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Theorem dchrelbas2 20476
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/n `  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 ` fld )
)  /\  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
StepHypRef Expression
1 dchrval.g . . 3  |-  G  =  (DChr `  N )
2 dchrval.z . . 3  |-  Z  =  (ℤ/n `  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
)
71, 2, 3, 4, 5, 6dchrelbas 20475 . 2  |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  ( ( B  \  U )  X. 
{ 0 } ) 
C_  X ) ) )
8 eqid 2283 . . . . . . . . . . 11  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
98, 3mgpbas 15331 . . . . . . . . . 10  |-  B  =  ( Base `  (mulGrp `  Z ) )
10 eqid 2283 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
11 cnfldbas 16383 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
1210, 11mgpbas 15331 . . . . . . . . . 10  |-  CC  =  ( Base `  (mulGrp ` fld ) )
139, 12mhmf 14420 . . . . . . . . 9  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  X : B --> CC )
1413adantl 452 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  ->  X : B --> CC )
15 ffun 5391 . . . . . . . 8  |-  ( X : B --> CC  ->  Fun 
X )
1614, 15syl 15 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  ->  Fun  X )
17 funssres 5294 . . . . . . 7  |-  ( ( Fun  X  /\  (
( B  \  U
)  X.  { 0 } )  C_  X
)  ->  ( X  |` 
dom  ( ( B 
\  U )  X. 
{ 0 } ) )  =  ( ( B  \  U )  X.  { 0 } ) )
1816, 17sylan 457 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )  /\  ( ( B  \  U )  X.  {
0 } )  C_  X )  ->  ( X  |`  dom  ( ( B  \  U )  X.  { 0 } ) )  =  ( ( B  \  U
)  X.  { 0 } ) )
19 resss 4979 . . . . . . 7  |-  ( X  |`  dom  ( ( B 
\  U )  X. 
{ 0 } ) )  C_  X
20 simpr 447 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )  /\  ( X  |`  dom  (
( B  \  U
)  X.  { 0 } ) )  =  ( ( B  \  U )  X.  {
0 } ) )  ->  ( X  |`  dom  ( ( B  \  U )  X.  {
0 } ) )  =  ( ( B 
\  U )  X. 
{ 0 } ) )
2120sseq1d 3205 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )  /\  ( X  |`  dom  (
( B  \  U
)  X.  { 0 } ) )  =  ( ( B  \  U )  X.  {
0 } ) )  ->  ( ( X  |`  dom  ( ( B 
\  U )  X. 
{ 0 } ) )  C_  X  <->  ( ( B  \  U )  X. 
{ 0 } ) 
C_  X ) )
2219, 21mpbii 202 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )  /\  ( X  |`  dom  (
( B  \  U
)  X.  { 0 } ) )  =  ( ( B  \  U )  X.  {
0 } ) )  ->  ( ( B 
\  U )  X. 
{ 0 } ) 
C_  X )
2318, 22impbida 805 . . . . 5  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( ( B 
\  U )  X. 
{ 0 } ) 
C_  X  <->  ( X  |` 
dom  ( ( B 
\  U )  X. 
{ 0 } ) )  =  ( ( B  \  U )  X.  { 0 } ) ) )
24 0cn 8831 . . . . . . . . 9  |-  0  e.  CC
25 fconst6g 5430 . . . . . . . . 9  |-  ( 0  e.  CC  ->  (
( B  \  U
)  X.  { 0 } ) : ( B  \  U ) --> CC )
2624, 25mp1i 11 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( B  \  U )  X.  {
0 } ) : ( B  \  U
) --> CC )
27 fdm 5393 . . . . . . . 8  |-  ( ( ( B  \  U
)  X.  { 0 } ) : ( B  \  U ) --> CC  ->  dom  ( ( B  \  U )  X.  { 0 } )  =  ( B 
\  U ) )
2826, 27syl 15 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  ->  dom  ( ( B  \  U )  X.  {
0 } )  =  ( B  \  U
) )
2928reseq2d 4955 . . . . . 6  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( X  |`  dom  (
( B  \  U
)  X.  { 0 } ) )  =  ( X  |`  ( B  \  U ) ) )
3029eqeq1d 2291 . . . . 5  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( X  |`  dom  ( ( B  \  U )  X.  {
0 } ) )  =  ( ( B 
\  U )  X. 
{ 0 } )  <-> 
( X  |`  ( B  \  U ) )  =  ( ( B 
\  U )  X. 
{ 0 } ) ) )
3123, 30bitrd 244 . . . 4  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( ( B 
\  U )  X. 
{ 0 } ) 
C_  X  <->  ( X  |`  ( B  \  U
) )  =  ( ( B  \  U
)  X.  { 0 } ) ) )
32 difss 3303 . . . . . . . 8  |-  ( B 
\  U )  C_  B
33 fssres 5408 . . . . . . . 8  |-  ( ( X : B --> CC  /\  ( B  \  U ) 
C_  B )  -> 
( X  |`  ( B  \  U ) ) : ( B  \  U ) --> CC )
3414, 32, 33sylancl 643 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( X  |`  ( B  \  U ) ) : ( B  \  U ) --> CC )
35 ffn 5389 . . . . . . 7  |-  ( ( X  |`  ( B  \  U ) ) : ( B  \  U
) --> CC  ->  ( X  |`  ( B  \  U ) )  Fn  ( B  \  U
) )
3634, 35syl 15 . . . . . 6  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( X  |`  ( B  \  U ) )  Fn  ( B  \  U ) )
37 ffn 5389 . . . . . . 7  |-  ( ( ( B  \  U
)  X.  { 0 } ) : ( B  \  U ) --> CC  ->  ( ( B  \  U )  X. 
