Theorem dchrelbas2 24962

Description: A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of CC, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)

Hypotheses

Ref	Expression
dchrval.g	|- G = (DChr ` N )
dchrval.z	|- Z = (ℤ/nℤ ` N )
dchrval.b	|- B = ( Base ` Z )
dchrval.u	|- U = (Unit ` Z )
dchrval.n	|- ( ph -> N e. NN )
dchrbas.b	|- D = ( Base ` G )

Assertion

Ref	Expression
dchrelbas2	|- ( ph -> ( X e. D <-> ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. B ( ( X ` x ) =/= 0 -> x e. U ) ) ) )

Distinct variable groups:   x, B   x, N   x, U   ph, x   x, X   x, Z

Allowed substitution hints:   D( x)   G( x)

Proof of Theorem dchrelbas2

Step	Hyp	Ref	Expression
1		dchrval.g	. . 3 |- G = (DChr ` N )
2		dchrval.z	. . 3 |- Z = (ℤ/nℤ ` N )
3		dchrval.b	. . 3 |- B = ( Base ` Z )
4		dchrval.u	. . 3 |- U = (Unit ` Z )
5		dchrval.n	. . 3 |- ( ph -> N e. NN )
6		dchrbas.b	. . 3 |- D = ( Base ` G )
7	1, 2, 3, 4, 5, 6	dchrelbas 24961	. 2 |- ( ph -> ( X e. D <-> ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ ( ( B \ U ) X. { 0 } ) C_ X ) ) )
8		eqid 2622	. . . . . . . . . . 11 |- (mulGrp ` Z ) = (mulGrp ` Z )
9	8, 3	mgpbas 18495	. . . . . . . . . 10 |- B = ( Base ` (mulGrp ` Z ) )
10		eqid 2622	. . . . . . . . . . 11 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
11		cnfldbas 19750	. . . . . . . . . . 11 |- CC = ( Base ` ℂfld )
12	10, 11	mgpbas 18495	. . . . . . . . . 10 |- CC = ( Base ` (mulGrp ` ℂfld ) )
13	9, 12	mhmf 17340	. . . . . . . . 9 |- ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) -> X : B --> CC )
14	13	adantl 482	. . . . . . . 8 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> X : B --> CC )
15		ffun 6048	. . . . . . . 8 |- ( X : B --> CC -> Fun X )
16	14, 15	syl 17	. . . . . . 7 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> Fun X )
17		funssres 5930	. . . . . . 7 |- ( ( Fun X /\ ( ( B \ U ) X. { 0 } ) C_ X ) -> ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) )
18	16, 17	sylan 488	. . . . . 6 |- ( ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ ( ( B \ U ) X. { 0 } ) C_ X ) -> ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) )
19		simpr 477	. . . . . . 7 |- ( ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) ) -> ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) )
20		resss 5422	. . . . . . 7 |- ( X |` dom ( ( B \ U ) X. { 0 } ) ) C_ X
21	19, 20	syl6eqssr 3656	. . . . . 6 |- ( ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) ) -> ( ( B \ U ) X. { 0 } ) C_ X )
22	18, 21	impbida 877	. . . . 5 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( ( ( B \ U ) X. { 0 } ) C_ X <-> ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) ) )
23		0cn 10032	. . . . . . . . 9 |- 0 e. CC
24		fconst6g 6094	. . . . . . . . 9 |- ( 0 e. CC -> ( ( B \ U ) X. { 0 } ) : ( B \ U ) --> CC )
25	23, 24	mp1i 13	. . . . . . . 8 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( ( B \ U ) X. { 0 } ) : ( B \ U ) --> CC )
26		fdm 6051	. . . . . . . 8 |- ( ( ( B \ U ) X. { 0 } ) : ( B \ U ) --> CC -> dom ( ( B \ U ) X. { 0 } ) = ( B \ U ) )
27	25, 26	syl 17	. . . . . . 7 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> dom ( ( B \ U ) X. { 0 } ) = ( B \ U ) )
28	27	reseq2d 5396	. . . . . 6 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( X |` ( B \ U ) ) )
29	28	eqeq1d 2624	. . . . 5 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( ( X |` dom ( ( B \ U ) X. { 0 } ) ) = ( ( B \ U ) X. { 0 } ) <-> ( X |` ( B \ U ) ) = ( ( B \ U ) X. { 0 } ) ) )
30	22, 29	bitrd 268	. . . 4 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( ( ( B \ U ) X. { 0 } ) C_ X <-> ( X |` ( B \ U ) ) = ( ( B \ U ) X. { 0 } ) ) )
31		difss 3737	. . . . . . . 8 |- ( B \ U ) C_ B
32		fssres 6070	. . . . . . . 8 |- ( ( X : B --> CC /\ ( B \ U ) C_ B ) -> ( X |` ( B \ U ) ) : ( B \ U ) --> CC )
33	14, 31, 32	sylancl 694	. . . . . . 7 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( X |` ( B \ U ) ) : ( B \ U ) --> CC )
