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Theorem dchrelbas2 22581
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 22580 . 2  |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  ( ( B  \  U )  X. 
{ 0 } ) 
C_  X ) ) )
8 eqid 2443 . . . . . . . . . . 11  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
98, 3mgpbas 16602 . . . . . . . . . 10  |-  B  =  ( Base `  (mulGrp `  Z ) )
10 eqid 2443 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
11 cnfldbas 17827 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
1210, 11mgpbas 16602 . . . . . . . . . 10  |-  CC  =  ( Base `  (mulGrp ` fld ) )
139, 12mhmf 15474 . . . . . . . . 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 5566 . . . . . . . 8  |-  ( X : B --> CC  ->  Fun 
X )
1614, 15syl 16 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  ->  Fun  X )
17 funssres 5463 . . . . . . 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 5139 . . . . . . 7  |-  ( X  |`  dom  ( ( B 
\  U )  X. 
{ 0 } ) )  C_  X
2119, 20syl6eqssr 3412 . . . . . 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 828 . . . . 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 9383 . . . . . . . . 9  |-  0  e.  CC
24 fconst6g 5604 . . . . . . . . 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 5568 . . . . . . . 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 5115 . . . . . 6  |-  ( (
ph  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )  -> 
( X  |`  dom  (
( B  \  U
)  X.  { 0 } ) )  =  ( X  |`  ( B  \  U ) ) )
2928eqeq1d 2451 . . . . 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 3488 . . . . . . . 8  |-  ( B 
\  U )  C_  B
32 fssres 5583 . . . . . . . 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 5564 . . . . . . 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 5564 . . . . . . 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 5802 . . . . . 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 5709 . . . . . . . 8  |-  ( x  e.  ( B  \  U )  ->  (
( X  |`  ( B  \  U ) ) `
 x )  =  ( X `  x
) )
41 c0ex 9385 . . . . . . . . 9  |-  0  e.  _V
4241fvconst2 5938 . . . . . . . 8  |-  ( x  e.  ( B  \  U )  ->  (
( ( B  \  U )  X.  {
0 } ) `  x )  =  0 )
4340, 42eqeq12d 2457 . . . . . . 7  |-  ( x  e.  ( B  \  U )  ->  (
( ( X  |`  ( B  \  U ) ) `  x )  =  ( ( ( B  \  U )  X.  { 0 } ) `  x )  <-> 
( X `  x
)  =  0 ) )
4443ralbiia 2752 . . . . . 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 3343 . . . . . . . . 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 2613 . . . . . . . . . . 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 2748 . . . . . 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 1369    e. wcel 1756    =/= wne 2611   A.wral 2720    \ cdif 3330    C_ wss 3333   {csn 3882    X. cxp 4843   dom cdm 4845    |` cres 4847   Fun wfun 5417    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   CCcc 9285   0cc0 9287   NNcn 10327   Basecbs 14179   MndHom cmhm 15467  mulGrpcmgp 16596  Unitcui 16736  ℂfldccnfld 17823  ℤ/nczn 17939  DChrcdchr 22576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-plusg 14256  df-mulr 14257  df-starv 14258  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-mhm 15469  df-mgp 16597  df-cnfld 17824  df-dchr 22577
This theorem is referenced by:  dchrelbas3  22582  dchrelbas4  22587  dchrmulcl  22593  dchrn0  22594  dchrmulid2  22596
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