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Theorem dchrelbas2 24028
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 24027 . 2  |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  ( ( B  \  U )  X. 
{ 0 } ) 
C_  X ) ) )
8 eqid 2429 . . . . . . . . . . 11  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
98, 3mgpbas 17664 . . . . . . . . . 10  |-  B  =  ( Base `  (mulGrp `  Z ) )
10 eqid 2429 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
11 cnfldbas 18909 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
1210, 11mgpbas 17664 . . . . . . . . . 10  |-  CC  =  ( Base `  (mulGrp ` fld ) )
139, 12mhmf 16538 . . . . . . . . 9  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  X : B --> CC )
1413adantl 467 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  ->  X : B --> CC )
15 ffun 5748 . . . . . . . 8  |-  ( X : B --> CC  ->  Fun 
X )
1614, 15syl 17 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  ->  Fun  X )
17 funssres 5641 . . . . . . 7  |-  ( ( Fun  X  /\  (
( B  \  U
)  X.  { 0 } )  C_  X
)  ->  ( X  |` 
dom  ( ( B 
\  U )  X. 
{ 0 } ) )  =  ( ( B  \  U )  X.  { 0 } ) )
1816, 17sylan 473 . . . . . 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 simpr 462 . . . . . . 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 } ) )  =  ( ( B 
\  U )  X. 
{ 0 } ) )
20 resss 5148 . . . . . . 7  |-  ( X  |`  dom  ( ( B 
\  U )  X. 
{ 0 } ) )  C_  X
2119, 20syl6eqssr 3521 . . . . . 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 )
2218, 21impbida 840 . . . . 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 } ) ) )
23 0cn 9634 . . . . . . . . 9  |-  0  e.  CC
24 fconst6g 5789 . . . . . . . . 9  |-  ( 0  e.  CC  ->  (
( B  \  U
)  X.  { 0 } ) : ( B  \  U ) --> CC )
2523, 24mp1i 13 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( B  \  U )  X.  {
0 } ) : ( B  \  U
) --> CC )
26 fdm 5750 . . . . . . . 8  |-  ( ( ( B  \  U
)  X.  { 0 } ) : ( B  \  U ) --> CC  ->  dom  ( ( B  \  U )  X.  { 0 } )  =  ( B 
\  U ) )
2725, 26syl 17 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  ->  dom  ( ( B  \  U )  X.  {
0 } )  =  ( B  \  U
) )
2827reseq2d 5125 . . . . . 6  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( X  |`  dom  (
( B  \  U
)  X.  { 0 } ) )  =  ( X  |`  ( B  \  U ) ) )
2928eqeq1d 2431 . . . . 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 } ) ) )
3022, 29bitrd 256 . . . 4  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( ( B 
\  U )  X. 
{ 0 } ) 
C_  X  <->  ( X  |`  ( B  \  U
) )  =  ( ( B  \  U
)  X.  { 0 } ) ) )
31 difss 3598 . . . . . . . 8  |-  ( B 
\  U )  C_  B
32 fssres 5766 . . . . . . . 8  |-  ( ( X : B --> CC  /\  ( B  \  U ) 
C_  B )  -> 
( X  |`  ( B  \  U ) ) : ( B  \  U ) --> CC )
3314, 31, 32sylancl 666 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( X  |`  ( B  \  U ) ) : ( B  \  U ) --> CC )
34 ffn 5746 . . . . . . 7  |-  ( ( X  |`  ( B  \  U ) ) : ( B  \  U
) --> CC  ->  ( X  |`  ( B  \  U ) )  Fn  ( B  \  U
) )
3533, 34syl 17 . . . . . 6  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( X  |`  ( B  \  U ) )  Fn  ( B  \  U ) )
36 ffn 5746 . . . . . . 7  |-  ( ( ( B  \  U
)  X.  { 0 } ) : ( B  \  U ) --> CC  ->  ( ( B  \  U )  X. 
