . . . 6 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( X ( +g ` G ) ( * o. X ) ) = ( X oF x. ( * o. X ) ) )
81	80	fveq1d 6193	. . . . 5 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( X ( +g ` G ) ( * o. X ) ) ` x ) = ( ( X oF x. ( * o. X ) ) ` x ) )
82	23	sseli 3599	. . . . . . . . 9 |- ( x e. (Unit ` (ℤ/nℤ ` N ) ) -> x e. ( Base ` (ℤ/nℤ ` N ) ) )
83	82, 62	sylan2 491	. . . . . . . 8 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( * o. X ) ` x ) = ( * ` ( X ` x ) ) )
84	83	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( X ` x ) x. ( ( * o. X ) ` x ) ) = ( ( X ` x ) x. ( * ` ( X ` x ) ) ) )
85	82, 67	sylan2 491	. . . . . . . 8 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( X ` x ) e. CC )
86	85	absvalsqd 14181	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( abs ` ( X ` x ) ) ^ 2 ) = ( ( X ` x ) x. ( * ` ( X ` x ) ) ) )
87	5	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> X e. D )
88		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> x e. (Unit ` (ℤ/nℤ ` N ) ) )
89	1, 3, 87, 2, 11, 88	dchrabs 24985	. . . . . . . . 9 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( abs ` ( X ` x ) ) = 1 )
90	89	oveq1d 6665	. . . . . . . 8 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( abs ` ( X ` x ) ) ^ 2 ) = ( 1 ^ 2 ) )
91		sq1 12958	. . . . . . . 8 |- ( 1 ^ 2 ) = 1
92	90, 91	syl6eq 2672	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( abs ` ( X ` x ) ) ^ 2 ) = 1 )
93	84, 86, 92	3eqtr2d 2662	. . . . . 6 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( X ` x ) x. ( ( * o. X ) ` x ) ) = 1 )
94	8	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> X : ( Base ` (ℤ/nℤ ` N ) ) --> CC )
95		ffn 6045	. . . . . . . 8 |- ( X : ( Base ` (ℤ/nℤ ` N ) ) --> CC -> X Fn ( Base ` (ℤ/nℤ ` N ) ) )
96	94, 95	syl 17	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> X Fn ( Base ` (ℤ/nℤ ` N ) ) )
97		ffn 6045	. . . . . . . . 9 |- ( ( * o. X ) : ( Base ` (ℤ/nℤ ` N ) ) --> CC -> ( * o. X ) Fn ( Base ` (ℤ/nℤ ` N ) ) )
98	10, 97	syl 17	. . . . . . . 8 |- ( ph -> ( * o. X ) Fn ( Base ` (ℤ/nℤ ` N ) ) )
99	98	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( * o. X ) Fn ( Base ` (ℤ/nℤ ` N ) ) )
100		fvexd 6203	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( Base ` (ℤ/nℤ ` N ) ) e. _V )
101	82	adantl 482	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> x e. ( Base ` (ℤ/nℤ ` N ) ) )
102		fnfvof 6911	. . . . . . 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 oF x. ( * o. X ) ) ` x ) = ( ( X ` x ) x. ( ( * o. X ) ` x ) ) )
103	96, 99, 100, 101, 102	syl22anc 1327	. . . . . 6 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( X oF x. ( * o. X ) ) ` x ) = ( ( X ` x ) x. ( ( * o. X ) ` x ) ) )
104		eqid 2622	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
105	13	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> N e. NN )
106	1, 2, 104, 11, 105, 88	dchr1 24982	. . . . . 6 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( 0g ` G ) ` x ) = 1 )
107	93, 103, 106	3eqtr4d 2666	. . . . 5 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( X oF x. ( * o. X ) ) ` x ) = ( ( 0g ` G ) ` x ) )
108	81, 107	eqtrd 2656	. . . 4 |- ( ( ph /\ x e. (Unit ` (ℤ/nℤ ` N ) ) ) -> ( ( X ( +g ` G ) ( * o. X ) ) ` x ) = ( ( 0g ` G ) ` x ) )
109	108	ralrimiva 2966	. . 3 |- ( ph -> A. x e. (Unit ` (ℤ/nℤ ` N ) ) ( ( X ( +g ` G ) ( * o. X ) ) ` x ) = ( ( 0g ` G ) ` x ) )
110	1, 2, 3, 4, 5, 78	dchrmulcl 24974	. . . 4 |- ( ph -> ( X ( +g ` G ) ( * o. X ) ) e. D )
111	1	dchrabl 24979	. . . . . 6 |- ( N e. NN -> G e. Abel )
112		ablgrp 18198	. . . . . 6 |- ( G e. Abel -> G e. Grp )
113	13, 111, 112	3syl 18	. . . . 5 |- ( ph -> G e. Grp )
114	3, 104	grpidcl 17450	. . . . 5 |- ( G e. Grp -> ( 0g ` G ) e. D )
115	113, 114	syl 17	. . . 4 |- ( ph -> ( 0g ` G ) e. D )
116	1, 2, 3, 11, 110, 115	dchreq 24983	. . 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 ) ) )
117	109, 116	mpbird 247	. 2 |- ( ph -> ( X ( +g ` G ) ( * o. X ) ) = ( 0g ` G ) )
118		dchrinv.i	. . . 4 |- I = ( invg ` G )
119	3, 4, 104, 118	grpinvid1 17470	. . 3 |- ( ( G e. Grp /\ X e. D /\ ( * o. X ) e. D ) -> ( ( I ` X ) = ( * o. X ) <-> ( X ( +g ` G ) ( * o. X ) ) = ( 0g ` G ) ) )
120	113, 5, 78, 119	syl3anc 1326	. 2 |- ( ph -> ( ( I ` X ) = ( * o. X ) <-> ( X ( +g ` G ) ( * o. X ) ) = ( 0g ` G ) ) )
121	117, 120	mpbird 247	1 |- ( ph -> ( I ` X ) = ( * o. X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   NNcn 11020   2c2 11070   NN0cn0 11292   ^cexp 12860   *ccj 13836   abscabs 13974   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423   Abelcabl 18194   1rcur 18501   Ringcrg 18547   CRingccrg 18548  Unitcui 18639  ℤ/nℤczn 19851  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-iin 4523  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-supp 7296  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-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  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-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  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-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-qus 16169  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  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-cntz 17750  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-dvr 18683  df-rnghom 18715  df-drng 18749  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-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-zn 19855  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304  df-dchr 24958

This theorem is referenced by:  dchr2sum  24998  dchrisum0re  25202