{ 0 } )  Fn  ( B  \  U ) )
3826, 37syl 15 . . . . . 6  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( B  \  U )  X.  {
0 } )  Fn  ( B  \  U
) )
39 eqfnfv 5622 . . . . . 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
) ) )
4036, 38, 39syl2anc 642 . . . . 5  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( X  |`  ( B  \  U ) )  =  ( ( B  \  U )  X.  { 0 } )  <->  A. x  e.  ( B  \  U ) ( ( X  |`  ( B  \  U ) ) `  x )  =  ( ( ( B  \  U )  X.  { 0 } ) `  x ) ) )
41 fvres 5542 . . . . . . . 8  |-  ( x  e.  ( B  \  U )  ->  (
( X  |`  ( B  \  U ) ) `
 x )  =  ( X `  x
) )
42 c0ex 8832 . . . . . . . . 9  |-  0  e.  _V
4342fvconst2 5729 . . . . . . . 8  |-  ( x  e.  ( B  \  U )  ->  (
( ( B  \  U )  X.  {
0 } ) `  x )  =  0 )
4441, 43eqeq12d 2297 . . . . . . 7  |-  ( x  e.  ( B  \  U )  ->  (
( ( X  |`  ( B  \  U ) ) `  x )  =  ( ( ( B  \  U )  X.  { 0 } ) `  x )  <-> 
( X `  x
)  =  0 ) )
4544ralbiia 2575 . . . . . 6  |-  ( A. x  e.  ( B  \  U ) ( ( X  |`  ( B  \  U ) ) `  x )  =  ( ( ( B  \  U )  X.  {
0 } ) `  x )  <->  A. x  e.  ( B  \  U
) ( X `  x )  =  0 )
46 eldif 3162 . . . . . . . . 9  |-  ( x  e.  ( B  \  U )  <->  ( x  e.  B  /\  -.  x  e.  U ) )
4746imbi1i 315 . . . . . . . 8  |-  ( ( x  e.  ( B 
\  U )  -> 
( X `  x
)  =  0 )  <-> 
( ( x  e.  B  /\  -.  x  e.  U )  ->  ( X `  x )  =  0 ) )
48 impexp 433 . . . . . . . 8  |-  ( ( ( x  e.  B  /\  -.  x  e.  U
)  ->  ( X `  x )  =  0 )  <->  ( x  e.  B  ->  ( -.  x  e.  U  ->  ( X `  x )  =  0 ) ) )
49 con1b 323 . . . . . . . . . 10  |-  ( ( -.  x  e.  U  ->  ( X `  x
)  =  0 )  <-> 
( -.  ( X `
 x )  =  0  ->  x  e.  U ) )
50 df-ne 2448 . . . . . . . . . . 11  |-  ( ( X `  x )  =/=  0  <->  -.  ( X `  x )  =  0 )
5150imbi1i 315 . . . . . . . . . 10  |-  ( ( ( X `  x
)  =/=  0  ->  x  e.  U )  <->  ( -.  ( X `  x )  =  0  ->  x  e.  U
) )
5249, 51bitr4i 243 . . . . . . . . 9  |-  ( ( -.  x  e.  U  ->  ( X `  x
)  =  0 )  <-> 
( ( X `  x )  =/=  0  ->  x  e.  U ) )
5352imbi2i 303 . . . . . . . 8  |-  ( ( x  e.  B  -> 
( -.  x  e.  U  ->  ( X `  x )  =  0 ) )  <->  ( x  e.  B  ->  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) )
5447, 48, 533bitri 262 . . . . . . 7  |-  ( ( x  e.  ( B 
\  U )  -> 
( X `  x
)  =  0 )  <-> 
( x  e.  B  ->  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) )
5554ralbii2 2571 . . . . . 6  |-  ( A. x  e.  ( B  \  U ) ( X `
 x )  =  0  <->  A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U ) )
5645, 55bitri 240 . . . . 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 ) )
5740, 56syl6bb 252 . . . 4  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( X  |`  ( B  \  U ) )  =  ( ( B  \  U )  X.  { 0 } )  <->  A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) )
5831, 57bitrd 244 . . 3  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( ( B 
\  U )  X. 
{ 0 } ) 
C_  X  <->  A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) )
5958pm5.32da 622 . 2  |-  ( ph  ->  ( ( X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  /\  (
( B  \  U
)  X.  { 0 } )  C_  X
)  <->  ( X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  B  (
( X `  x
)  =/=  0  ->  x  e.  U )
) ) )
607, 59bitrd 244 1  |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149    C_ wss 3152   {csn 3640    X. cxp 4687   dom cdm 4689    |` cres 4691   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   NNcn 9746   Basecbs 13148   MndHom cmhm 14413  mulGrpcmgp 15325  Unitcui 15421  ℂfldccnfld 16377  ℤ/nczn 16454  DChrcdchr 20471
This theorem is referenced by:  dchrelbas3  20477  dchrelbas4  20482  dchrmulcl  20488  dchrn0  20489  dchrmulid2  20491
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-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
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-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-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  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-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-mhm 14415  df-mgp 15326  df-cnfld 16378  df-dchr 20472
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