Theorem dchrmulcl 24974

Description: Closure of the group operation on Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)

Hypotheses

Ref	Expression
dchrmhm.g	|- G = (DChr ` N )
dchrmhm.z	|- Z = (ℤ/nℤ ` N )
dchrmhm.b	|- D = ( Base ` G )
dchrmul.t	|- .x. = ( +g ` G )
dchrmul.x	|- ( ph -> X e. D )
dchrmul.y	|- ( ph -> Y e. D )

Assertion

Ref	Expression
dchrmulcl	|- ( ph -> ( X .x. Y ) e. D )

Proof of Theorem dchrmulcl

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

Step	Hyp	Ref	Expression
1		dchrmhm.g	. . 3 |- G = (DChr ` N )
2		dchrmhm.z	. . 3 |- Z = (ℤ/nℤ ` N )
3		dchrmhm.b	. . 3 |- D = ( Base ` G )
4		dchrmul.t	. . 3 |- .x. = ( +g ` G )
5		dchrmul.x	. . 3 |- ( ph -> X e. D )
6		dchrmul.y	. . 3 |- ( ph -> Y e. D )
7	1, 2, 3, 4, 5, 6	dchrmul 24973	. 2 |- ( ph -> ( X .x. Y ) = ( X oF x. Y ) )
8		mulcl 10020	. . . . 5 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
9	8	adantl 482	. . . 4 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
10		eqid 2622	. . . . 5 |- ( Base ` Z ) = ( Base ` Z )
11	1, 2, 3, 10, 5	dchrf 24967	. . . 4 |- ( ph -> X : ( Base ` Z ) --> CC )
12	1, 2, 3, 10, 6	dchrf 24967	. . . 4 |- ( ph -> Y : ( Base ` Z ) --> CC )
13		fvexd 6203	. . . 4 |- ( ph -> ( Base ` Z ) e. _V )
14		inidm 3822	. . . 4 |- ( ( Base ` Z ) i^i ( Base ` Z ) ) = ( Base ` Z )
15	9, 11, 12, 13, 13, 14	off 6912	. . 3 |- ( ph -> ( X oF x. Y ) : ( Base ` Z ) --> CC )
16		eqid 2622	. . . . . . . 8 |- (Unit ` Z ) = (Unit ` Z )
17	10, 16	unitcl 18659	. . . . . . 7 |- ( x e. (Unit ` Z ) -> x e. ( Base ` Z ) )
18	10, 16	unitcl 18659	. . . . . . 7 |- ( y e. (Unit ` Z ) -> y e. ( Base ` Z ) )
19	17, 18	anim12i 590	. . . . . 6 |- ( ( x e. (Unit ` Z ) /\ y e. (Unit ` Z ) ) -> ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) )
20	1, 3	dchrrcl 24965	. . . . . . . . . . . . . 14 |- ( X e. D -> N e. NN )
21	5, 20	syl 17	. . . . . . . . . . . . 13 |- ( ph -> N e. NN )
22	1, 2, 10, 16, 21, 3	dchrelbas2 24962	. . . . . . . . . . . 12 |- ( ph -> ( X e. D <-> ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. ( Base ` Z ) ( ( X ` x ) =/= 0 -> x e. (Unit ` Z ) ) ) ) )
23	5, 22	mpbid 222	. . . . . . . . . . 11 |- ( ph -> ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. ( Base ` Z ) ( ( X ` x ) =/= 0 -> x e. (Unit ` Z ) ) ) )
24	23	simpld 475	. . . . . . . . . 10 |- ( ph -> X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) )
25		eqid 2622	. . . . . . . . . . . . 13 |- (mulGrp ` Z ) = (mulGrp ` Z )
26	25, 10	mgpbas 18495	. . . . . . . . . . . 12 |- ( Base ` Z ) = ( Base ` (mulGrp ` Z ) )
27		eqid 2622	. . . . . . . . . . . . 13 |- ( .r ` Z ) = ( .r ` Z )
28	25, 27	mgpplusg 18493	. . . . . . . . . . . 12 |- ( .r ` Z ) = ( +g ` (mulGrp ` Z ) )
