Theorem dchrsum2 24993

Description: An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character X is 0 if X is non-principal and phi ( n ) otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)

Hypotheses

Ref	Expression
dchrsum.g	|- G = (DChr ` N )
dchrsum.z	|- Z = (ℤ/nℤ ` N )
dchrsum.d	|- D = ( Base ` G )
dchrsum.1	|- .1. = ( 0g ` G )
dchrsum.x	|- ( ph -> X e. D )
dchrsum2.u	|- U = (Unit ` Z )

Assertion

Ref	Expression
dchrsum2	|- ( ph -> sum_ a e. U ( X ` a ) = if ( X = .1. , ( phi ` N ) , 0 ) )

Distinct variable groups:   .1. , a   ph, a   U, a   X, a   Z, a

Allowed substitution hints:   D( a)   G( a)   N( a)

Proof of Theorem dchrsum2

Dummy variables k x b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2633	. 2 |- ( ( phi ` N ) = if ( X = .1. , ( phi ` N ) , 0 ) -> ( sum_ a e. U ( X ` a ) = ( phi ` N ) <-> sum_ a e. U ( X ` a ) = if ( X = .1. , ( phi ` N ) , 0 ) ) )
2		eqeq2 2633	. 2 |- ( 0 = if ( X = .1. , ( phi ` N ) , 0 ) -> ( sum_ a e. U ( X ` a ) = 0 <-> sum_ a e. U ( X ` a ) = if ( X = .1. , ( phi ` N ) , 0 ) ) )
3		fveq1 6190	. . . . . 6 |- ( X = .1. -> ( X ` a ) = ( .1. ` a ) )
4		dchrsum.g	. . . . . . 7 |- G = (DChr ` N )
5		dchrsum.z	. . . . . . 7 |- Z = (ℤ/nℤ ` N )
6		dchrsum.1	. . . . . . 7 |- .1. = ( 0g ` G )
7		dchrsum2.u	. . . . . . 7 |- U = (Unit ` Z )
8		dchrsum.x	. . . . . . . . 9 |- ( ph -> X e. D )
9		dchrsum.d	. . . . . . . . . 10 |- D = ( Base ` G )
10	4, 9	dchrrcl 24965	. . . . . . . . 9 |- ( X e. D -> N e. NN )
11	8, 10	syl 17	. . . . . . . 8 |- ( ph -> N e. NN )
12	11	adantr 481	. . . . . . 7 |- ( ( ph /\ a e. U ) -> N e. NN )
13		simpr 477	. . . . . . 7 |- ( ( ph /\ a e. U ) -> a e. U )
14	4, 5, 6, 7, 12, 13	dchr1 24982	. . . . . 6 |- ( ( ph /\ a e. U ) -> ( .1. ` a ) = 1 )
15	3, 14	sylan9eqr 2678	. . . . 5 |- ( ( ( ph /\ a e. U ) /\ X = .1. ) -> ( X ` a ) = 1 )
16	15	an32s 846	. . . 4 |- ( ( ( ph /\ X = .1. ) /\ a e. U ) -> ( X ` a ) = 1 )
17	16	sumeq2dv 14433	. . 3 |- ( ( ph /\ X = .1. ) -> sum_ a e. U ( X ` a ) = sum_ a e. U 1 )
18	5, 7	znunithash 19913	. . . . . . . . 9 |- ( N e. NN -> ( # ` U ) = ( phi ` N ) )
19	11, 18	syl 17	. . . . . . . 8 |- ( ph -> ( # ` U ) = ( phi ` N ) )
20	11	phicld 15477	. . . . . . . . 9 |- ( ph -> ( phi ` N ) e. NN )
21	20	nnnn0d 11351	. . . . . . . 8 |- ( ph -> ( phi ` N ) e. NN0 )
22	19, 21	eqeltrd 2701	. . . . . . 7 |- ( ph -> ( # ` U ) e. NN0 )
23		fvex 6201	. . . . . . . . 9 |- (Unit ` Z ) e. _V
24	7, 23	eqeltri 2697	. . . . . . . 8 |- U e. _V
25		hashclb 13149	. . . . . . . 8 |- ( U e. _V -> ( U e. Fin <-> ( # ` U ) e. NN0 ) )
26	24, 25	ax-mp 5	. . . . . . 7 |- ( U e. Fin <-> ( # ` U ) e. NN0 )
27	22, 26	sylibr 224	. . . . . 6 |- ( ph -> U e. Fin )
28		ax-1cn 9994	. . . . . 6 |- 1 e. CC
29		fsumconst 14522	. . . . . 6 |- ( ( U e. Fin /\ 1 e. CC ) -> sum_ a e. U 1 = ( ( # ` U ) x. 1 ) )
30	27, 28, 29	sylancl 694	. . . . 5 |- ( ph -> sum_ a e. U 1 = ( ( # ` U ) x. 1 ) )
