MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dchrghm Unicode version

Theorem dchrghm 20511
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 20496 . . . . 5  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
5 dchrghm.x . . . . 5  |-  ( ph  ->  X  e.  D )
64, 5sseldi 3191 . . . 4  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
71, 3dchrrcl 20495 . . . . . . . . 9  |-  ( X  e.  D  ->  N  e.  NN )
85, 7syl 15 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
98nnnn0d 10034 . . . . . . 7  |-  ( ph  ->  N  e.  NN0 )
102zncrng 16514 . . . . . . 7  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
119, 10syl 15 . . . . . 6  |-  ( ph  ->  Z  e.  CRing )
12 crngrng 15367 . . . . . 6  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
1311, 12syl 15 . . . . 5  |-  ( ph  ->  Z  e.  Ring )
14 dchrghm.u . . . . . 6  |-  U  =  (Unit `  Z )
15 eqid 2296 . . . . . 6  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
1614, 15unitsubm 15468 . . . . 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 14452 . . . 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 16412 . . . . 5  |-fld  e.  Ring
22 cnfldbas 16399 . . . . . . 7  |-  CC  =  ( Base ` fld )
23 cnfld0 16414 . . . . . . 7  |-  0  =  ( 0g ` fld )
24 cndrng 16419 . . . . . . 7  |-fld  e.  DivRing
2522, 23, 24drngui 15534 . . . . . 6  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
26 eqid 2296 . . . . . 6  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
2725, 26unitsubm 15468 . . . . 5  |-  (fld  e.  Ring  -> 
( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) ) )
2821, 27ax-mp 8 . . . 4  |-  ( CC 
\  { 0 } )  e.  (SubMnd `  (mulGrp ` fld ) )
29 df-ima 4718 . . . . 5  |-  ( X
" U )  =  ran  ( X  |`  U )
30 eqid 2296 . . . . . . . . . 10  |-  ( Base `  Z )  =  (
Base `  Z )
311, 2, 3, 30, 5dchrf 20497 . . . . . . . . 9  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
3230, 14unitss 15458 . . . . . . . . . 10  |-  U  C_  ( Base `  Z )
3332sseli 3189 . . . . . . . . 9  |-  ( x  e.  U  ->  x  e.  ( Base `  Z
) )
34 ffvelrn 5679 . . . . . . . . 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 20505 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  U )  ->  (
( X `  x
)  =/=  0  <->  x  e.  U ) )
4036, 39mpbird 223 . . . . . . . 8  |-  ( (
ph  /\  x  e.  U )  ->  ( X `  x )  =/=  0 )
41 eldifsn 3762 . . . . . . . 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 2639 . . . . . 6  |-  ( ph  ->  A. x  e.  U  ( X `  x )  e.  ( CC  \  { 0 } ) )
44 ffun 5407 . . . . . . . 8  |-  ( X : ( Base `  Z
) --> CC  ->  Fun  X )
4531, 44syl 15 . . . . . . 7  |-  ( ph  ->  Fun  X )
46 fdm 5409 . . . . . . . . 9  |-  ( X : ( Base `  Z
) --> CC  ->  dom  X  =  ( Base `  Z
) )
4731, 46syl 15 . . . . . . . 8  |-  ( ph  ->  dom  X  =  (
Base `  Z )
)
4832, 47syl5sseqr 3240 . . . . . . 7  |-  ( ph  ->  U  C_  dom  X )
49 funimass4 5589 . . . . . . 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 3236 . . . 4  |-  ( ph  ->  ran  ( X  |`  U )  C_  ( CC  \  { 0 } ) )
53 dchrghm.m . . . . 5  |-  M  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
5453resmhm2b 14454 . . . 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 15465 . . . 4  |-  ( Z  e.  Ring  ->  H  e. 
Grp )
5813, 57syl 15 . . 3  |-  ( ph  ->  H  e.  Grp )
5953cnmgpabl 16449 . . . 4  |-  M  e. 
Abel
60 ablgrp 15110 . . . 4  |-  ( M  e.  Abel  ->  M  e. 
Grp )
6159, 60ax-mp 8 . . 3  |-  M  e. 
Grp
62 ghmmhmb 14710 . . 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 2373 1  |-  ( ph  ->  ( X  |`  U )  e.  ( H  GrpHom  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    \ cdif 3162    C_ wss 3165   {csn 3653   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   NNcn 9762   NN0cn0 9981   Basecbs 13164   ↾s cress 13165   Grpcgrp 14378   MndHom cmhm 14429  SubMndcsubmnd 14430    GrpHom cghm 14696   Abelcabel 15106  mulGrpcmgp 15341   Ringcrg 15353   CRingccrg 15354  Unitcui 15437  ℂfldccnfld 16393  ℤ/nczn 16470  DChrcdchr 20487
This theorem is referenced by:  dchrabs  20515  sum2dchr  20529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-imas 13427  df-divs 13428  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-nsg 14635  df-eqg 14636  df-ghm 14697  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-sra 15941  df-rgmod 15942  df-lidl 15943  df-rsp 15944  df-2idl 16000  df-cnfld 16394  df-zn 16474  df-dchr 20488
  Copyright terms: Public domain W3C validator