29		eqid 2622	. . . . . . . . . . . . 13 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
30		cnfldmul 19752	. . . . . . . . . . . . 13 |- x. = ( .r ` ℂfld )
31	29, 30	mgpplusg 18493	. . . . . . . . . . . 12 |- x. = ( +g ` (mulGrp ` ℂfld ) )
32	26, 28, 31	mhmlin 17342	. . . . . . . . . . 11 |- ( ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) )
33	32	3expb 1266	. . . . . . . . . 10 |- ( ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) )
34	24, 33	sylan 488	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) )
35	1, 2, 10, 16, 21, 3	dchrelbas2 24962	. . . . . . . . . . . 12 |- ( ph -> ( Y e. D <-> ( Y e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. ( Base ` Z ) ( ( Y ` x ) =/= 0 -> x e. (Unit ` Z ) ) ) ) )
36	6, 35	mpbid 222	. . . . . . . . . . 11 |- ( ph -> ( Y e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. ( Base ` Z ) ( ( Y ` x ) =/= 0 -> x e. (Unit ` Z ) ) ) )
37	36	simpld 475	. . . . . . . . . 10 |- ( ph -> Y e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) )
38	26, 28, 31	mhmlin 17342	. . . . . . . . . . 11 |- ( ( Y e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) -> ( Y ` ( x ( .r ` Z ) y ) ) = ( ( Y ` x ) x. ( Y ` y ) ) )
39	38	3expb 1266	. . . . . . . . . 10 |- ( ( Y e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( Y ` ( x ( .r ` Z ) y ) ) = ( ( Y ` x ) x. ( Y ` y ) ) )
40	37, 39	sylan 488	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( Y ` ( x ( .r ` Z ) y ) ) = ( ( Y ` x ) x. ( Y ` y ) ) )
41	34, 40	oveq12d 6668	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( ( X ` ( x ( .r ` Z ) y ) ) x. ( Y ` ( x ( .r ` Z ) y ) ) ) = ( ( ( X ` x ) x. ( X ` y ) ) x. ( ( Y ` x ) x. ( Y ` y ) ) ) )
42	11	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ x e. ( Base ` Z ) ) -> ( X ` x ) e. CC )
43	42	adantrr 753	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( X ` x ) e. CC )
44		simpr 477	. . . . . . . . . 10 |- ( ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) -> y e. ( Base ` Z ) )
45		ffvelrn 6357	. . . . . . . . . 10 |- ( ( X : ( Base ` Z ) --> CC /\ y e. ( Base ` Z ) ) -> ( X ` y ) e. CC )
46	11, 44, 45	syl2an 494	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( X ` y ) e. CC )
47	12	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ x e. ( Base ` Z ) ) -> ( Y ` x ) e. CC )
48	47	adantrr 753	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( Y ` x ) e. CC )
49		ffvelrn 6357	. . . . . . . . . 10 |- ( ( Y : ( Base ` Z ) --> CC /\ y e. ( Base ` Z ) ) -> ( Y ` y ) e. CC )
50	12, 44, 49	syl2an 494	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( Y ` y ) e. CC )
51	43, 46, 48, 50	mul4d 10248	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( ( ( X ` x ) x. ( X ` y ) ) x. ( ( Y ` x ) x. ( Y ` y ) ) ) = ( ( ( X ` x ) x. ( Y ` x ) ) x. ( ( X ` y ) x. ( Y ` y ) ) ) )