31	19	oveq1d 6665	. . . . 5 |- ( ph -> ( ( # ` U ) x. 1 ) = ( ( phi ` N ) x. 1 ) )
32	20	nncnd 11036	. . . . . 6 |- ( ph -> ( phi ` N ) e. CC )
33	32	mulid1d 10057	. . . . 5 |- ( ph -> ( ( phi ` N ) x. 1 ) = ( phi ` N ) )
34	30, 31, 33	3eqtrd 2660	. . . 4 |- ( ph -> sum_ a e. U 1 = ( phi ` N ) )
35	34	adantr 481	. . 3 |- ( ( ph /\ X = .1. ) -> sum_ a e. U 1 = ( phi ` N ) )
36	17, 35	eqtrd 2656	. 2 |- ( ( ph /\ X = .1. ) -> sum_ a e. U ( X ` a ) = ( phi ` N ) )
37	4	dchrabl 24979	. . . . . . . 8 |- ( N e. NN -> G e. Abel )
38		ablgrp 18198	. . . . . . . 8 |- ( G e. Abel -> G e. Grp )
39	9, 6	grpidcl 17450	. . . . . . . 8 |- ( G e. Grp -> .1. e. D )
40	11, 37, 38, 39	4syl 19	. . . . . . 7 |- ( ph -> .1. e. D )
41	4, 5, 9, 7, 8, 40	dchreq 24983	. . . . . 6 |- ( ph -> ( X = .1. <-> A. k e. U ( X ` k ) = ( .1. ` k ) ) )
42	41	notbid 308	. . . . 5 |- ( ph -> ( -. X = .1. <-> -. A. k e. U ( X ` k ) = ( .1. ` k ) ) )
43		rexnal 2995	. . . . 5 |- ( E. k e. U -. ( X ` k ) = ( .1. ` k ) <-> -. A. k e. U ( X ` k ) = ( .1. ` k ) )
44	42, 43	syl6bbr 278	. . . 4 |- ( ph -> ( -. X = .1. <-> E. k e. U -. ( X ` k ) = ( .1. ` k ) ) )
45		df-ne 2795	. . . . . 6 |- ( ( X ` k ) =/= ( .1. ` k ) <-> -. ( X ` k ) = ( .1. ` k ) )
46	11	adantr 481	. . . . . . . . 9 |- ( ( ph /\ k e. U ) -> N e. NN )
47		simpr 477	. . . . . . . . 9 |- ( ( ph /\ k e. U ) -> k e. U )
48	4, 5, 6, 7, 46, 47	dchr1 24982	. . . . . . . 8 |- ( ( ph /\ k e. U ) -> ( .1. ` k ) = 1 )
49	48	neeq2d 2854	. . . . . . 7 |- ( ( ph /\ k e. U ) -> ( ( X ` k ) =/= ( .1. ` k ) <-> ( X ` k ) =/= 1 ) )
50	27	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> U e. Fin )
51		eqid 2622	. . . . . . . . . . . . 13 |- ( Base ` Z ) = ( Base ` Z )
52	4, 5, 9, 51, 8	dchrf 24967	. . . . . . . . . . . 12 |- ( ph -> X : ( Base ` Z ) --> CC )
53	51, 7	unitss 18660	. . . . . . . . . . . . 13 |- U C_ ( Base ` Z )
54	53	sseli 3599	. . . . . . . . . . . 12 |- ( a e. U -> a e. ( Base ` Z ) )
55		ffvelrn 6357	. . . . . . . . . . . 12 |- ( ( X : ( Base ` Z ) --> CC /\ a e. ( Base ` Z ) ) -> ( X ` a ) e. CC )
56	52, 54, 55	syl2an 494	. . . . . . . . . . 11 |- ( ( ph /\ a e. U ) -> ( X ` a ) e. CC )
57	56	adantlr 751	. . . . . . . . . 10 |- ( ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) /\ a e. U ) -> ( X ` a ) e. CC )
58	50, 57	fsumcl 14464	. . . . . . . . 9 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> sum_ a e. U ( X ` a ) e. CC )
59		0cnd 10033	. . . . . . . . 9 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> 0 e. CC )
60	52	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> X : ( Base ` Z ) --> CC )
61		simprl 794	. . . . . . . . . . . 12 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> k e. U )
62	53, 61	sseldi 3601	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> k e. ( Base ` Z ) )
63	60, 62	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( X ` k ) e. CC )
