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Theorem dchrghm 23400
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 23385 . . . . 5  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
5 dchrghm.x . . . . 5  |-  ( ph  ->  X  e.  D )
64, 5sseldi 3485 . . . 4  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
71, 3dchrrcl 23384 . . . . . . . . 9  |-  ( X  e.  D  ->  N  e.  NN )
85, 7syl 16 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
98nnnn0d 10855 . . . . . . 7  |-  ( ph  ->  N  e.  NN0 )
102zncrng 18453 . . . . . . 7  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
119, 10syl 16 . . . . . 6  |-  ( ph  ->  Z  e.  CRing )
12 crngring 17080 . . . . . 6  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
1311, 12syl 16 . . . . 5  |-  ( ph  ->  Z  e.  Ring )
14 dchrghm.u . . . . . 6  |-  U  =  (Unit `  Z )
15 eqid 2441 . . . . . 6  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
1614, 15unitsubm 17190 . . . . 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 15861 . . . 4  |-  ( ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  U  e.  (SubMnd `  (mulGrp `  Z
) ) )  -> 
( X  |`  U )  e.  ( H MndHom  (mulGrp ` fld ) ) )
206, 17, 19syl2anc 661 . . 3  |-  ( ph  ->  ( X  |`  U )  e.  ( H MndHom  (mulGrp ` fld ) ) )
21 cnring 18311 . . . . 5  |-fld  e.  Ring
22 cnfldbas 18295 . . . . . . 7  |-  CC  =  ( Base ` fld )
23 cnfld0 18313 . . . . . . 7  |-  0  =  ( 0g ` fld )
24 cndrng 18318 . . . . . . 7  |-fld  e.  DivRing
2522, 23, 24drngui 17273 . . . . . 6  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
26 eqid 2441 . . . . . 6  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
2725, 26unitsubm 17190 . . . . 5  |-  (fld  e.  Ring  -> 
( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) ) )
2821, 27ax-mp 5 . . . 4  |-  ( CC 
\  { 0 } )  e.  (SubMnd `  (mulGrp ` fld ) )
29 df-ima 4999 . . . . 5  |-  ( X
" U )  =  ran  ( X  |`  U )
30 eqid 2441 . . . . . . . . . 10  |-  ( Base `  Z )  =  (
Base `  Z )
311, 2, 3, 30, 5dchrf 23386 . . . . . . . . 9  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
3230, 14unitss 17180 . . . . . . . . . 10  |-  U  C_  ( Base `  Z )
3332sseli 3483 . . . . . . . . 9  |-  ( x  e.  U  ->  x  e.  ( Base `  Z
) )
34 ffvelrn 6011 . . . . . . . . 9  |-  ( ( X : ( Base `  Z ) --> CC  /\  x  e.  ( Base `  Z ) )  -> 
( X `  x
)  e.  CC )
3531, 33, 34syl2an 477 . . . . . . . 8  |-  ( (
ph  /\  x  e.  U )  ->  ( X `  x )  e.  CC )
36 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  U )  ->  x  e.  U )
375adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  U )  ->  X  e.  D )
3833adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  U )  ->  x  e.  ( Base `  Z
) )
391, 2, 3, 30, 14, 37, 38dchrn0 23394 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  U )  ->  (
( X `  x
)  =/=  0  <->  x  e.  U ) )
4036, 39mpbird 232 . . . . . . . 8  |-  ( (
ph  /\  x  e.  U )  ->  ( X `  x )  =/=  0 )
41 eldifsn 4137 . . . . . . . 8  |-  ( ( X `  x )  e.  ( CC  \  { 0 } )  <-> 
( ( X `  x )  e.  CC  /\  ( X `  x
)  =/=  0 ) )
4235, 40, 41sylanbrc 664 . . . . . . 7  |-  ( (
ph  /\  x  e.  U )  ->  ( X `  x )  e.  ( CC  \  {
0 } ) )
4342ralrimiva 2855 . . . . . 6  |-  ( ph  ->  A. x  e.  U  ( X `  x )  e.  ( CC  \  { 0 } ) )
44 ffun 5720 . . . . . . . 8  |-  ( X : ( Base `  Z
) --> CC  ->  Fun  X )
4531, 44syl 16 . . . . . . 7  |-  ( ph  ->  Fun  X )
