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Theorem dchrghm 21042
Description: A Dirichlet character restricted to the unit group of ℤ/nℤ is a group homomorphism into the multiplicative group of nonzero complex numbers. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
dchrghm.g  |-  G  =  (DChr `  N )
dchrghm.z  |-  Z  =  (ℤ/n `  N )
dchrghm.b  |-  D  =  ( Base `  G
)
dchrghm.u  |-  U  =  (Unit `  Z )
dchrghm.h  |-  H  =  ( (mulGrp `  Z
)s 
U )
dchrghm.m  |-  M  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
dchrghm.x  |-  ( ph  ->  X  e.  D )
Assertion
Ref Expression
dchrghm  |-  ( ph  ->  ( X  |`  U )  e.  ( H  GrpHom  M ) )

Proof of Theorem dchrghm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dchrghm.g . . . . . 6  |-  G  =  (DChr `  N )
2 dchrghm.z . . . . . 6  |-  Z  =  (ℤ/n `  N )
3 dchrghm.b . . . . . 6  |-  D  =  ( Base `  G
)
41, 2, 3dchrmhm 21027 . . . . 5  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
5 dchrghm.x . . . . 5  |-  ( ph  ->  X  e.  D )
64, 5sseldi 3348 . . . 4  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
71, 3dchrrcl 21026 . . . . . . . . 9  |-  ( X  e.  D  ->  N  e.  NN )
85, 7syl 16 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
98nnnn0d 10276 . . . . . . 7  |-  ( ph  ->  N  e.  NN0 )
102zncrng 16827 . . . . . . 7  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
119, 10syl 16 . . . . . 6  |-  ( ph  ->  Z  e.  CRing )
12 crngrng 15676 . . . . . 6  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
1311, 12syl 16 . . . . 5  |-  ( ph  ->  Z  e.  Ring )
14 dchrghm.u . . . . . 6  |-  U  =  (Unit `  Z )
15 eqid 2438 . . . . . 6  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
1614, 15unitsubm 15777 . . . . 5  |-  ( Z  e.  Ring  ->  U  e.  (SubMnd `  (mulGrp `  Z
) ) )
1713, 16syl 16 . . . 4  |-  ( ph  ->  U  e.  (SubMnd `  (mulGrp `  Z ) ) )
18 dchrghm.h . . . . 5  |-  H  =  ( (mulGrp `  Z
)s 
U )
1918resmhm 14761 . . . 4  |-  ( ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  U  e.  (SubMnd `  (mulGrp `  Z
) ) )  -> 
( X  |`  U )  e.  ( H MndHom  (mulGrp ` fld ) ) )
206, 17, 19syl2anc 644 . . 3  |-  ( ph  ->  ( X  |`  U )  e.  ( H MndHom  (mulGrp ` fld ) ) )
21 cnrng 16725 . . . . 5  |-fld  e.  Ring
22 cnfldbas 16709 . . . . . . 7  |-  CC  =  ( Base ` fld )
23 cnfld0 16727 . . . . . . 7  |-  0  =  ( 0g ` fld )
24 cndrng 16732 . . . . . . 7  |-fld  e.  DivRing
2522, 23, 24drngui 15843 . . . . . 6  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
26 eqid 2438 . . . . . 6  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
2725, 26unitsubm 15777 . . . . 5  |-  (fld  e.  Ring  -> 
( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) ) )
2821, 27ax-mp 8 . . . 4  |-  ( CC 
\  { 0 } )  e.  (SubMnd `  (mulGrp ` fld ) )
29 df-ima 4893 . . . . 5  |-  ( X
" U )  =  ran  ( X  |`  U )
30 eqid 2438 . . . . . . . . . 10  |-  ( Base `  Z )  =  (
Base `  Z )
311, 2, 3, 30, 5dchrf 21028 . . . . . . . . 9  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
3230, 14unitss 15767 . . . . . . . . . 10  |-  U  C_  ( Base `  Z )
3332sseli 3346 . . . . . . . . 9  |-  ( x  e.  U  ->  x  e.  ( Base `  Z
) )
34 ffvelrn 5870 . . . . . . . . 9  |-  ( ( X : ( Base `  Z ) --> CC  /\  x  e.  ( Base `  Z ) )  -> 
( X `  x
)  e.  CC )
3531, 33, 34syl2an 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  U )  ->  ( X `  x )  e.  CC )
36 simpr 449 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  U )  ->  x  e.  U )
375adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  U )  ->  X  e.  D )
3833adantl 454 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  U )  ->  x  e.  ( Base `  Z
) )
391, 2, 3, 30, 14, 37, 38dchrn0 21036 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  U )  ->  (