64		subcl 10280	. . . . . . . . . 10 |- ( ( ( X ` k ) e. CC /\ 1 e. CC ) -> ( ( X ` k ) - 1 ) e. CC )
65	63, 28, 64	sylancl 694	. . . . . . . . 9 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( ( X ` k ) - 1 ) e. CC )
66		simprr 796	. . . . . . . . . 10 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( X ` k ) =/= 1 )
67		subeq0 10307	. . . . . . . . . . . 12 |- ( ( ( X ` k ) e. CC /\ 1 e. CC ) -> ( ( ( X ` k ) - 1 ) = 0 <-> ( X ` k ) = 1 ) )
68	63, 28, 67	sylancl 694	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( ( ( X ` k ) - 1 ) = 0 <-> ( X ` k ) = 1 ) )
69	68	necon3bid 2838	. . . . . . . . . 10 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( ( ( X ` k ) - 1 ) =/= 0 <-> ( X ` k ) =/= 1 ) )
70	66, 69	mpbird 247	. . . . . . . . 9 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( ( X ` k ) - 1 ) =/= 0 )
71		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( x = a -> ( k ( .r ` Z ) x ) = ( k ( .r ` Z ) a ) )
72	71	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( x = a -> ( X ` ( k ( .r ` Z ) x ) ) = ( X ` ( k ( .r ` Z ) a ) ) )
73	72	cbvsumv 14426	. . . . . . . . . . . . . 14 |- sum_ x e. U ( X ` ( k ( .r ` Z ) x ) ) = sum_ a e. U ( X ` ( k ( .r ` Z ) a ) )
74	4, 5, 9	dchrmhm 24966	. . . . . . . . . . . . . . . . . 18 |- D C_ ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) )
75	74, 8	sseldi 3601	. . . . . . . . . . . . . . . . 17 |- ( ph -> X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) )
76	75	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) /\ a e. U ) -> X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) )
77	62	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) /\ a e. U ) -> k e. ( Base ` Z ) )
78	54	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) /\ a e. U ) -> a e. ( Base ` Z ) )
79		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- (mulGrp ` Z ) = (mulGrp ` Z )
80	79, 51	mgpbas 18495	. . . . . . . . . . . . . . . . 17 |- ( Base ` Z ) = ( Base ` (mulGrp ` Z ) )
81		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( .r ` Z ) = ( .r ` Z )
82	79, 81	mgpplusg 18493	. . . . . . . . . . . . . . . . 17 |- ( .r ` Z ) = ( +g ` (mulGrp ` Z ) )
83		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
84		cnfldmul 19752	. . . . . . . . . . . . . . . . . 18 |- x. = ( .r ` ℂfld )
85	83, 84	mgpplusg 18493	. . . . . . . . . . . . . . . . 17 |- x. = ( +g ` (mulGrp ` ℂfld ) )
86	80, 82, 85	mhmlin 17342	. . . . . . . . . . . . . . . 16 |- ( ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ k e. ( Base ` Z ) /\ a e. ( Base ` Z ) ) -> ( X ` ( k ( .r ` Z ) a ) ) = ( ( X ` k ) x. ( X ` a ) ) )
87	76, 77, 78, 86	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) /\ a e. U ) -> ( X ` ( k ( .r ` Z ) a ) ) = ( ( X ` k ) x. ( X ` a ) ) )
88	87	sumeq2dv 14433	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> sum_ a e. U ( X ` ( k ( .r ` Z ) a ) ) = sum_ a e. U ( ( X ` k ) x. ( X ` a ) ) )
