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Theorem dchrelbas4 22597
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 22594 . . 3  |-  ( X  e.  D  ->  N  e.  NN )
4 dchrmhm.z . . . . 5  |-  Z  =  (ℤ/n `  N )
5 eqid 2443 . . . . 5  |-  ( Base `  Z )  =  (
Base `  Z )
6 eqid 2443 . . . . 5  |-  (Unit `  Z )  =  (Unit `  Z )
7 id 22 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN )
81, 4, 5, 6, 7, 2dchrelbas2 22591 . . . 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 10601 . . . . . . . 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 17995 . . . . . . 7  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
13 fveq2 5706 . . . . . . . . . 10  |-  ( ( L `  x )  =  y  ->  ( X `  ( L `  x ) )  =  ( X `  y
) )
1413neeq1d 2636 . . . . . . . . 9  |-  ( ( L `  x )  =  y  ->  (
( X `  ( L `  x )
)  =/=  0  <->  ( X `  y )  =/=  0 ) )
15 eleq1 2503 . . . . . . . . 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 6008 . . . . . . 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 20 . . . . . 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 2622 . . . . . . . . . 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 18011 . . . . . . . . . . 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 9416 . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  N  e.  NN )
2625nnzd 10761 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  N  e.  ZZ )
27 nnne0 10369 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  ->  N  =/=  0 )
28 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  0  /\  N  =  0 )  ->  N  =  0 )
2928necon3ai 2666 . . . . . . . . . . . . . . 15  |-  ( N  =/=  0  ->  -.  ( x  =  0  /\  N  =  0
) )
3025, 27, 293syl 20 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  -.  ( x  =  0  /\  N  =  0
) )
31 gcdn0cl 13713 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( x  =  0  /\  N  =  0 ) )  ->  ( x  gcd  N )  e.  NN )
3224, 26, 30, 31syl21anc 1217 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
x  gcd  N )  e.  NN )
3332nnred 10352 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
x  gcd  N )  e.  RR )
3432nnge1d 10379 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  1  <_  ( x  gcd  N
) )
3523, 33, 34leltned 9540 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
1  <  ( x  gcd  N )  <->  ( x  gcd  N )  =/=  1
) )
3635necon2bbid 2684 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )  /\  x  e.  ZZ )  ->  (
( x  gcd  N
)  =  1  <->  -.  1  <  ( x  gcd  N ) ) )
3722, 36bitrd 253 . . . . . . . . 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 263 . . . . . . 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 2746 . . . . . 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 255 . . . . 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 253 . . 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 969 . 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 252 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2730   class class class wbr 4307   -onto->wfo 5431   ` cfv 5433  (class class class)co 6106   0cc0 9297   1c1 9298    < clt 9433   NNcn 10337   NN0cn0 10594   ZZcz 10661    gcd cgcd 13705   Basecbs 14189   MndHom cmhm 15477  mulGrpcmgp 16606  Unitcui 16746  ℂfldccnfld 17833   ZRHomczrh 17946  ℤ/nczn 17949  DChrcdchr 22586
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-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375  ax-addf 9376  ax-mulf 9377
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 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-tpos 6760  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-ec 7118  df-qs 7122  df-map 7231  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-sup 7706  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-7 10400  df-8 10401  df-9 10402  df-10 10403  df-n0 10595  df-z 10662  df-dec 10771  df-uz 10877  df-rp 11007  df-fz 11453  df-fl 11657  df-mod 11724  df-seq 11822  df-exp 11881  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-dvds 13551  df-gcd 13706  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-sets 14195  df-ress 14196  df-plusg 14266  df-mulr 14267  df-starv 14268  df-sca 14269  df-vsca 14270  df-ip 14271  df-tset 14272  df-ple 14273  df-ds 14275  df-unif 14276  df-0g 14395  df-imas 14461  df-divs 14462  df-mnd 15430  df-mhm 15479  df-grp 15560  df-minusg 15561  df-sbg 15562  df-mulg 15563  df-subg 15693  df-nsg 15694  df-eqg 15695  df-ghm 15760  df-cmn 16294  df-abl 16295  df-mgp 16607  df-ur 16619  df-rng 16662  df-cring 16663  df-oppr 16730  df-dvdsr 16748  df-unit 16749  df-rnghom 16821  df-subrg 16878  df-lmod 16965  df-lss 17029  df-lsp 17068  df-sra 17268  df-rgmod 17269  df-lidl 17270  df-rsp 17271  df-2idl 17329  df-cnfld 17834  df-zring 17899  df-zrh 17950  df-zn 17953  df-dchr 22587
This theorem is referenced by: (None)
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