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Theorem dchrelbas4 20498
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 20495 . . 3  |-  ( X  e.  D  ->  N  e.  NN )
4 dchrmhm.z . . . . 5  |-  Z  =  (ℤ/n `  N )
5 eqid 2296 . . . . 5  |-  ( Base `  Z )  =  (
Base `  Z )
6 eqid 2296 . . . . 5  |-  (Unit `  Z )  =  (Unit `  Z )
7 id 19 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN )
81, 4, 5, 6, 7, 2dchrelbas2 20492 . . . 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 9988 . . . . . . . 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 16517 . . . . . . 7  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
13 fveq2 5541 . . . . . . . . . 10  |-  ( ( L `  x )  =  y  ->  ( X `  ( L `  x ) )  =  ( X `  y
) )
1413neeq1d 2472 . . . . . . . . 9  |-  ( ( L `  x )  =  y  ->  (
( X `  ( L `  x )
)  =/=  0  <->  ( X `  y )  =/=  0 ) )
15 eleq1 2356 . . . . . . . . 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 5815 . . . . . . 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 2461 . . . . . . . . . 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 16533 . . . . . . . . . . 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 8853 . . . . . . . . . . . . 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 10132 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  N  e.  ZZ )
28 nnne0 9794 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  ->  N  =/=  0 )
29 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  0  /\  N  =  0 )  ->  N  =  0 )
3029necon3ai 2499 . . . . . . . . . . . . . . 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 12709 . . . . . . . . . . . . . 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 9777 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
x  gcd  N )  e.  RR )
3533nnge1d 9804 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  1  <_  ( x  gcd  N
) )
3624, 34, 35leltned 8986 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
1  <  ( x  gcd  N )  <->  ( x  gcd  N )  =/=  1
) )
3736necon2bbid 2517 . . . . . . . . . 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 2572 . . . . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   class class class wbr 4039   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    < clt 8883   NNcn 9762   NN0cn0 9981   ZZcz 10040    gcd cgcd 12701   Basecbs 13164   MndHom cmhm 14429  mulGrpcmgp 15341  Unitcui 15437  ℂfldccnfld 16393   ZRHomczrh 16467  ℤ/nczn 16470  DChrcdchr 20487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-rp 10371  df-fz 10799  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-imas 13427  df-divs 13428  df-mnd 14383  df-mhm 14431  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-nsg 14635  df-eqg 14636  df-ghm 14697  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-rnghom 15512  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-sra 15941  df-rgmod 15942  df-lidl 15943  df-rsp 15944  df-2idl 16000  df-cnfld 16394  df-zrh 16471  df-zn 16474  df-dchr 20488
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