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Theorem dchrelbas4 21032
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 21029 . . 3  |-  ( X  e.  D  ->  N  e.  NN )
4 dchrmhm.z . . . . 5  |-  Z  =  (ℤ/n `  N )
5 eqid 2438 . . . . 5  |-  ( Base `  Z )  =  (
Base `  Z )
6 eqid 2438 . . . . 5  |-  (Unit `  Z )  =  (Unit `  Z )
7 id 21 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN )
81, 4, 5, 6, 7, 2dchrelbas2 21026 . . . 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 10233 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  NN0 )
109adantr 453 . . . . . . 7  |-  ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  ->  N  e.  NN0 )
11 dchrelbas4.l . . . . . . . 8  |-  L  =  ( ZRHom `  Z
)
124, 5, 11znzrhfo 16833 . . . . . . 7  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
13 fveq2 5731 . . . . . . . . . 10  |-  ( ( L `  x )  =  y  ->  ( X `  ( L `  x ) )  =  ( X `  y
) )
1413neeq1d 2616 . . . . . . . . 9  |-  ( ( L `  x )  =  y  ->  (
( X `  ( L `  x )
)  =/=  0  <->  ( X `  y )  =/=  0 ) )
15 eleq1 2498 . . . . . . . . 9  |-  ( ( L `  x )  =  y  ->  (
( L `  x
)  e.  (Unit `  Z )  <->  y  e.  (Unit `  Z ) ) )
1614, 15imbi12d 313 . . . . . . . 8  |-  ( ( L `  x )  =  y  ->  (
( ( X `  ( L `  x ) )  =/=  0  -> 
( L `  x
)  e.  (Unit `  Z ) )  <->  ( ( X `  y )  =/=  0  ->  y  e.  (Unit `  Z )
) ) )
1716cbvfo 6025 . . . . . . 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 19 . . . . . 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 2603 . . . . . . . . . 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 16849 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( L `  x )  e.  (Unit `  Z )  <->  ( x  gcd  N )  =  1 ) )
2210, 21sylan 459 . . . . . . . . . 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 9095 . . . . . . . . . . . . 13  |-  1  e.  RR
2423a1i 11 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  1  e.  RR )
25 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
26 simpll 732 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  N  e.  NN )
2726nnzd 10379 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  N  e.  ZZ )
28 nnne0 10037 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  ->  N  =/=  0 )
29 simpr 449 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  0  /\  N  =  0 )  ->  N  =  0 )
3029necon3ai 2646 . . . . . . . . . . . . . . 15  |-  ( N  =/=  0  ->  -.  ( x  =  0  /\  N  =  0
) )
3126, 28, 303syl 19 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  -.  ( x  =  0  /\  N  =  0
) )
32 gcdn0cl 13019 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( x  =  0  /\  N  =  0 ) )  ->  ( x  gcd  N )  e.  NN )
3325, 27, 31, 32syl21anc 1184 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
x  gcd  N )  e.  NN )
3433nnred 10020 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
x  gcd  N )  e.  RR )
3533nnge1d 10047 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  1  <_  ( x  gcd  N
) )
3624, 34, 35leltned 9229 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
1  <  ( x  gcd  N )  <->  ( x  gcd  N )  =/=  1
) )
3736necon2bbid 2664 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
( x  gcd  N
)  =  1  <->  -.  1  <  ( x  gcd  N ) ) )
3822, 37bitrd 246 . . . . . . . . 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 313 . . . . . . . 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 285 . . . . . . . 8  |-  ( ( 1  <  ( x  gcd  N )  -> 
( X `  ( L `  x )
)  =  0 )  <-> 
( -.  ( X `
 ( L `  x ) )  =  0  ->  -.  1  <  ( x  gcd  N
) ) )
4139, 40syl6bbr 256 . . . . . . 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 2723 . . . . . 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 248 . . . . 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 624 . . . 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 246 . . 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 625 . 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 941 . 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 245 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   class class class wbr 4215   -onto->wfo 5455   ` cfv 5457  (class class class)co 6084   RRcr 8994   0cc0 8995   1c1 8996    < clt 9125   NNcn 10005   NN0cn0 10226   ZZcz 10287    gcd cgcd 13011   Basecbs 13474   MndHom cmhm 14741  mulGrpcmgp 15653  Unitcui 15749  ℂfldccnfld 16708   ZRHomczrh 16783  ℤ/nczn 16786  DChrcdchr 21021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-ec 6910  df-qs 6914  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-rp 10618  df-fz 11049  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-dvds 12858  df-gcd 13012  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-0g 13732  df-imas 13739  df-divs 13740  df-mnd 14695  df-mhm 14743  df-grp 14817  df-minusg 14818  df-sbg 14819  df-mulg 14820  df-subg 14946  df-nsg 14947  df-eqg 14948  df-ghm 15009  df-cmn 15419  df-abl 15420  df-mgp 15654  df-rng 15668  df-cring 15669  df-ur 15670  df-oppr 15733  df-dvdsr 15751  df-unit 15752  df-rnghom 15824  df-subrg 15871  df-lmod 15957  df-lss 16014  df-lsp 16053  df-sra 16249  df-rgmod 16250  df-lidl 16251  df-rsp 16252  df-2idl 16308  df-cnfld 16709  df-zrh 16787  df-zn 16790  df-dchr 21022
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