MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dchrelbas4 Structured version   Unicode version

Theorem dchrelbas4 23901
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
dchrmhm.g  |-  G  =  (DChr `  N )
dchrmhm.z  |-  Z  =  (ℤ/n `  N )
dchrmhm.b  |-  D  =  ( Base `  G
)
dchrelbas4.l  |-  L  =  ( ZRHom `  Z
)
Assertion
Ref Expression
dchrelbas4  |-  ( X  e.  D  <->  ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ZZ  (
1  <  ( x  gcd  N )  ->  ( X `  ( L `  x ) )  =  0 ) ) )
Distinct variable groups:    x, L    x, N    x, X    x, Z    x, D
Allowed substitution hint:    G( x)

Proof of Theorem dchrelbas4
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dchrmhm.g . . . 4  |-  G  =  (DChr `  N )
2 dchrmhm.b . . . 4  |-  D  =  ( Base `  G
)
31, 2dchrrcl 23898 . . 3  |-  ( X  e.  D  ->  N  e.  NN )
4 dchrmhm.z . . . . 5  |-  Z  =  (ℤ/n `  N )
5 eqid 2404 . . . . 5  |-  ( Base `  Z )  =  (
Base `  Z )
6 eqid 2404 . . . . 5  |-  (Unit `  Z )  =  (Unit `  Z )
7 id 23 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN )
81, 4, 5, 6, 7, 2dchrelbas2 23895 . . . 4  |-  ( N  e.  NN  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  /\  A. y  e.  ( Base `  Z ) ( ( X `  y )  =/=  0  ->  y  e.  (Unit `  Z )
) ) ) )
9 nnnn0 10845 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  NN0 )
109adantr 465 . . . . . . 7  |-  ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  ->  N  e.  NN0 )
11 dchrelbas4.l . . . . . . . 8  |-  L  =  ( ZRHom `  Z
)
124, 5, 11znzrhfo 18886 . . . . . . 7  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
13 fveq2 5851 . . . . . . . . . 10  |-  ( ( L `  x )  =  y  ->  ( X `  ( L `  x ) )  =  ( X `  y
) )
1413neeq1d 2682 . . . . . . . . 9  |-  ( ( L `  x )  =  y  ->  (
( X `  ( L `  x )
)  =/=  0  <->  ( X `  y )  =/=  0 ) )
15 eleq1 2476 . . . . . . . . 9  |-  ( ( L `  x )  =  y  ->  (
( L `  x
)  e.  (Unit `  Z )  <->  y  e.  (Unit `  Z ) ) )
1614, 15imbi12d 320 . . . . . . . 8  |-  ( ( L `  x )  =  y  ->  (
( ( X `  ( L `  x ) )  =/=  0  -> 
( L `  x
)  e.  (Unit `  Z ) )  <->  ( ( X `  y )  =/=  0  ->  y  e.  (Unit `  Z )
) ) )
1716cbvfo 6177 . . . . . . 7  |-  ( L : ZZ -onto-> ( Base `  Z )  ->  ( A. x  e.  ZZ  ( ( X `  ( L `  x ) )  =/=  0  -> 
( L `  x
)  e.  (Unit `  Z ) )  <->  A. y  e.  ( Base `  Z
) ( ( X `
 y )  =/=  0  ->  y  e.  (Unit `  Z ) ) ) )
1810, 12, 173syl 18 . . . . . 6  |-  ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  ->  ( A. x  e.  ZZ  ( ( X `  ( L `  x ) )  =/=  0  -> 
( L `  x
)  e.  (Unit `  Z ) )  <->  A. y  e.  ( Base `  Z
) ( ( X `
 y )  =/=  0  ->  y  e.  (Unit `  Z ) ) ) )
19 df-ne 2602 . . . . . . . . . 10  |-  ( ( X `  ( L `
 x ) )  =/=  0  <->  -.  ( X `  ( L `  x ) )  =  0 )
2019a1i 11 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
( X `  ( L `  x )
)  =/=  0  <->  -.  ( X `  ( L `
 x ) )  =  0 ) )
214, 6, 11znunit 18902 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( L `  x )  e.  (Unit `  Z )  <->  ( x  gcd  N )  =  1 ) )
2210, 21sylan 471 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
( L `  x
)  e.  (Unit `  Z )  <->  ( x  gcd  N )  =  1 ) )
23 1red 9643 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  1  e.  RR )
24 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
25 simpll 754 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  N  e.  NN )
2625nnzd 11009 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  N  e.  ZZ )
27 nnne0 10611 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  ->  N  =/=  0 )
28 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  0  /\  N  =  0 )  ->  N  =  0 )
2928necon3ai 2633 . . . . . . . . . . . . . . 15  |-  ( N  =/=  0  ->  -.  ( x  =  0  /\  N  =  0
) )
3025, 27, 293syl 18 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  -.  ( x  =  0  /\  N  =  0
) )
31 gcdn0cl 14363 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( x  =  0  /\  N  =  0 ) )  ->  ( x  gcd  N )  e.  NN )
3224, 26, 30, 31syl21anc 1231 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
x  gcd  N )  e.  NN )
3332nnred 10593 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
x  gcd  N )  e.  RR )
3432nnge1d 10621 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  1  <_  ( x  gcd  N
) )
