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Theorem dchrelbas2 23268
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 23267 . 2  |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  ( ( B  \  U )  X. 
{ 0 } ) 
C_  X ) ) )
8 eqid 2467 . . . . . . . . . . 11  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
98, 3mgpbas 16949 . . . . . . . . . 10  |-  B  =  ( Base `  (mulGrp `  Z ) )
10 eqid 2467 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
11 cnfldbas 18223 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
1210, 11mgpbas 16949 . . . . . . . . . 10  |-  CC  =  ( Base `  (mulGrp ` fld ) )
139, 12mhmf 15791 . . . . . . . . 9  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  X : B --> CC )
1413adantl 466 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  ->  X : B --> CC )
15 ffun 5733 . . . . . . . 8  |-  ( X : B --> CC  ->  Fun 
X )
1614, 15syl 16 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  ->  Fun  X )
17 funssres 5628 . . . . . . 7  |-  ( ( Fun  X  /\  (
( B  \  U
)  X.  { 0 } )  C_  X
)  ->  ( X  |` 
dom  ( ( B 
\  U )  X. 
{ 0 } ) )  =  ( ( B  \  U )  X.  { 0 } ) )
1816, 17sylan 471 . . . . . 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 461 . . . . . . 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 5297 . . . . . . 7  |-  ( X  |`  dom  ( ( B 
\  U )  X. 
{ 0 } ) )  C_  X
2119, 20syl6eqssr 3555 . . . . . 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 830 . . . . 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 9588 . . . . . . . . 9  |-  0  e.  CC
24 fconst6g 5774 . . . . . . . . 9  |-  ( 0  e.  CC  ->  (
( B  \  U
)  X.  { 0 } ) : ( B  \  U ) --> CC )
2523, 24mp1i 12 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( B  \  U )  X.  {
0 } ) : ( B  \  U
) --> CC )
26 fdm 5735 . . . . . . . 8  |-  ( ( ( B  \  U
)  X.  { 0 } ) : ( B  \  U ) --> CC  ->  dom  ( ( B  \  U )  X.  { 0 } )  =  ( B 
\  U ) )
2725, 26syl 16 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  ->  dom  ( ( B  \  U )  X.  {
0 } )  =  ( B  \  U
) )
2827reseq2d 5273 . . . . . 6  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( X  |`  dom  (
( B  \  U
)  X.  { 0 } ) )  =  ( X  |`  ( B  \  U ) ) )
2928eqeq1d 2469 . . . . 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 253 . . . 4  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( ( B 
\  U )  X. 
{ 0 } ) 
C_  X  <->  ( X  |`  ( B  \  U
) )  =  ( ( B  \  U
)  X.  { 0 } ) ) )
31 difss 3631 . . . . . . . 8  |-  ( B 
\  U )  C_  B
32 fssres 5751 . . . . . . . 8  |-  ( ( X : B --> CC  /\  ( B  \  U ) 
C_  B )  -> 
( X  |`  ( B  \  U ) ) : ( B  \  U ) --> CC )
3314, 31, 32sylancl 662 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( X  |`  ( B  \  U ) ) : ( B  \  U ) --> CC )
34 ffn 5731 . . . . . . 7  |-  ( ( X  |`  ( B  \  U ) ) : ( B  \  U
) --> CC  ->  ( X  |`  ( B  \  U ) )  Fn  ( B  \  U
) )
3533, 34syl 16 . . . . . 6  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( X  |`  ( B  \  U ) )  Fn  ( B  \  U ) )
36 ffn 5731 . . . . . . 7  |-  ( ( ( B  \  U
)  X.  { 0 } ) : ( B  \  U ) --> CC  ->  ( ( B  \  U )  X. 