( X `  x
)  =/=  0  <->  x  e.  U ) )
4036, 39mpbird 225 . . . . . . . 8  |-  ( (
ph  /\  x  e.  U )  ->  ( X `  x )  =/=  0 )
41 eldifsn 3929 . . . . . . . 8  |-  ( ( X `  x )  e.  ( CC  \  { 0 } )  <-> 
( ( X `  x )  e.  CC  /\  ( X `  x
)  =/=  0 ) )
4235, 40, 41sylanbrc 647 . . . . . . 7  |-  ( (
ph  /\  x  e.  U )  ->  ( X `  x )  e.  ( CC  \  {
0 } ) )
4342ralrimiva 2791 . . . . . 6  |-  ( ph  ->  A. x  e.  U  ( X `  x )  e.  ( CC  \  { 0 } ) )
44 ffun 5595 . . . . . . . 8  |-  ( X : ( Base `  Z
) --> CC  ->  Fun  X )
4531, 44syl 16 . . . . . . 7  |-  ( ph  ->  Fun  X )
46 fdm 5597 . . . . . . . . 9  |-  ( X : ( Base `  Z
) --> CC  ->  dom  X  =  ( Base `  Z
) )
4731, 46syl 16 . . . . . . . 8  |-  ( ph  ->  dom  X  =  (
Base `  Z )
)
4832, 47syl5sseqr 3399 . . . . . . 7  |-  ( ph  ->  U  C_  dom  X )
49 funimass4 5779 . . . . . . 7  |-  ( ( Fun  X  /\  U  C_ 
dom  X )  -> 
( ( X " U )  C_  ( CC  \  { 0 } )  <->  A. x  e.  U  ( X `  x )  e.  ( CC  \  { 0 } ) ) )
5045, 48, 49syl2anc 644 . . . . . 6  |-  ( ph  ->  ( ( X " U )  C_  ( CC  \  { 0 } )  <->  A. x  e.  U  ( X `  x )  e.  ( CC  \  { 0 } ) ) )
5143, 50mpbird 225 . . . . 5  |-  ( ph  ->  ( X " U
)  C_  ( CC  \  { 0 } ) )
5229, 51syl5eqssr 3395 . . . 4  |-  ( ph  ->  ran  ( X  |`  U )  C_  ( CC  \  { 0 } ) )
53 dchrghm.m . . . . 5  |-  M  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
5453resmhm2b 14763 . . . 4  |-  ( ( ( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) )  /\  ran  ( X  |`  U ) 
C_  ( CC  \  { 0 } ) )  ->  ( ( X  |`  U )  e.  ( H MndHom  (mulGrp ` fld )
)  <->  ( X  |`  U )  e.  ( H MndHom  M ) ) )
5528, 52, 54sylancr 646 . . 3  |-  ( ph  ->  ( ( X  |`  U )  e.  ( H MndHom  (mulGrp ` fld ) )  <->  ( X  |`  U )  e.  ( H MndHom  M ) ) )
5620, 55mpbid 203 . 2  |-  ( ph  ->  ( X  |`  U )  e.  ( H MndHom  M
) )
5714, 18unitgrp 15774 . . . 4  |-  ( Z  e.  Ring  ->  H  e. 
Grp )
5813, 57syl 16 . . 3  |-  ( ph  ->  H  e.  Grp )
5953cnmgpabl 16762 . . . 4  |-  M  e. 
Abel
60 ablgrp 15419 . . . 4  |-  ( M  e.  Abel  ->  M  e. 
Grp )
6159, 60ax-mp 8 . . 3  |-  M  e. 
Grp
62 ghmmhmb 15019 . . 3  |-  ( ( H  e.  Grp  /\  M  e.  Grp )  ->  ( H  GrpHom  M )  =  ( H MndHom  M
) )
6358, 61, 62sylancl 645 . 2  |-  ( ph  ->  ( H  GrpHom  M )  =  ( H MndHom  M
) )
6456, 63eleqtrrd 2515 1  |-  ( ph  ->  ( X  |`  U )  e.  ( H  GrpHom  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    \ cdif 3319    C_ wss 3322   {csn 3816   dom cdm 4880   ran crn 4881    |` cres 4882   "cima 4883   Fun wfun 5450   -->wf 5452   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992   NNcn 10002   NN0cn0 10223   Basecbs 13471   ↾s cress 13472   Grpcgrp 14687   MndHom cmhm 14738  SubMndcsubmnd 14739    GrpHom cghm 15005   Abelcabel 15415  mulGrpcmgp 15650   Ringcrg 15662   CRingccrg 15663  Unitcui 15746  ℂfldccnfld 16705  ℤ/nczn 16783  DChrcdchr 21018
This theorem is referenced by:  dchrabs  21046  sum2dchr  21060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-addf 9071  ax-mulf 9072
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-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-ec 6909  df-qs 6913  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-0g 13729  df-imas 13736  df-divs 13737  df-mnd 14692  df-mhm 14740  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-nsg 14944  df-eqg 14945  df-ghm 15006  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-cring 15666  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-dvr 15790  df-drng 15839  df-subrg 15868  df-lmod 15954  df-lss 16011  df-lsp 16050  df-sra 16246  df-rgmod 16247  df-lidl 16248  df-rsp 16249  df-2idl 16305  df-cnfld 16706  df-zn 16787  df-dchr 21019
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