52	41, 51	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( ( X ` ( x ( .r ` Z ) y ) ) x. ( Y ` ( x ( .r ` Z ) y ) ) ) = ( ( ( X ` x ) x. ( Y ` x ) ) x. ( ( X ` y ) x. ( Y ` y ) ) ) )
53		ffn 6045	. . . . . . . . . 10 |- ( X : ( Base ` Z ) --> CC -> X Fn ( Base ` Z ) )
54	11, 53	syl 17	. . . . . . . . 9 |- ( ph -> X Fn ( Base ` Z ) )
55	54	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> X Fn ( Base ` Z ) )
56		ffn 6045	. . . . . . . . . 10 |- ( Y : ( Base ` Z ) --> CC -> Y Fn ( Base ` Z ) )
57	12, 56	syl 17	. . . . . . . . 9 |- ( ph -> Y Fn ( Base ` Z ) )
58	57	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> Y Fn ( Base ` Z ) )
59		fvexd 6203	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( Base ` Z ) e. _V )
60	21	nnnn0d 11351	. . . . . . . . . 10 |- ( ph -> N e. NN0 )
61	2	zncrng 19893	. . . . . . . . . 10 |- ( N e. NN0 -> Z e. CRing )
62		crngring 18558	. . . . . . . . . 10 |- ( Z e. CRing -> Z e. Ring )
63	60, 61, 62	3syl 18	. . . . . . . . 9 |- ( ph -> Z e. Ring )
64	10, 27	ringcl 18561	. . . . . . . . . 10 |- ( ( Z e. Ring /\ x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) -> ( x ( .r ` Z ) y ) e. ( Base ` Z ) )
65	64	3expb 1266	. . . . . . . . 9 |- ( ( Z e. Ring /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( x ( .r ` Z ) y ) e. ( Base ` Z ) )
66	63, 65	sylan 488	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( x ( .r ` Z ) y ) e. ( Base ` Z ) )
67		fnfvof 6911	. . . . . . . 8 |- ( ( ( X Fn ( Base ` Z ) /\ Y Fn ( Base ` Z ) ) /\ ( ( Base ` Z ) e. _V /\ ( x ( .r ` Z ) y ) e. ( Base ` Z ) ) ) -> ( ( X oF x. Y ) ` ( x ( .r ` Z ) y ) ) = ( ( X ` ( x ( .r ` Z ) y ) ) x. ( Y ` ( x ( .r ` Z ) y ) ) ) )
68	55, 58, 59, 66, 67	syl22anc 1327	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( ( X oF x. Y ) ` ( x ( .r ` Z ) y ) ) = ( ( X ` ( x ( .r ` Z ) y ) ) x. ( Y ` ( x ( .r ` Z ) y ) ) ) )
69	54	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( Base ` Z ) ) -> X Fn ( Base ` Z ) )
70	57	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( Base ` Z ) ) -> Y Fn ( Base ` Z ) )
71		fvexd 6203	. . . . . . . . . 10 |- ( ( ph /\ x e. ( Base ` Z ) ) -> ( Base ` Z ) e. _V )
72		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ x e. ( Base ` Z ) ) -> x e. ( Base ` Z ) )
73		fnfvof 6911	. . . . . . . . . 10 |- ( ( ( X Fn ( Base ` Z ) /\ Y Fn ( Base ` Z ) ) /\ ( ( Base ` Z ) e. _V /\ x e. ( Base ` Z ) ) ) -> ( ( X oF x. Y ) ` x ) = ( ( X ` x ) x. ( Y ` x ) ) )
74	69, 70, 71, 72, 73	syl22anc 1327	. . . . . . . . 9 |- ( ( ph /\ x e. ( Base ` Z ) ) -> ( ( X oF x. Y ) ` x ) = ( ( X ` x ) x. ( Y ` x ) ) )
75	74	adantrr 753	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( ( X oF x. Y ) ` x ) = ( ( X ` x ) x. ( Y ` x ) ) )
