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Theorem dchrghm 20495
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 20480 . . . . 5  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
5 dchrghm.x . . . . 5  |-  ( ph  ->  X  e.  D )
64, 5sseldi 3178 . . . 4  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
71, 3dchrrcl 20479 . . . . . . . . 9  |-  ( X  e.  D  ->  N  e.  NN )
85, 7syl 15 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
98nnnn0d 10018 . . . . . . 7  |-  ( ph  ->  N  e.  NN0 )
102zncrng 16498 . . . . . . 7  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
119, 10syl 15 . . . . . 6  |-  ( ph  ->  Z  e.  CRing )
12 crngrng 15351 . . . . . 6  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
1311, 12syl 15 . . . . 5  |-  ( ph  ->  Z  e.  Ring )
14 dchrghm.u . . . . . 6  |-  U  =  (Unit `  Z )
15 eqid 2283 . . . . . 6  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
1614, 15unitsubm 15452 . . . . 5  |-  ( Z  e.  Ring  ->  U  e.  (SubMnd `  (mulGrp `  Z
) ) )
1713, 16syl 15 . . . 4  |-  ( ph  ->  U  e.  (SubMnd `  (mulGrp `  Z ) ) )
18 dchrghm.h . . . . 5  |-  H  =  ( (mulGrp `  Z
)s 
U )
1918resmhm 14436 . . . 4  |-  ( ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  U  e.  (SubMnd `  (mulGrp `  Z
) ) )  -> 
( X  |`  U )  e.  ( H MndHom  (mulGrp ` fld ) ) )
206, 17, 19syl2anc 642 . . 3  |-  ( ph  ->  ( X  |`  U )  e.  ( H MndHom  (mulGrp ` fld ) ) )
21 cnrng 16396 . . . . 5  |-fld  e.  Ring
22 cnfldbas 16383 . . . . . . 7  |-  CC  =  ( Base ` fld )
23 cnfld0 16398 . . . . . . 7  |-  0  =  ( 0g ` fld )
24 cndrng 16403 . . . . . . 7  |-fld  e.  DivRing
2522, 23, 24drngui 15518 . . . . . 6  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
26 eqid 2283 . . . . . 6  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
2725, 26unitsubm 15452 . . . . 5  |-  (fld  e.  Ring  -> 
( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) ) )
2821, 27ax-mp 8 . . . 4  |-  ( CC 
\  { 0 } )  e.  (SubMnd `  (mulGrp ` fld ) )
29 df-ima 4702 . . . . 5  |-  ( X
" U )  =  ran  ( X  |`  U )
30 eqid 2283 . . . . . . . . . 10  |-  ( Base `  Z )  =  (
Base `  Z )
311, 2, 3, 30, 5dchrf 20481 . . . . . . . . 9  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
3230, 14unitss 15442 . . . . . . . . . 10  |-  U  C_  ( Base `  Z )
3332sseli 3176 . . . . . . . . 9  |-  ( x  e.  U  ->  x  e.  ( Base `  Z
) )
34 ffvelrn 5663 . . . . . . . . 9  |-  ( ( X : ( Base `  Z ) --> CC  /\  x  e.  ( Base `  Z ) )  -> 
( X `  x
)  e.  CC )
3531, 33, 34syl2an 463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  U )  ->  ( X `  x )  e.  CC )
36 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  U )  ->  x  e.  U )
375adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  U )  ->  X  e.  D )
3833adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  U )  ->  x  e.  ( Base `  Z
) )
391, 2, 3, 30, 14, 37, 38dchrn0 20489 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  U )  ->  (
