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Theorem dchrn0 21026
Description: A Dirichlet character is nonzero on the 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
)
dchrn0.b  |-  B  =  ( Base `  Z
)
dchrn0.u  |-  U  =  (Unit `  Z )
dchrn0.x  |-  ( ph  ->  X  e.  D )
dchrn0.a  |-  ( ph  ->  A  e.  B )
Assertion
Ref Expression
dchrn0  |-  ( ph  ->  ( ( X `  A )  =/=  0  <->  A  e.  U ) )

Proof of Theorem dchrn0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dchrn0.a . . . 4  |-  ( ph  ->  A  e.  B )
2 dchrn0.x . . . . . 6  |-  ( ph  ->  X  e.  D )
3 dchrmhm.g . . . . . . 7  |-  G  =  (DChr `  N )
4 dchrmhm.z . . . . . . 7  |-  Z  =  (ℤ/n `  N )
5 dchrn0.b . . . . . . 7  |-  B  =  ( Base `  Z
)
6 dchrn0.u . . . . . . 7  |-  U  =  (Unit `  Z )
7 dchrmhm.b . . . . . . . . 9  |-  D  =  ( Base `  G
)
83, 7dchrrcl 21016 . . . . . . . 8  |-  ( X  e.  D  ->  N  e.  NN )
92, 8syl 16 . . . . . . 7  |-  ( ph  ->  N  e.  NN )
103, 4, 5, 6, 9, 7dchrelbas2 21013 . . . . . 6  |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) ) )
112, 10mpbid 202 . . . . 5  |-  ( ph  ->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  B  (
( X `  x
)  =/=  0  ->  x  e.  U )
) )
1211simprd 450 . . . 4  |-  ( ph  ->  A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U ) )
13 fveq2 5720 . . . . . . 7  |-  ( x  =  A  ->  ( X `  x )  =  ( X `  A ) )
1413neeq1d 2611 . . . . . 6  |-  ( x  =  A  ->  (
( X `  x
)  =/=  0  <->  ( X `  A )  =/=  0 ) )
15 eleq1 2495 . . . . . 6  |-  ( x  =  A  ->  (
x  e.  U  <->  A  e.  U ) )
1614, 15imbi12d 312 . . . . 5  |-  ( x  =  A  ->  (
( ( X `  x )  =/=  0  ->  x  e.  U )  <-> 
( ( X `  A )  =/=  0  ->  A  e.  U ) ) )
1716rspcv 3040 . . . 4  |-  ( A  e.  B  ->  ( A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U )  ->  ( ( X `
 A )  =/=  0  ->  A  e.  U ) ) )
181, 12, 17sylc 58 . . 3  |-  ( ph  ->  ( ( X `  A )  =/=  0  ->  A  e.  U ) )
1918imp 419 . 2  |-  ( (
ph  /\  ( X `  A )  =/=  0
)  ->  A  e.  U )
20 ax-1ne0 9051 . . . . 5  |-  1  =/=  0
2120a1i 11 . . . 4  |-  ( (
ph  /\  A  e.  U )  ->  1  =/=  0 )
229nnnn0d 10266 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
234zncrng 16817 . . . . . . . 8  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
24 crngrng 15666 . . . . . . . 8  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
2522, 23, 243syl 19 . . . . . . 7  |-  ( ph  ->  Z  e.  Ring )
26 eqid 2435 . . . . . . . 8  |-  ( invr `  Z )  =  (
invr `  Z )
27 eqid 2435 . . . . . . . 8  |-  ( .r
`  Z )  =  ( .r `  Z
)
28 eqid 2435 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
296, 26, 27, 28unitrinv 15775 . . . . . . 7  |-  ( ( Z  e.  Ring  /\  A  e.  U )  ->  ( A ( .r `  Z ) ( (
invr `  Z ) `  A ) )  =  ( 1r `  Z
) )
3025, 29sylan 458 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  ( A ( .r `  Z ) ( (
invr `  Z ) `  A ) )  =  ( 1r `  Z
) )
3130fveq2d 5724 . . . . 5  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  ( A
( .r `  Z
) ( ( invr `  Z ) `  A
) ) )  =  ( X `  ( 1r `  Z ) ) )
3211simpld 446 . . . . . . 7  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
3332adantr 452 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )
341adantr 452 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  A  e.  B )
356, 26, 5rnginvcl 15773 . . . . . . 7  |-  ( ( Z  e.  Ring  /\  A  e.  U )  ->  (
( invr `  Z ) `  A )  e.  B
)
3625, 35sylan 458 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  (
( invr `  Z ) `  A )  e.  B
)
37 eqid 2435 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
3837, 5mgpbas 15646 . . . . . . 7  |-  B  =  ( Base `  (mulGrp `  Z ) )
