MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dchrn0 Structured version   Unicode version

Theorem dchrn0 23250
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 23240 . . . . . . . 8  |-  ( X  e.  D  ->  N  e.  NN )
92, 8syl 16 . . . . . . 7  |-  ( ph  ->  N  e.  NN )
103, 4, 5, 6, 9, 7dchrelbas2 23237 . . . . . 6  |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) ) )
112, 10mpbid 210 . . . . 5  |-  ( ph  ->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  B  (
( X `  x
)  =/=  0  ->  x  e.  U )
) )
1211simprd 463 . . . 4  |-  ( ph  ->  A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U ) )
13 fveq2 5864 . . . . . . 7  |-  ( x  =  A  ->  ( X `  x )  =  ( X `  A ) )
1413neeq1d 2744 . . . . . 6  |-  ( x  =  A  ->  (
( X `  x
)  =/=  0  <->  ( X `  A )  =/=  0 ) )
15 eleq1 2539 . . . . . 6  |-  ( x  =  A  ->  (
x  e.  U  <->  A  e.  U ) )
1614, 15imbi12d 320 . . . . 5  |-  ( x  =  A  ->  (
( ( X `  x )  =/=  0  ->  x  e.  U )  <-> 
( ( X `  A )  =/=  0  ->  A  e.  U ) ) )
1716rspcv 3210 . . . 4  |-  ( A  e.  B  ->  ( A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U )  ->  ( ( X `
 A )  =/=  0  ->  A  e.  U ) ) )
181, 12, 17sylc 60 . . 3  |-  ( ph  ->  ( ( X `  A )  =/=  0  ->  A  e.  U ) )
1918imp 429 . 2  |-  ( (
ph  /\  ( X `  A )  =/=  0
)  ->  A  e.  U )
20 ax-1ne0 9557 . . . . 5  |-  1  =/=  0
2120a1i 11 . . . 4  |-  ( (
ph  /\  A  e.  U )  ->  1  =/=  0 )
229nnnn0d 10848 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
234zncrng 18347 . . . . . . . 8  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
24 crngrng 16993 . . . . . . . 8  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
2522, 23, 243syl 20 . . . . . . 7  |-  ( ph  ->  Z  e.  Ring )
26 eqid 2467 . . . . . . . 8  |-  ( invr `  Z )  =  (
invr `  Z )
27 eqid 2467 . . . . . . . 8  |-  ( .r
`  Z )  =  ( .r `  Z
)
28 eqid 2467 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
296, 26, 27, 28unitrinv 17108 . . . . . . 7  |-  ( ( Z  e.  Ring  /\  A  e.  U )  ->  ( A ( .r `  Z ) ( (
invr `  Z ) `  A ) )  =  ( 1r `  Z
) )
3025, 29sylan 471 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  ( A ( .r `  Z ) ( (
invr `  Z ) `  A ) )  =  ( 1r `  Z
) )
3130fveq2d 5868 . . . . 5  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  ( A
( .r `  Z
) ( ( invr `  Z ) `  A
) ) )  =  ( X `  ( 1r `  Z ) ) )
3211simpld 459 . . . . . . 7  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
3332adantr 465 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )
341adantr 465 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  A  e.  B )
356, 26, 5rnginvcl 17106 . . . . . . 7  |-  ( ( Z  e.  Ring  /\  A  e.  U )  ->  (
( invr `  Z ) `  A )  e.  B
)
3625, 35sylan 471 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  (
( invr `  Z ) `  A )  e.  B
)
37 eqid 2467 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
3837, 5mgpbas 16934 . . . . . . 7  |-  B  =  ( Base `  (mulGrp `  Z ) )
3937, 27mgpplusg 16932 . . . . . . 7  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
40 eqid 2467 . . . . . . . 8  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
41 cnfldmul 18194 . . . . . . . 8  |-  x.  =  ( .r ` fld )
4240, 41mgpplusg 16932 . . . . . . 7  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
4338, 39, 42mhmlin 15781 . . . . . 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 1228 . . . . 5  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  ( A
( .r `  Z
) ( ( invr `  Z ) `  A
) ) )  =  ( ( X `  A )  x.  ( X `  ( ( invr `  Z ) `  A ) ) ) )
4537, 28rngidval 16942 . . . . . . 7  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
46 cnfld1 18211 . . . . . . . 8  |-  1  =  ( 1r ` fld )
4740, 46rngidval 16942 . . . . . . 7  |-  1  =  ( 0g `  (mulGrp ` fld ) )
4845, 47mhm0 15782 . . . . . 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 2516 . . . 4  |-  ( (
ph  /\  A  e.  U )  ->  (
( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =  1 )
51 cnfldbas 18192 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
5240, 51mgpbas 16934 . . . . . . . 8  |-  CC  =  ( Base `  (mulGrp ` fld ) )
5338, 52mhmf 15779 . . . . . . 7  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  X : B --> CC )
5433, 53syl 16 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  X : B --> CC )
5554, 36ffvelrnd 6020 . . . . 5  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  ( ( invr `  Z ) `  A ) )  e.  CC )
5655mul02d 9773 . . . 4  |-  ( (
ph  /\  A  e.  U )  ->  (
0  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =  0 )
5721, 50, 563netr4d 2772 . . 3  |-  ( (
ph  /\  A  e.  U )  ->  (
( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =/=  ( 0  x.  ( X `  ( ( invr `  Z ) `  A ) ) ) )
58 oveq1 6289 . . . 4  |-  ( ( X `  A )  =  0  ->  (
( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =  ( 0  x.  ( X `  ( ( invr `  Z ) `  A ) ) ) )
5958necon3i 2707 . . 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 830 1  |-  ( ph  ->  ( ( X `  A )  =/=  0  <->  A  e.  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    x. cmul 9493   NNcn 10532   NN0cn0 10791   Basecbs 14483   .rcmulr 14549   MndHom cmhm 15772  mulGrpcmgp 16928   1rcur 16940   Ringcrg 16983   CRingccrg 16984  Unitcui 17069   invrcinvr 17101  ℂfldccnfld 18188  ℤ/nczn 18304  DChrcdchr 23232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-ec 7310  df-qs 7314  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-0g 14690  df-imas 14756  df-divs 14757  df-mnd 15725  df-mhm 15774  df-grp 15855  df-minusg 15856  df-sbg 15857  df-subg 15990  df-nsg 15991  df-eqg 15992  df-cmn 16593  df-abl 16594  df-mgp 16929  df-ur 16941  df-rng 16985  df-cring 16986  df-oppr 17053  df-dvdsr 17071  df-unit 17072  df-invr 17102  df-subrg 17207  df-lmod 17294  df-lss 17359  df-lsp 17398  df-sra 17598  df-rgmod 17599  df-lidl 17600  df-rsp 17601  df-2idl 17659  df-cnfld 18189  df-zring 18254  df-zn 18308  df-dchr 23233
This theorem is referenced by:  dchrinvcl  23253  dchrfi  23255  dchrghm  23256  dchreq  23258  dchrabs  23260  dchrabs2  23262  dchr1re  23263  dchrpt  23267  dchrsum  23269  sum2dchr  23274  rpvmasumlem  23397  dchrisum0flblem1  23418
  Copyright terms: Public domain W3C validator