( X `  x
)  =/=  0  <->  x  e.  U ) )
4036, 39mpbird 223 . . . . . . . 8  |-  ( (
ph  /\  x  e.  U )  ->  ( X `  x )  =/=  0 )
41 eldifsn 3749 . . . . . . . 8  |-  ( ( X `  x )  e.  ( CC  \  { 0 } )  <-> 
( ( X `  x )  e.  CC  /\  ( X `  x
)  =/=  0 ) )
4235, 40, 41sylanbrc 645 . . . . . . 7  |-  ( (
ph  /\  x  e.  U )  ->  ( X `  x )  e.  ( CC  \  {
0 } ) )
4342ralrimiva 2626 . . . . . 6  |-  ( ph  ->  A. x  e.  U  ( X `  x )  e.  ( CC  \  { 0 } ) )
44 ffun 5391 . . . . . . . 8  |-  ( X : ( Base `  Z
) --> CC  ->  Fun  X )
4531, 44syl 15 . . . . . . 7  |-  ( ph  ->  Fun  X )
46 fdm 5393 . . . . . . . . 9  |-  ( X : ( Base `  Z
) --> CC  ->  dom  X  =  ( Base `  Z
) )
4731, 46syl 15 . . . . . . . 8  |-  ( ph  ->  dom  X  =  (
Base `  Z )
)
4832, 47syl5sseqr 3227 . . . . . . 7  |-  ( ph  ->  U  C_  dom  X )
49 funimass4 5573 . . . . . . 7  |-  ( ( Fun  X  /\  U  C_ 
dom  X )  -> 
( ( X " U )  C_  ( CC  \  { 0 } )  <->  A. x  e.  U  ( X `  x )  e.  ( CC  \  { 0 } ) ) )
5045, 48, 49syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( X " U )  C_  ( CC  \  { 0 } )  <->  A. x  e.  U  ( X `  x )  e.  ( CC  \  { 0 } ) ) )
5143, 50mpbird 223 . . . . 5  |-  ( ph  ->  ( X " U
)  C_  ( CC  \  { 0 } ) )
5229, 51syl5eqssr 3223 . . . 4  |-  ( ph  ->  ran  ( X  |`  U )  C_  ( CC  \  { 0 } ) )
53 dchrghm.m . . . . 5  |-  M  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
5453resmhm2b 14438 . . . 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 644 . . 3  |-  ( ph  ->  ( ( X  |`  U )  e.  ( H MndHom  (mulGrp ` fld ) )  <->  ( X  |`  U )  e.  ( H MndHom  M ) ) )
5620, 55mpbid 201 . 2  |-  ( ph  ->  ( X  |`  U )  e.  ( H MndHom  M
) )
5714, 18unitgrp 15449 . . . 4  |-  ( Z  e.  Ring  ->  H  e. 
Grp )
5813, 57syl 15 . . 3  |-  ( ph  ->  H  e.  Grp )
5953cnmgpabl 16433 . . . 4  |-  M  e. 
Abel
60 ablgrp 15094 . . . 4  |-  ( M  e.  Abel  ->  M  e. 
Grp )
6159, 60ax-mp 8 . . 3  |-  M  e. 
Grp
62 ghmmhmb 14694 . . 3  |-  ( ( H  e.  Grp  /\  M  e.  Grp )  ->  ( H  GrpHom  M )  =  ( H MndHom  M
) )
6358, 61, 62sylancl 643 . 2  |-  ( ph  ->  ( H  GrpHom  M )  =  ( H MndHom  M
) )
6456, 63eleqtrrd 2360 1  |-  ( ph  ->  ( X  |`  U )  e.  ( H  GrpHom  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149    C_ wss 3152   {csn 3640   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   NNcn 9746   NN0cn0 9965   Basecbs 13148   ↾s cress 13149   Grpcgrp 14362   MndHom cmhm 14413  SubMndcsubmnd 14414    GrpHom cghm 14680   Abelcabel 15090  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338  Unitcui 15421  ℂfldccnfld 16377  ℤ/nczn 16454  DChrcdchr 20471
This theorem is referenced by:  dchrabs  20499  sum2dchr  20513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-imas 13411  df-divs 13412  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-cnfld 16378  df-zn 16458  df-dchr 20472
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