46 fdm 5722 . . . . . . . . 9  |-  ( X : ( Base `  Z
) --> CC  ->  dom  X  =  ( Base `  Z
) )
4731, 46syl 16 . . . . . . . 8  |-  ( ph  ->  dom  X  =  (
Base `  Z )
)
4832, 47syl5sseqr 3536 . . . . . . 7  |-  ( ph  ->  U  C_  dom  X )
49 funimass4 5906 . . . . . . 7  |-  ( ( Fun  X  /\  U  C_ 
dom  X )  -> 
( ( X " U )  C_  ( CC  \  { 0 } )  <->  A. x  e.  U  ( X `  x )  e.  ( CC  \  { 0 } ) ) )
5045, 48, 49syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( X " U )  C_  ( CC  \  { 0 } )  <->  A. x  e.  U  ( X `  x )  e.  ( CC  \  { 0 } ) ) )
5143, 50mpbird 232 . . . . 5  |-  ( ph  ->  ( X " U
)  C_  ( CC  \  { 0 } ) )
5229, 51syl5eqssr 3532 . . . 4  |-  ( ph  ->  ran  ( X  |`  U )  C_  ( CC  \  { 0 } ) )
53 dchrghm.m . . . . 5  |-  M  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
5453resmhm2b 15863 . . . 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 663 . . 3  |-  ( ph  ->  ( ( X  |`  U )  e.  ( H MndHom  (mulGrp ` fld ) )  <->  ( X  |`  U )  e.  ( H MndHom  M ) ) )
5620, 55mpbid 210 . 2  |-  ( ph  ->  ( X  |`  U )  e.  ( H MndHom  M
) )
5714, 18unitgrp 17187 . . . 4  |-  ( Z  e.  Ring  ->  H  e. 
Grp )
5813, 57syl 16 . . 3  |-  ( ph  ->  H  e.  Grp )
5953cnmgpabl 18350 . . . 4  |-  M  e. 
Abel
60 ablgrp 16674 . . . 4  |-  ( M  e.  Abel  ->  M  e. 
Grp )
6159, 60ax-mp 5 . . 3  |-  M  e. 
Grp
62 ghmmhmb 16149 . . 3  |-  ( ( H  e.  Grp  /\  M  e.  Grp )  ->  ( H  GrpHom  M )  =  ( H MndHom  M
) )
6358, 61, 62sylancl 662 . 2  |-  ( ph  ->  ( H  GrpHom  M )  =  ( H MndHom  M
) )
6456, 63eleqtrrd 2532 1  |-  ( ph  ->  ( X  |`  U )  e.  ( H  GrpHom  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791    \ cdif 3456    C_ wss 3459   {csn 4011   dom cdm 4986   ran crn 4987    |` cres 4988   "cima 4989   Fun wfun 5569   -->wf 5571   ` cfv 5575  (class class class)co 6278   CCcc 9490   0cc0 9492   NNcn 10539   NN0cn0 10798   Basecbs 14506   ↾s cress 14507   MndHom cmhm 15835  SubMndcsubmnd 15836   Grpcgrp 15924    GrpHom cghm 16135   Abelcabl 16670  mulGrpcmgp 17012   Ringcrg 17069   CRingccrg 17070  Unitcui 17159  ℂfldccnfld 18291  ℤ/nczn 18410  DChrcdchr 23376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-tpos 6954  df-recs 7041  df-rdg 7075  df-1o 7129  df-oadd 7133  df-er 7310  df-ec 7312  df-qs 7316  df-map 7421  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-sup 7900  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-div 10210  df-nn 10540  df-2 10597  df-3 10598  df-4 10599  df-5 10600  df-6 10601  df-7 10602  df-8 10603  df-9 10604  df-10 10605  df-n0 10799  df-z 10868  df-dec 10982  df-uz 11088  df-fz 11679  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-0g 14713  df-imas 14779  df-qus 14780  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-mhm 15837  df-submnd 15838  df-grp 15928  df-minusg 15929  df-sbg 15930  df-subg 16069  df-nsg 16070  df-eqg 16071  df-ghm 16136  df-cmn 16671  df-abl 16672  df-mgp 17013  df-ur 17025  df-ring 17071  df-cring 17072  df-oppr 17143  df-dvdsr 17161  df-unit 17162  df-invr 17192  df-dvr 17203  df-drng 17269  df-subrg 17298  df-lmod 17385  df-lss 17450  df-lsp 17489  df-sra 17689  df-rgmod 17690  df-lidl 17691  df-rsp 17692  df-2idl 17751  df-cnfld 18292  df-zring 18360  df-zn 18414  df-dchr 23377
This theorem is referenced by:  dchrabs  23404  sum2dchr  23418
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