{ 0 } )  Fn  ( B  \  U ) )
3725, 36syl 16 . . . . . 6  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( B  \  U )  X.  {
0 } )  Fn  ( B  \  U
) )
38 eqfnfv 5975 . . . . . 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 661 . . . . 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 5880 . . . . . . . 8  |-  ( x  e.  ( B  \  U )  ->  (
( X  |`  ( B  \  U ) ) `
 x )  =  ( X `  x
) )
41 c0ex 9590 . . . . . . . . 9  |-  0  e.  _V
4241fvconst2 6116 . . . . . . . 8  |-  ( x  e.  ( B  \  U )  ->  (
( ( B  \  U )  X.  {
0 } ) `  x )  =  0 )
4340, 42eqeq12d 2489 . . . . . . 7  |-  ( x  e.  ( B  \  U )  ->  (
( ( X  |`  ( B  \  U ) ) `  x )  =  ( ( ( B  \  U )  X.  { 0 } ) `  x )  <-> 
( X `  x
)  =  0 ) )
4443ralbiia 2894 . . . . . 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 3486 . . . . . . . . 9  |-  ( x  e.  ( B  \  U )  <->  ( x  e.  B  /\  -.  x  e.  U ) )
4645imbi1i 325 . . . . . . . 8  |-  ( ( x  e.  ( B 
\  U )  -> 
( X `  x
)  =  0 )  <-> 
( ( x  e.  B  /\  -.  x  e.  U )  ->  ( X `  x )  =  0 ) )
47 impexp 446 . . . . . . . 8  |-  ( ( ( x  e.  B  /\  -.  x  e.  U
)  ->  ( X `  x )  =  0 )  <->  ( x  e.  B  ->  ( -.  x  e.  U  ->  ( X `  x )  =  0 ) ) )
48 con1b 333 . . . . . . . . . 10  |-  ( ( -.  x  e.  U  ->  ( X `  x
)  =  0 )  <-> 
( -.  ( X `
 x )  =  0  ->  x  e.  U ) )
49 df-ne 2664 . . . . . . . . . . 11  |-  ( ( X `  x )  =/=  0  <->  -.  ( X `  x )  =  0 )
5049imbi1i 325 . . . . . . . . . 10  |-  ( ( ( X `  x
)  =/=  0  ->  x  e.  U )  <->  ( -.  ( X `  x )  =  0  ->  x  e.  U
) )
5148, 50bitr4i 252 . . . . . . . . 9  |-  ( ( -.  x  e.  U  ->  ( X `  x
)  =  0 )  <-> 
( ( X `  x )  =/=  0  ->  x  e.  U ) )
5251imbi2i 312 . . . . . . . 8  |-  ( ( x  e.  B  -> 
( -.  x  e.  U  ->  ( X `  x )  =  0 ) )  <->  ( x  e.  B  ->  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) )
5346, 47, 523bitri 271 . . . . . . 7  |-  ( ( x  e.  ( B 
\  U )  -> 
( X `  x
)  =  0 )  <-> 
( x  e.  B  ->  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) )
5453ralbii2 2893 . . . . . 6  |-  ( A. x  e.  ( B  \  U ) ( X `
 x )  =  0  <->  A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U ) )
5544, 54bitri 249 . . . . 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 261 . . . 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 253 . . 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 641 . 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 253 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 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    \ cdif 3473    C_ wss 3476   {csn 4027    X. cxp 4997   dom cdm 4999    |` cres 5001   Fun wfun 5582    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   CCcc 9490   0cc0 9492   NNcn 10536   Basecbs 14490   MndHom cmhm 15784  mulGrpcmgp 16943  Unitcui 17089  ℂfldccnfld 18219  ℤ/nczn 18335  DChrcdchr 23263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-fz 11673  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-plusg 14568  df-mulr 14569  df-starv 14570  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-mhm 15786  df-mgp 16944  df-cnfld 18220  df-dchr 23264
This theorem is referenced by:  dchrelbas3  23269  dchrelbas4  23274  dchrmulcl  23280  dchrn0  23281  dchrmulid2  23283
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