76		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> y e. ( Base ` Z ) )
77		fnfvof 6911	. . . . . . . . 9 |- ( ( ( X Fn ( Base ` Z ) /\ Y Fn ( Base ` Z ) ) /\ ( ( Base ` Z ) e. _V /\ y e. ( Base ` Z ) ) ) -> ( ( X oF x. Y ) ` y ) = ( ( X ` y ) x. ( Y ` y ) ) )
78	55, 58, 59, 76, 77	syl22anc 1327	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( ( X oF x. Y ) ` y ) = ( ( X ` y ) x. ( Y ` y ) ) )
79	75, 78	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( ( ( X oF x. Y ) ` x ) x. ( ( X oF x. Y ) ` y ) ) = ( ( ( X ` x ) x. ( Y ` x ) ) x. ( ( X ` y ) x. ( Y ` y ) ) ) )
80	52, 68, 79	3eqtr4d 2666	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` Z ) /\ y e. ( Base ` Z ) ) ) -> ( ( X oF x. Y ) ` ( x ( .r ` Z ) y ) ) = ( ( ( X oF x. Y ) ` x ) x. ( ( X oF x. Y ) ` y ) ) )
81	19, 80	sylan2 491	. . . . 5 |- ( ( ph /\ ( x e. (Unit ` Z ) /\ y e. (Unit ` Z ) ) ) -> ( ( X oF x. Y ) ` ( x ( .r ` Z ) y ) ) = ( ( ( X oF x. Y ) ` x ) x. ( ( X oF x. Y ) ` y ) ) )
82	81	ralrimivva 2971	. . . 4 |- ( ph -> A. x e. (Unit ` Z ) A. y e. (Unit ` Z ) ( ( X oF x. Y ) ` ( x ( .r ` Z ) y ) ) = ( ( ( X oF x. Y ) ` x ) x. ( ( X oF x. Y ) ` y ) ) )
83		eqid 2622	. . . . . . . 8 |- ( 1r ` Z ) = ( 1r ` Z )
84	10, 83	ringidcl 18568	. . . . . . 7 |- ( Z e. Ring -> ( 1r ` Z ) e. ( Base ` Z ) )
85	63, 84	syl 17	. . . . . 6 |- ( ph -> ( 1r ` Z ) e. ( Base ` Z ) )
86		fnfvof 6911	. . . . . 6 |- ( ( ( X Fn ( Base ` Z ) /\ Y Fn ( Base ` Z ) ) /\ ( ( Base ` Z ) e. _V /\ ( 1r ` Z ) e. ( Base ` Z ) ) ) -> ( ( X oF x. Y ) ` ( 1r ` Z ) ) = ( ( X ` ( 1r ` Z ) ) x. ( Y ` ( 1r ` Z ) ) ) )
87	54, 57, 13, 85, 86	syl22anc 1327	. . . . 5 |- ( ph -> ( ( X oF x. Y ) ` ( 1r ` Z ) ) = ( ( X ` ( 1r ` Z ) ) x. ( Y ` ( 1r ` Z ) ) ) )
88	25, 83	ringidval 18503	. . . . . . . . 9 |- ( 1r ` Z ) = ( 0g ` (mulGrp ` Z ) )
89		cnfld1 19771	. . . . . . . . . 10 |- 1 = ( 1r ` ℂfld )
90	29, 89	ringidval 18503	. . . . . . . . 9 |- 1 = ( 0g ` (mulGrp ` ℂfld ) )
91	88, 90	mhm0 17343	. . . . . . . 8 |- ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) -> ( X ` ( 1r ` Z ) ) = 1 )
92	24, 91	syl 17	. . . . . . 7 |- ( ph -> ( X ` ( 1r ` Z ) ) = 1 )
93	88, 90	mhm0 17343	. . . . . . . 8 |- ( Y e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) -> ( Y ` ( 1r ` Z ) ) = 1 )
94	37, 93	syl 17	. . . . . . 7 |- ( ph -> ( Y ` ( 1r ` Z ) ) = 1 )
95	92, 94	oveq12d 6668	. . . . . 6 |- ( ph -> ( ( X ` ( 1r ` Z ) ) x. ( Y ` ( 1r ` Z ) ) ) = ( 1 x. 1 ) )
96		1t1e1 11175	. . . . . 6 |- ( 1 x. 1 ) = 1
97	95, 96	syl6eq 2672	. . . . 5 |- ( ph -> ( ( X ` ( 1r ` Z ) ) x. ( Y ` ( 1r ` Z ) ) ) = 1 )
98	87, 97	eqtrd 2656	. . . 4 |- ( ph -> ( ( X oF x. Y ) ` ( 1r ` Z ) ) = 1 )