89	73, 88	syl5eq 2668	. . . . . . . . . . . . 13 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> sum_ x e. U ( X ` ( k ( .r ` Z ) x ) ) = sum_ a e. U ( ( X ` k ) x. ( X ` a ) ) )
90		fveq2 6191	. . . . . . . . . . . . . 14 |- ( a = ( k ( .r ` Z ) x ) -> ( X ` a ) = ( X ` ( k ( .r ` Z ) x ) ) )
91	11	nnnn0d 11351	. . . . . . . . . . . . . . . . 17 |- ( ph -> N e. NN0 )
92	5	zncrng 19893	. . . . . . . . . . . . . . . . 17 |- ( N e. NN0 -> Z e. CRing )
93		crngring 18558	. . . . . . . . . . . . . . . . 17 |- ( Z e. CRing -> Z e. Ring )
94		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( (mulGrp ` Z ) ↾s U ) = ( (mulGrp ` Z ) ↾s U )
95	7, 94	unitgrp 18667	. . . . . . . . . . . . . . . . 17 |- ( Z e. Ring -> ( (mulGrp ` Z ) ↾s U ) e. Grp )
96	91, 92, 93, 95	4syl 19	. . . . . . . . . . . . . . . 16 |- ( ph -> ( (mulGrp ` Z ) ↾s U ) e. Grp )
97	96	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( (mulGrp ` Z ) ↾s U ) e. Grp )
98		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( b e. U |-> ( c e. U |-> ( b ( .r ` Z ) c ) ) ) = ( b e. U |-> ( c e. U |-> ( b ( .r ` Z ) c ) ) )
99	7, 94	unitgrpbas 18666	. . . . . . . . . . . . . . . 16 |- U = ( Base ` ( (mulGrp ` Z ) ↾s U ) )
100	94, 82	ressplusg 15993	. . . . . . . . . . . . . . . . 17 |- ( U e. _V -> ( .r ` Z ) = ( +g ` ( (mulGrp ` Z ) ↾s U ) ) )
101	24, 100	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( .r ` Z ) = ( +g ` ( (mulGrp ` Z ) ↾s U ) )
102	98, 99, 101	grplactf1o 17519	. . . . . . . . . . . . . . 15 |- ( ( ( (mulGrp ` Z ) ↾s U ) e. Grp /\ k e. U ) -> ( ( b e. U |-> ( c e. U |-> ( b ( .r ` Z ) c ) ) ) ` k ) : U -1-1-onto-> U )
103	97, 61, 102	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( ( b e. U |-> ( c e. U |-> ( b ( .r ` Z ) c ) ) ) ` k ) : U -1-1-onto-> U )
104	98, 99	grplactval 17517	. . . . . . . . . . . . . . 15 |- ( ( k e. U /\ x e. U ) -> ( ( ( b e. U |-> ( c e. U |-> ( b ( .r ` Z ) c ) ) ) ` k ) ` x ) = ( k ( .r ` Z ) x ) )
105	61, 104	sylan 488	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) /\ x e. U ) -> ( ( ( b e. U |-> ( c e. U |-> ( b ( .r ` Z ) c ) ) ) ` k ) ` x ) = ( k ( .r ` Z ) x ) )
106	90, 50, 103, 105, 57	fsumf1o 14454	. . . . . . . . . . . . 13 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> sum_ a e. U ( X ` a ) = sum_ x e. U ( X ` ( k ( .r ` Z ) x ) ) )
107	50, 63, 57	fsummulc2 14516	. . . . . . . . . . . . 13 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( ( X ` k ) x. sum_ a e. U ( X ` a ) ) = sum_ a e. U ( ( X ` k ) x. ( X ` a ) ) )
108	89, 106, 107	3eqtr4rd 2667	. . . . . . . . . . . 12 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( ( X ` k ) x. sum_ a e. U ( X ` a ) ) = sum_ a e. U ( X ` a ) )
109	58	mulid2d 10058	. . . . . . . . . . . 12 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( 1 x. sum_ a e. U ( X ` a ) ) = sum_ a e. U ( X ` a ) )