3937, 27mgpplusg 15644 . . . . . . 7  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
40 eqid 2435 . . . . . . . 8  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
41 cnfldmul 16701 . . . . . . . 8  |-  x.  =  ( .r ` fld )
4240, 41mgpplusg 15644 . . . . . . 7  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
4338, 39, 42mhmlin 14737 . . . . . 6  |-  ( ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  A  e.  B  /\  ( ( invr `  Z ) `  A
)  e.  B )  ->  ( X `  ( A ( .r `  Z ) ( (
invr `  Z ) `  A ) ) )  =  ( ( X `
 A )  x.  ( X `  (
( invr `  Z ) `  A ) ) ) )
4433, 34, 36, 43syl3anc 1184 . . . . 5  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  ( A
( .r `  Z
) ( ( invr `  Z ) `  A
) ) )  =  ( ( X `  A )  x.  ( X `  ( ( invr `  Z ) `  A ) ) ) )
4537, 28rngidval 15658 . . . . . . 7  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
46 cnfld1 16718 . . . . . . . 8  |-  1  =  ( 1r ` fld )
4740, 46rngidval 15658 . . . . . . 7  |-  1  =  ( 0g `  (mulGrp ` fld ) )
4845, 47mhm0 14738 . . . . . 6  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( X `  ( 1r `  Z ) )  =  1 )
4933, 48syl 16 . . . . 5  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  ( 1r `  Z ) )  =  1 )
5031, 44, 493eqtr3d 2475 . . . 4  |-  ( (
ph  /\  A  e.  U )  ->  (
( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =  1 )
51 cnfldbas 16699 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
5240, 51mgpbas 15646 . . . . . . . 8  |-  CC  =  ( Base `  (mulGrp ` fld ) )
5338, 52mhmf 14735 . . . . . . 7  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  X : B --> CC )
5433, 53syl 16 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  X : B --> CC )
5554, 36ffvelrnd 5863 . . . . 5  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  ( ( invr `  Z ) `  A ) )  e.  CC )
5655mul02d 9256 . . . 4  |-  ( (
ph  /\  A  e.  U )  ->  (
0  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =  0 )
5721, 50, 563netr4d 2625 . . 3  |-  ( (
ph  /\  A  e.  U )  ->  (
( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =/=  ( 0  x.  ( X `  ( ( invr `  Z ) `  A ) ) ) )
58 oveq1 6080 . . . 4  |-  ( ( X `  A )  =  0  ->  (
( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =  ( 0  x.  ( X `  ( ( invr `  Z ) `  A ) ) ) )
5958necon3i 2637 . . 3  |-  ( ( ( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =/=  ( 0  x.  ( X `  ( ( invr `  Z ) `  A ) ) )  ->  ( X `  A )  =/=  0
)
6057, 59syl 16 . 2  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  A )  =/=  0 )
6119, 60impbida 806 1  |-  ( ph  ->  ( ( X `  A )  =/=  0  <->  A  e.  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    x. cmul 8987   NNcn 9992   NN0cn0 10213   Basecbs 13461   .rcmulr 13522   MndHom cmhm 14728  mulGrpcmgp 15640   Ringcrg 15652   CRingccrg 15653   1rcur 15654  Unitcui 15736   invrcinvr 15768  ℂfldccnfld 16695  ℤ/nczn 16773  DChrcdchr 21008
This theorem is referenced by:  dchrinvcl  21029  dchrfi  21031  dchrghm  21032  dchreq  21034  dchrabs  21036  dchrabs2  21038  dchr1re  21039  dchrpt  21043  dchrsum  21045  sum2dchr  21050  rpvmasumlem  21173  dchrisum0flblem1  21194
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-ec 6899  df-qs 6903  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-imas 13726  df-divs 13727  df-mnd 14682  df-mhm 14730  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-nsg 14934  df-eqg 14935  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-subrg 15858  df-lmod 15944  df-lss 16001  df-lsp 16040  df-sra 16236  df-rgmod 16237  df-lidl 16238  df-rsp 16239  df-2idl 16295  df-cnfld 16696  df-zn 16777  df-dchr 21009
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