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Theorem dchrelbas4 20482
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 20479 . . 3  |-  ( X  e.  D  ->  N  e.  NN )
4 dchrmhm.z . . . . 5  |-  Z  =  (ℤ/n `  N )
5 eqid 2283 . . . . 5  |-  ( Base `  Z )  =  (
Base `  Z )
6 eqid 2283 . . . . 5  |-  (Unit `  Z )  =  (Unit `  Z )
7 id 19 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN )
81, 4, 5, 6, 7, 2dchrelbas2 20476 . . . 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 9972 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  NN0 )
109adantr 451 . . . . . . 7  |-  ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  ->  N  e.  NN0 )
11 dchrelbas4.l . . . . . . . 8  |-  L  =  ( ZRHom `  Z
)
124, 5, 11znzrhfo 16501 . . . . . . 7  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
13 fveq2 5525 . . . . . . . . . 10  |-  ( ( L `  x )  =  y  ->  ( X `  ( L `  x ) )  =  ( X `  y
) )
1413neeq1d 2459 . . . . . . . . 9  |-  ( ( L `  x )  =  y  ->  (
( X `  ( L `  x )
)  =/=  0  <->  ( X `  y )  =/=  0 ) )
15 eleq1 2343 . . . . . . . . 9  |-  ( ( L `  x )  =  y  ->  (
( L `  x
)  e.  (Unit `  Z )  <->  y  e.  (Unit `  Z ) ) )
1614, 15imbi12d 311 . . . . . . . 8  |-  ( ( L `  x )  =  y  ->  (
( ( X `  ( L `  x ) )  =/=  0  -> 
( L `  x
)  e.  (Unit `  Z ) )  <->  ( ( X `  y )  =/=  0  ->  y  e.  (Unit `  Z )
) ) )
1716cbvfo 5799 . . . . . . 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 2448 . . . . . . . . . 10  |-  ( ( X `  ( L `
 x ) )  =/=  0  <->  -.  ( X `  ( L `  x ) )  =  0 )
2019a1i 10 . . . . . . . . 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 16517 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( L `  x )  e.  (Unit `  Z )  <->  ( x  gcd  N )  =  1 ) )
2210, 21sylan 457 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
( L `  x
)  e.  (Unit `  Z )  <->  ( x  gcd  N )  =  1 ) )
23 1re 8837 . . . . . . . . . . . . 13  |-  1  e.  RR
2423a1i 10 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  1  e.  RR )
25 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
26 simpll 730 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  N  e.  NN )
2726nnzd 10116 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  N  e.  ZZ )
28 nnne0 9778 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  ->  N  =/=  0 )
29 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  0  /\  N  =  0 )  ->  N  =  0 )
3029necon3ai 2486 . . . . . . . . . . . . . . 15  |-  ( N  =/=  0  ->  -.  ( x  =  0  /\  N  =  0
) )
3126, 28, 303syl 18 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  -.  ( x  =  0  /\  N  =  0
) )
32 gcdn0cl 12693 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( x  =  0  /\  N  =  0 ) )  ->  ( x  gcd  N )  e.  NN )
3325, 27, 31, 32syl21anc 1181 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
x  gcd  N )  e.  NN )
3433nnred 9761 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
x  gcd  N )  e.  RR )
3533nnge1d 9788 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  1  <_  ( x  gcd  N
) )
3624, 34, 35leltned 8970 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
1  <  ( x  gcd  N )  <->  ( x  gcd  N )  =/=  1
) )
3736necon2bbid 2504 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
( x  gcd  N
)  =  1  <->  -.  1  <  ( x  gcd  N ) ) )
3822, 37bitrd 244 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
( L `  x
)  e.  (Unit `  Z )  <->  -.  1  <  ( x  gcd  N
) ) )
3920, 38imbi12d 311 . . . . . . . 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 ) ) ) )
40 con34b 283 . . . . . . . 8  |-  ( ( 1  <  ( x  gcd  N )  -> 
( X `  ( L `  x )
)  =  0 )  <-> 
( -.  ( X `
 ( L `  x ) )  =  0  ->  -.  1  <  ( x  gcd  N
) ) )
4139, 40syl6bbr 254 . . . . . . 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 ) ) )
4241ralbidva 2559 . . . . . 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 ) ) )
4318, 42bitr3d 246 . . . . 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 ) ) )
4443pm5.32da 622 . . . 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 ) ) ) )
458, 44bitrd 244 . . 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 ) ) ) )
463, 45biadan2 623 . 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 ) ) ) )
47 3anass 938 . 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 ) ) ) )
4846, 47bitr4i 243 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   class class class wbr 4023   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    < clt 8867   NNcn 9746   NN0cn0 9965   ZZcz 10024    gcd cgcd 12685   Basecbs 13148   MndHom cmhm 14413  mulGrpcmgp 15325  Unitcui 15421  ℂfldccnfld 16377   ZRHomczrh 16451  ℤ/nczn 16454  DChrcdchr 20471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-imas 13411  df-divs 13412  df-mnd 14367  df-mhm 14415  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-rnghom 15496  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-cnfld 16378  df-zrh 16455  df-zn 16458  df-dchr 20472
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