110	108, 109	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( ( ( X ` k ) x. sum_ a e. U ( X ` a ) ) - ( 1 x. sum_ a e. U ( X ` a ) ) ) = ( sum_ a e. U ( X ` a ) - sum_ a e. U ( X ` a ) ) )
111	58	subidd 10380	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( sum_ a e. U ( X ` a ) - sum_ a e. U ( X ` a ) ) = 0 )
112	110, 111	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( ( ( X ` k ) x. sum_ a e. U ( X ` a ) ) - ( 1 x. sum_ a e. U ( X ` a ) ) ) = 0 )
113		1cnd 10056	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> 1 e. CC )
114	63, 113, 58	subdird 10487	. . . . . . . . . 10 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( ( ( X ` k ) - 1 ) x. sum_ a e. U ( X ` a ) ) = ( ( ( X ` k ) x. sum_ a e. U ( X ` a ) ) - ( 1 x. sum_ a e. U ( X ` a ) ) ) )
115	65	mul01d 10235	. . . . . . . . . 10 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( ( ( X ` k ) - 1 ) x. 0 ) = 0 )
116	112, 114, 115	3eqtr4d 2666	. . . . . . . . 9 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> ( ( ( X ` k ) - 1 ) x. sum_ a e. U ( X ` a ) ) = ( ( ( X ` k ) - 1 ) x. 0 ) )
117	58, 59, 65, 70, 116	mulcanad 10662	. . . . . . . 8 |- ( ( ph /\ ( k e. U /\ ( X ` k ) =/= 1 ) ) -> sum_ a e. U ( X ` a ) = 0 )
118	117	expr 643	. . . . . . 7 |- ( ( ph /\ k e. U ) -> ( ( X ` k ) =/= 1 -> sum_ a e. U ( X ` a ) = 0 ) )
119	49, 118	sylbid 230	. . . . . 6 |- ( ( ph /\ k e. U ) -> ( ( X ` k ) =/= ( .1. ` k ) -> sum_ a e. U ( X ` a ) = 0 ) )
120	45, 119	syl5bir 233	. . . . 5 |- ( ( ph /\ k e. U ) -> ( -. ( X ` k ) = ( .1. ` k ) -> sum_ a e. U ( X ` a ) = 0 ) )
121	120	rexlimdva 3031	. . . 4 |- ( ph -> ( E. k e. U -. ( X ` k ) = ( .1. ` k ) -> sum_ a e. U ( X ` a ) = 0 ) )
122	44, 121	sylbid 230	. . 3 |- ( ph -> ( -. X = .1. -> sum_ a e. U ( X ` a ) = 0 ) )
123	122	imp 445	. 2 |- ( ( ph /\ -. X = .1. ) -> sum_ a e. U ( X ` a ) = 0 )
124	1, 2, 36, 123	ifbothda 4123	1 |- ( ph -> sum_ a e. U ( X ` a ) = if ( X = .1. , ( phi ` N ) , 0 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   ifcif 4086   |-> cmpt 4729   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   - cmin 10266   NNcn 11020   NN0cn0 11292   #chash 13117   sum_csu 14416   phicphi 15469   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   MndHom cmhm 17333   Grpcgrp 17422   Abelcabl 18194  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548  Unitcui 18639  ℂ_fldccnfld 19746  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-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-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  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-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-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-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-nsg 17592  df-eqg 17593  df-ghm 17658  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-dchr 24958

This theorem is referenced by:  dchrsum  24994