99	74	neeq1d 2853	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` Z ) ) -> ( ( ( X oF x. Y ) ` x ) =/= 0 <-> ( ( X ` x ) x. ( Y ` x ) ) =/= 0 ) )
100	42, 47	mulne0bd 10678	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` Z ) ) -> ( ( ( X ` x ) =/= 0 /\ ( Y ` x ) =/= 0 ) <-> ( ( X ` x ) x. ( Y ` x ) ) =/= 0 ) )
101	99, 100	bitr4d 271	. . . . . 6 |- ( ( ph /\ x e. ( Base ` Z ) ) -> ( ( ( X oF x. Y ) ` x ) =/= 0 <-> ( ( X ` x ) =/= 0 /\ ( Y ` x ) =/= 0 ) ) )
102	23	simprd 479	. . . . . . . 8 |- ( ph -> A. x e. ( Base ` Z ) ( ( X ` x ) =/= 0 -> x e. (Unit ` Z ) ) )
103	102	r19.21bi 2932	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` Z ) ) -> ( ( X ` x ) =/= 0 -> x e. (Unit ` Z ) ) )
104	103	adantrd 484	. . . . . 6 |- ( ( ph /\ x e. ( Base ` Z ) ) -> ( ( ( X ` x ) =/= 0 /\ ( Y ` x ) =/= 0 ) -> x e. (Unit ` Z ) ) )
105	101, 104	sylbid 230	. . . . 5 |- ( ( ph /\ x e. ( Base ` Z ) ) -> ( ( ( X oF x. Y ) ` x ) =/= 0 -> x e. (Unit ` Z ) ) )
106	105	ralrimiva 2966	. . . 4 |- ( ph -> A. x e. ( Base ` Z ) ( ( ( X oF x. Y ) ` x ) =/= 0 -> x e. (Unit ` Z ) ) )
107	82, 98, 106	3jca 1242	. . 3 |- ( ph -> ( A. x e. (Unit ` Z ) A. y e. (Unit ` Z ) ( ( X oF x. Y ) ` ( x ( .r ` Z ) y ) ) = ( ( ( X oF x. Y ) ` x ) x. ( ( X oF x. Y ) ` y ) ) /\ ( ( X oF x. Y ) ` ( 1r ` Z ) ) = 1 /\ A. x e. ( Base ` Z ) ( ( ( X oF x. Y ) ` x ) =/= 0 -> x e. (Unit ` Z ) ) ) )
108	1, 2, 10, 16, 21, 3	dchrelbas3 24963	. . 3 |- ( ph -> ( ( X oF x. Y ) e. D <-> ( ( X oF x. Y ) : ( Base ` Z ) --> CC /\ ( A. x e. (Unit ` Z ) A. y e. (Unit ` Z ) ( ( X oF x. Y ) ` ( x ( .r ` Z ) y ) ) = ( ( ( X oF x. Y ) ` x ) x. ( ( X oF x. Y ) ` y ) ) /\ ( ( X oF x. Y ) ` ( 1r ` Z ) ) = 1 /\ A. x e. ( Base ` Z ) ( ( ( X oF x. Y ) ` x ) =/= 0 -> x e. (Unit ` Z ) ) ) ) ) )
109	15, 107, 108	mpbir2and 957	. 2 |- ( ph -> ( X oF x. Y ) e. D )
110	7, 109	eqeltrd 2701	1 |- ( ph -> ( X .x. Y ) e. D )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   NNcn 11020   NN0cn0 11292   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   MndHom cmhm 17333  mulGrpcmgp 18489   1rcur 18501   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-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-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-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-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-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-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  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-fz 12327  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-subg 17591  df-nsg 17592  df-eqg 17593  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-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-zn 19855  df-dchr 24958

This theorem is referenced by:  dchrabl  24979  dchrinv  24986