{ 0 } )  Fn  ( B  \  U ) )
3725, 36syl 17 . . . . . 6  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( B  \  U )  X.  {
0 } )  Fn  ( B  \  U
) )
38 eqfnfv 5991 . . . . . 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
) ) )
3935, 37, 38syl2anc 665 . . . . 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 ) ) )
40 fvres 5895 . . . . . . . 8  |-  ( x  e.  ( B  \  U )  ->  (
( X  |`  ( B  \  U ) ) `
 x )  =  ( X `  x
) )
41 c0ex 9636 . . . . . . . . 9  |-  0  e.  _V
4241fvconst2 6135 . . . . . . . 8  |-  ( x  e.  ( B  \  U )  ->  (
( ( B  \  U )  X.  {
0 } ) `  x )  =  0 )
4340, 42eqeq12d 2451 . . . . . . 7  |-  ( x  e.  ( B  \  U )  ->  (
( ( X  |`  ( B  \  U ) ) `  x )  =  ( ( ( B  \  U )  X.  { 0 } ) `  x )  <-> 
( X `  x
)  =  0 ) )
4443ralbiia 2862 . . . . . 6  |-  ( A. x  e.  ( B  \  U ) ( ( X  |`  ( B  \  U ) ) `  x )  =  ( ( ( B  \  U )  X.  {
0 } ) `  x )  <->  A. x  e.  ( B  \  U
) ( X `  x )  =  0 )
45 eldif 3452 . . . . . . . . 9  |-  ( x  e.  ( B  \  U )  <->  ( x  e.  B  /\  -.  x  e.  U ) )
4645imbi1i 326 . . . . . . . 8  |-  ( ( x  e.  ( B 
\  U )  -> 
( X `  x
)  =  0 )  <-> 
( ( x  e.  B  /\  -.  x  e.  U )  ->  ( X `  x )  =  0 ) )
47 impexp 447 . . . . . . . 8  |-  ( ( ( x  e.  B  /\  -.  x  e.  U
)  ->  ( X `  x )  =  0 )  <->  ( x  e.  B  ->  ( -.  x  e.  U  ->  ( X `  x )  =  0 ) ) )
48 con1b 334 . . . . . . . . . 10  |-  ( ( -.  x  e.  U  ->  ( X `  x
)  =  0 )  <-> 
( -.  ( X `
 x )  =  0  ->  x  e.  U ) )
49 df-ne 2627 . . . . . . . . . . 11  |-  ( ( X `  x )  =/=  0  <->  -.  ( X `  x )  =  0 )
5049imbi1i 326 . . . . . . . . . 10  |-  ( ( ( X `  x
)  =/=  0  ->  x  e.  U )  <->  ( -.  ( X `  x )  =  0  ->  x  e.  U
) )
5148, 50bitr4i 255 . . . . . . . . 9  |-  ( ( -.  x  e.  U  ->  ( X `  x
)  =  0 )  <-> 
( ( X `  x )  =/=  0  ->  x  e.  U ) )
5251imbi2i 313 . . . . . . . 8  |-  ( ( x  e.  B  -> 
( -.  x  e.  U  ->  ( X `  x )  =  0 ) )  <->  ( x  e.  B  ->  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) )
5346, 47, 523bitri 274 . . . . . . 7  |-  ( ( x  e.  ( B 
\  U )  -> 
( X `  x
)  =  0 )  <-> 
( x  e.  B  ->  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) )
5453ralbii2 2861 . . . . . 6  |-  ( A. x  e.  ( B  \  U ) ( X `
 x )  =  0  <->  A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U ) )
5544, 54bitri 252 . . . . 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 ) )
5639, 55syl6bb 264 . . . 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 ) ) )
5730, 56bitrd 256 . . 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 ) ) )
5857pm5.32da 645 . 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 )
) ) )
597, 58bitrd 256 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782    \ cdif 3439    C_ wss 3442   {csn 4002    X. cxp 4852   dom cdm 4854    |` cres 4856   Fun wfun 5595    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9536   0cc0 9538   NNcn 10609   Basecbs 15084   MndHom cmhm 16531  mulGrpcmgp 17658  Unitcui 17802  ℂfldccnfld 18905  ℤ/nczn 19005  DChrcdchr 24023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-plusg 15165  df-mulr 15166  df-starv 15167  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-mhm 16533  df-mgp 17659  df-cnfld 18906  df-dchr 24024
This theorem is referenced by:  dchrelbas3  24029  dchrelbas4  24034  dchrmulcl  24040  dchrn0  24041  dchrmulid2  24043
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