34		ffn 6045	. . . . . . 7 |- ( ( X |` ( B \ U ) ) : ( B \ U ) --> CC -> ( X |` ( B \ U ) ) Fn ( B \ U ) )
35	33, 34	syl 17	. . . . . 6 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( X |` ( B \ U ) ) Fn ( B \ U ) )
36		ffn 6045	. . . . . . 7 |- ( ( ( B \ U ) X. { 0 } ) : ( B \ U ) --> CC -> ( ( B \ U ) X. { 0 } ) Fn ( B \ U ) )
37	25, 36	syl 17	. . . . . 6 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( ( B \ U ) X. { 0 } ) Fn ( B \ U ) )
38		eqfnfv 6311	. . . . . 6 |- ( ( ( X |` ( B \ U ) ) Fn ( B \ U ) /\ ( ( B \ U ) X. { 0 } ) Fn ( B \ U ) ) -> ( ( X |` ( B \ U ) ) = ( ( B \ U ) X. { 0 } ) <-> A. x e. ( B \ U ) ( ( X |` ( B \ U ) ) ` x ) = ( ( ( B \ U ) X. { 0 } ) ` x ) ) )
39	35, 37, 38	syl2anc 693	. . . . 5 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( ( X |` ( B \ U ) ) = ( ( B \ U ) X. { 0 } ) <-> A. x e. ( B \ U ) ( ( X |` ( B \ U ) ) ` x ) = ( ( ( B \ U ) X. { 0 } ) ` x ) ) )
40		fvres 6207	. . . . . . . 8 |- ( x e. ( B \ U ) -> ( ( X |` ( B \ U ) ) ` x ) = ( X ` x ) )
41		c0ex 10034	. . . . . . . . 9 |- 0 e. _V
42	41	fvconst2 6469	. . . . . . . 8 |- ( x e. ( B \ U ) -> ( ( ( B \ U ) X. { 0 } ) ` x ) = 0 )
43	40, 42	eqeq12d 2637	. . . . . . 7 |- ( x e. ( B \ U ) -> ( ( ( X |` ( B \ U ) ) ` x ) = ( ( ( B \ U ) X. { 0 } ) ` x ) <-> ( X ` x ) = 0 ) )
44	43	ralbiia 2979	. . . . . 6 |- ( A. x e. ( B \ U ) ( ( X |` ( B \ U ) ) ` x ) = ( ( ( B \ U ) X. { 0 } ) ` x ) <-> A. x e. ( B \ U ) ( X ` x ) = 0 )
45		eldif 3584	. . . . . . . . 9 |- ( x e. ( B \ U ) <-> ( x e. B /\ -. x e. U ) )
46	45	imbi1i 339	. . . . . . . 8 |- ( ( x e. ( B \ U ) -> ( X ` x ) = 0 ) <-> ( ( x e. B /\ -. x e. U ) -> ( X ` x ) = 0 ) )
47		impexp 462	. . . . . . . 8 |- ( ( ( x e. B /\ -. x e. U ) -> ( X ` x ) = 0 ) <-> ( x e. B -> ( -. x e. U -> ( X ` x ) = 0 ) ) )
48		con1b 348	. . . . . . . . . 10 |- ( ( -. x e. U -> ( X ` x ) = 0 ) <-> ( -. ( X ` x ) = 0 -> x e. U ) )
49		df-ne 2795	. . . . . . . . . . 11 |- ( ( X ` x ) =/= 0 <-> -. ( X ` x ) = 0 )
50	49	imbi1i 339	. . . . . . . . . 10 |- ( ( ( X ` x ) =/= 0 -> x e. U ) <-> ( -. ( X ` x ) = 0 -> x e. U ) )
51	48, 50	bitr4i 267	. . . . . . . . 9 |- ( ( -. x e. U -> ( X ` x ) = 0 ) <-> ( ( X ` x ) =/= 0 -> x e. U ) )
52	51	imbi2i 326	. . . . . . . 8 |- ( ( x e. B -> ( -. x e. U -> ( X ` x ) = 0 ) ) <-> ( x e. B -> ( ( X ` x ) =/= 0 -> x e. U ) ) )
53	46, 47, 52	3bitri 286	. . . . . . 7 |- ( ( x e. ( B \ U ) -> ( X ` x ) = 0 ) <-> ( x e. B -> ( ( X ` x ) =/= 0 -> x e. U ) ) )
54	53	ralbii2 2978	. . . . . 6 |- ( A. x e. ( B \ U ) ( X ` x ) = 0 <-> A. x e. B ( ( X ` x ) =/= 0 -> x e. U ) )
55	44, 54	bitri 264	. . . . 5 |- ( A. x e. ( B \ U ) ( ( X |` ( B \ U ) ) ` x ) = ( ( ( B \ U ) X. { 0 } ) ` x ) <-> A. x e. B ( ( X ` x ) =/= 0 -> x e. U ) )
56	39, 55	syl6bb 276	. . . 4 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( ( X |` ( B \ U ) ) = ( ( B \ U ) X. { 0 } ) <-> A. x e. B ( ( X ` x ) =/= 0 -> x e. U ) ) )
57	30, 56	bitrd 268	. . 3 |- ( ( ph /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( ( ( B \ U ) X. { 0 } ) C_ X <-> A. x e. B ( ( X ` x ) =/= 0 -> x e. U ) ) )
58	57	pm5.32da 673	. 2 |- ( ph -> ( ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ ( ( B \ U ) X. { 0 } ) C_ X ) <-> ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. B ( ( X ` x ) =/= 0 -> x e. U ) ) ) )
59	7, 58	bitrd 268	1 |- ( ph -> ( X e. D <-> ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. B ( ( X ` x ) =/= 0 -> x e. U ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   C_ wss 3574   {csn 4177   X. cxp 5112   dom cdm 5114   |` cres 5116   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   NNcn 11020   Basecbs 15857   MndHom cmhm 17333  mulGrpcmgp 18489  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-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

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-mhm 17335  df-mgp 18490  df-cnfld 19747  df-dchr 24958

This theorem is referenced by:  dchrelbas3  24963  dchrelbas4  24968  dchrmulcl  24974  dchrn0  24975  dchrmulid2  24977