3523, 33, 34leltned 9772 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
1  <  ( x  gcd  N )  <->  ( x  gcd  N )  =/=  1
) )
3635necon2bbid 2661 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
( x  gcd  N
)  =  1  <->  -.  1  <  ( x  gcd  N ) ) )
3722, 36bitrd 255 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
( L `  x
)  e.  (Unit `  Z )  <->  -.  1  <  ( x  gcd  N
) ) )
3820, 37imbi12d 320 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
( ( X `  ( L `  x ) )  =/=  0  -> 
( L `  x
)  e.  (Unit `  Z ) )  <->  ( -.  ( X `  ( L `
 x ) )  =  0  ->  -.  1  <  ( x  gcd  N ) ) ) )
39 con34b 292 . . . . . . . 8  |-  ( ( 1  <  ( x  gcd  N )  -> 
( X `  ( L `  x )
)  =  0 )  <-> 
( -.  ( X `
 ( L `  x ) )  =  0  ->  -.  1  <  ( x  gcd  N
) ) )
4038, 39syl6bbr 265 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
( ( X `  ( L `  x ) )  =/=  0  -> 
( L `  x
)  e.  (Unit `  Z ) )  <->  ( 1  <  ( x  gcd  N )  ->  ( X `  ( L `  x
) )  =  0 ) ) )
4140ralbidva 2842 . . . . . 6  |-  ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  ->  ( A. x  e.  ZZ  ( ( X `  ( L `  x ) )  =/=  0  -> 
( L `  x
)  e.  (Unit `  Z ) )  <->  A. x  e.  ZZ  ( 1  < 
( x  gcd  N
)  ->  ( X `  ( L `  x
) )  =  0 ) ) )
4218, 41bitr3d 257 . . . . 5  |-  ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  ->  ( A. y  e.  ( Base `  Z ) ( ( X `  y
)  =/=  0  -> 
y  e.  (Unit `  Z ) )  <->  A. x  e.  ZZ  ( 1  < 
( x  gcd  N
)  ->  ( X `  ( L `  x
) )  =  0 ) ) )
4342pm5.32da 641 . . . 4  |-  ( N  e.  NN  ->  (
( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. y  e.  ( Base `  Z ) ( ( X `  y )  =/=  0  ->  y  e.  (Unit `  Z )
) )  <->  ( X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ZZ  (
1  <  ( x  gcd  N )  ->  ( X `  ( L `  x ) )  =  0 ) ) ) )
448, 43bitrd 255 . . 3  |-  ( N  e.  NN  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ZZ  (
1  <  ( x  gcd  N )  ->  ( X `  ( L `  x ) )  =  0 ) ) ) )
453, 44biadan2 642 . 2  |-  ( X  e.  D  <->  ( N  e.  NN  /\  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ZZ  (
1  <  ( x  gcd  N )  ->  ( X `  ( L `  x ) )  =  0 ) ) ) )
46 3anass 980 . 2  |-  ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  A. x  e.  ZZ  ( 1  < 
( x  gcd  N
)  ->  ( X `  ( L `  x
) )  =  0 ) )  <->  ( N  e.  NN  /\  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ZZ  (
1  <  ( x  gcd  N )  ->  ( X `  ( L `  x ) )  =  0 ) ) ) )
4745, 46bitr4i 254 1  |-  ( X  e.  D  <->  ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  ZZ  (
1  <  ( x  gcd  N )  ->  ( X `  ( L `  x ) )  =  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    =/= wne 2600   A.wral 2756   class class class wbr 4397   -onto->wfo 5569   ` cfv 5571  (class class class)co 6280   0cc0 9524   1c1 9525    < clt 9660   NNcn 10578   NN0cn0 10838   ZZcz 10907    gcd cgcd 14355   Basecbs 14843   MndHom cmhm 16290  mulGrpcmgp 17463  Unitcui 17610  ℂfldccnfld 18742   ZRHomczrh 18839  ℤ/nczn 18842  DChrcdchr 23890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602  ax-addf 9603  ax-mulf 9604
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-tpos 6960  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-ec 7352  df-qs 7356  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7937  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-rp 11268  df-fz 11729  df-fl 11968  df-mod 12037  df-seq 12154  df-exp 12213  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-dvds 14198  df-gcd 14356  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-starv 14926  df-sca 14927  df-vsca 14928  df-ip 14929  df-tset 14930  df-ple 14931  df-ds 14933  df-unif 14934  df-0g 15058  df-imas 15124  df-qus 15125  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-mhm 16292  df-grp 16383  df-minusg 16384  df-sbg 16385  df-mulg 16386  df-subg 16524  df-nsg 16525  df-eqg 16526  df-ghm 16591  df-cmn 17126  df-abl 17127  df-mgp 17464  df-ur 17476  df-ring 17522  df-cring 17523  df-oppr 17594  df-dvdsr 17612  df-unit 17613  df-rnghom 17686  df-subrg 17749  df-lmod 17836  df-lss 17901  df-lsp 17940  df-sra 18140  df-rgmod 18141  df-lidl 18142  df-rsp 18143  df-2idl 18202  df-cnfld 18743  df-zring 18811  df-zrh 18843  df-zn 18846  df-dchr 23891
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator