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Theorem dchrelbas2 20888
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 20887 . 2  |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  ( ( B  \  U )  X. 
{ 0 } ) 
C_  X ) ) )
8 eqid 2387 . . . . . . . . . . 11  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
98, 3mgpbas 15581 . . . . . . . . . 10  |-  B  =  ( Base `  (mulGrp `  Z ) )
10 eqid 2387 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
11 cnfldbas 16630 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
1210, 11mgpbas 15581 . . . . . . . . . 10  |-  CC  =  ( Base `  (mulGrp ` fld ) )
139, 12mhmf 14670 . . . . . . . . 9  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  X : B --> CC )
1413adantl 453 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  ->  X : B --> CC )
15 ffun 5533 . . . . . . . 8  |-  ( X : B --> CC  ->  Fun 
X )
1614, 15syl 16 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  ->  Fun  X )
17 funssres 5433 . . . . . . 7  |-  ( ( Fun  X  /\  (
( B  \  U
)  X.  { 0 } )  C_  X
)  ->  ( X  |` 
dom  ( ( B 
\  U )  X. 
{ 0 } ) )  =  ( ( B  \  U )  X.  { 0 } ) )
1816, 17sylan 458 . . . . . 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 448 . . . . . . 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 5110 . . . . . . 7  |-  ( X  |`  dom  ( ( B 
\  U )  X. 
{ 0 } ) )  C_  X
2119, 20syl6eqssr 3342 . . . . . 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 806 . . . . 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 9017 . . . . . . . . 9  |-  0  e.  CC
24 fconst6g 5572 . . . . . . . . 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 5535 . . . . . . . 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 5086 . . . . . 6  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( X  |`  dom  (
( B  \  U
)  X.  { 0 } ) )  =  ( X  |`  ( B  \  U ) ) )
2928eqeq1d 2395 . . . . 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 245 . . . 4  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( ( ( B 
\  U )  X. 
{ 0 } ) 
C_  X  <->  ( X  |`  ( B  \  U
) )  =  ( ( B  \  U
)  X.  { 0 } ) ) )
31 difss 3417 . . . . . . . 8  |-  ( B 
\  U )  C_  B
32 fssres 5550 . . . . . . . 8  |-  ( ( X : B --> CC  /\  ( B  \  U ) 
C_  B )  -> 
( X  |`  ( B  \  U ) ) : ( B  \  U ) --> CC )
3314, 31, 32sylancl 644 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( X  |`  ( B  \  U ) ) : ( B  \  U ) --> CC )
34 ffn 5531 . . . . . . 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 5531 . . . . . . 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 5766 . . . . . 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 643 . . . . 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 5685 . . . . . . . 8  |-  ( x  e.  ( B  \  U )  ->  (
( X  |`  ( B  \  U ) ) `
 x )  =  ( X `  x
) )
41 c0ex 9018 . . . . . . . . 9  |-  0  e.  _V
4241fvconst2 5886 . . . . . . . 8  |-  ( x  e.  ( B  \  U )  ->  (
( ( B  \  U )  X.  {
0 } ) `  x )  =  0 )
4340, 42eqeq12d 2401 . . . . . . 7  |-  ( x  e.  ( B  \  U )  ->  (
( ( X  |`  ( B  \  U ) ) `  x )  =  ( ( ( B  \  U )  X.  { 0 } ) `  x )  <-> 
( X `  x
)  =  0 ) )
4443ralbiia 2681 . . . . . 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 3273 . . . . . . . . 9  |-  ( x  e.  ( B  \  U )  <->  ( x  e.  B  /\  -.  x  e.  U ) )
4645imbi1i 316 . . . . . . . 8  |-  ( ( x  e.  ( B 
\  U )  -> 
( X `  x
)  =  0 )  <-> 
( ( x  e.  B  /\  -.  x  e.  U )  ->  ( X `  x )  =  0 ) )
47 impexp 434 . . . . . . . 8  |-  ( ( ( x  e.  B  /\  -.  x  e.  U
)  ->  ( X `  x )  =  0 )  <->  ( x  e.  B  ->  ( -.  x  e.  U  ->  ( X `  x )  =  0 ) ) )
48 con1b 324 . . . . . . . . . 10  |-  ( ( -.  x  e.  U  ->  ( X `  x
)  =  0 )  <-> 
( -.  ( X `
 x )  =  0  ->  x  e.  U ) )
49 df-ne 2552 . . . . . . . . . . 11  |-  ( ( X `  x )  =/=  0  <->  -.  ( X `  x )  =  0 )
5049imbi1i 316 . . . . . . . . . 10  |-  ( ( ( X `  x
)  =/=  0  ->  x  e.  U )  <->  ( -.  ( X `  x )  =  0  ->  x  e.  U
) )
5148, 50bitr4i 244 . . . . . . . . 9  |-  ( ( -.  x  e.  U  ->  ( X `  x
)  =  0 )  <-> 
( ( X `  x )  =/=  0  ->  x  e.  U ) )
5251imbi2i 304 . . . . . . . 8  |-  ( ( x  e.  B  -> 
( -.  x  e.  U  ->  ( X `  x )  =  0 ) )  <->  ( x  e.  B  ->  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) )
5346, 47, 523bitri 263 . . . . . . 7  |-  ( ( x  e.  ( B 
\  U )  -> 
( X `  x
)  =  0 )  <-> 
( x  e.  B  ->  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) )
5453ralbii2 2677 . . . . . 6  |-  ( A. x  e.  ( B  \  U ) ( X `
 x )  =  0  <->  A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U ) )
5544, 54bitri 241 . . . . 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 253 . . . 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 245 . . 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 623 . 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 245 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 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649    \ cdif 3260    C_ wss 3263   {csn 3757    X. cxp 4816   dom cdm 4818    |` cres 4820   Fun wfun 5388    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   CCcc 8921   0cc0 8923   NNcn 9932   Basecbs 13396   MndHom cmhm 14663  mulGrpcmgp 15575  Unitcui 15671  ℂfldccnfld 16626  ℤ/nczn 16704  DChrcdchr 20883
This theorem is referenced by:  dchrelbas3  20889  dchrelbas4  20894  dchrmulcl  20900  dchrn0  20901  dchrmulid2  20903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-plusg 13469  df-mulr 13470  df-starv 13471  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-mhm 14665  df-mgp 15576  df-cnfld 16627  df-dchr 20884
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