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Theorem dchrn0 20489
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 20479 . . . . . . . 8  |-  ( X  e.  D  ->  N  e.  NN )
92, 8syl 15 . . . . . . 7  |-  ( ph  ->  N  e.  NN )
103, 4, 5, 6, 9, 7dchrelbas2 20476 . . . . . 6  |-  ( ph  ->  ( X  e.  D  <->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U ) ) ) )
112, 10mpbid 201 . . . . 5  |-  ( ph  ->  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  /\  A. x  e.  B  (
( X `  x
)  =/=  0  ->  x  e.  U )
) )
1211simprd 449 . . . 4  |-  ( ph  ->  A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U ) )
13 fveq2 5525 . . . . . . 7  |-  ( x  =  A  ->  ( X `  x )  =  ( X `  A ) )
1413neeq1d 2459 . . . . . 6  |-  ( x  =  A  ->  (
( X `  x
)  =/=  0  <->  ( X `  A )  =/=  0 ) )
15 eleq1 2343 . . . . . 6  |-  ( x  =  A  ->  (
x  e.  U  <->  A  e.  U ) )
1614, 15imbi12d 311 . . . . 5  |-  ( x  =  A  ->  (
( ( X `  x )  =/=  0  ->  x  e.  U )  <-> 
( ( X `  A )  =/=  0  ->  A  e.  U ) ) )
1716rspcv 2880 . . . 4  |-  ( A  e.  B  ->  ( A. x  e.  B  ( ( X `  x )  =/=  0  ->  x  e.  U )  ->  ( ( X `
 A )  =/=  0  ->  A  e.  U ) ) )
181, 12, 17sylc 56 . . 3  |-  ( ph  ->  ( ( X `  A )  =/=  0  ->  A  e.  U ) )
1918imp 418 . 2  |-  ( (
ph  /\  ( X `  A )  =/=  0
)  ->  A  e.  U )
20 ax-1ne0 8806 . . . . 5  |-  1  =/=  0
2120a1i 10 . . . 4  |-  ( (
ph  /\  A  e.  U )  ->  1  =/=  0 )
229nnnn0d 10018 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
234zncrng 16498 . . . . . . . 8  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
24 crngrng 15351 . . . . . . . 8  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
2522, 23, 243syl 18 . . . . . . 7  |-  ( ph  ->  Z  e.  Ring )
26 eqid 2283 . . . . . . . 8  |-  ( invr `  Z )  =  (
invr `  Z )
27 eqid 2283 . . . . . . . 8  |-  ( .r
`  Z )  =  ( .r `  Z
)
28 eqid 2283 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
296, 26, 27, 28unitrinv 15460 . . . . . . 7  |-  ( ( Z  e.  Ring  /\  A  e.  U )  ->  ( A ( .r `  Z ) ( (
invr `  Z ) `  A ) )  =  ( 1r `  Z
) )
3025, 29sylan 457 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  ( A ( .r `  Z ) ( (
invr `  Z ) `  A ) )  =  ( 1r `  Z
) )
3130fveq2d 5529 . . . . 5  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  ( A
( .r `  Z
) ( ( invr `  Z ) `  A
) ) )  =  ( X `  ( 1r `  Z ) ) )
3211simpld 445 . . . . . . 7  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
3332adantr 451 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  X  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )
341adantr 451 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  A  e.  B )
356, 26, 5rnginvcl 15458 . . . . . . 7  |-  ( ( Z  e.  Ring  /\  A  e.  U )  ->  (
( invr `  Z ) `  A )  e.  B
)
3625, 35sylan 457 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  (
( invr `  Z ) `  A )  e.  B
)
37 eqid 2283 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
3837, 5mgpbas 15331 . . . . . . 7  |-  B  =  ( Base `  (mulGrp `  Z ) )
3937, 27mgpplusg 15329 . . . . . . 7  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
40 eqid 2283 . . . . . . . 8  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
41 cnfldmul 16385 . . . . . . . 8  |-  x.  =  ( .r ` fld )
4240, 41mgpplusg 15329 . . . . . . 7  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
4338, 39, 42mhmlin 14422 . . . . . 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 1182 . . . . 5  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  ( A
( .r `  Z
) ( ( invr `  Z ) `  A
) ) )  =  ( ( X `  A )  x.  ( X `  ( ( invr `  Z ) `  A ) ) ) )
4537, 28rngidval 15343 . . . . . . 7  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
46 cnfld1 16399 . . . . . . . 8  |-  1  =  ( 1r ` fld )
4740, 46rngidval 15343 . . . . . . 7  |-  1  =  ( 0g `  (mulGrp ` fld ) )
4845, 47mhm0 14423 . . . . . 6  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( X `  ( 1r `  Z ) )  =  1 )
4933, 48syl 15 . . . . 5  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  ( 1r `  Z ) )  =  1 )
5031, 44, 493eqtr3d 2323 . . . 4  |-  ( (
ph  /\  A  e.  U )  ->  (
( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =  1 )
51 cnfldbas 16383 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
5240, 51mgpbas 15331 . . . . . . . 8  |-  CC  =  ( Base `  (mulGrp ` fld ) )
5338, 52mhmf 14420 . . . . . . 7  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  X : B --> CC )
5433, 53syl 15 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  X : B --> CC )
55 ffvelrn 5663 . . . . . 6  |-  ( ( X : B --> CC  /\  ( ( invr `  Z
) `  A )  e.  B )  ->  ( X `  ( ( invr `  Z ) `  A ) )  e.  CC )
5654, 36, 55syl2anc 642 . . . . 5  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  ( ( invr `  Z ) `  A ) )  e.  CC )
5756mul02d 9010 . . . 4  |-  ( (
ph  /\  A  e.  U )  ->  (
0  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =  0 )
5821, 50, 573netr4d 2473 . . 3  |-  ( (
ph  /\  A  e.  U )  ->  (
( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =/=  ( 0  x.  ( X `  ( ( invr `  Z ) `  A ) ) ) )
59 oveq1 5865 . . . 4  |-  ( ( X `  A )  =  0  ->  (
( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =  ( 0  x.  ( X `  ( ( invr `  Z ) `  A ) ) ) )
6059necon3i 2485 . . 3  |-  ( ( ( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =/=  ( 0  x.  ( X `  ( ( invr `  Z ) `  A ) ) )  ->  ( X `  A )  =/=  0
)
6158, 60syl 15 . 2  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  A )  =/=  0 )
6219, 61impbida 805 1  |-  ( ph  ->  ( ( X `  A )  =/=  0  <->  A  e.  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742   NNcn 9746   NN0cn0 9965   Basecbs 13148   .rcmulr 13209   MndHom cmhm 14413  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338   1rcur 15339  Unitcui 15421   invrcinvr 15453  ℂfldccnfld 16377  ℤ/nczn 16454  DChrcdchr 20471
This theorem is referenced by:  dchrinvcl  20492  dchrfi  20494  dchrghm  20495  dchreq  20497  dchrabs  20499  dchrabs2  20501  dchr1re  20502  dchrpt  20506  dchrsum  20508  sum2dchr  20513  rpvmasumlem  20636  dchrisum0flblem1  20657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-imas 13411  df-divs 13412  df-mnd 14367  df-mhm 14415  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-nsg 14619  df-eqg 14620  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-cnfld 16378  df-zn 16458  df-dchr 20472
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