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Theorem dchrn0 20901
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 20891 . . . . . . . 8  |-  ( X  e.  D  ->  N  e.  NN )
92, 8syl 16 . . . . . . 7  |-  ( ph  ->  N  e.  NN )
103, 4, 5, 6, 9, 7dchrelbas2 20888 . . . . . 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 5668 . . . . . . 7  |-  ( x  =  A  ->  ( X `  x )  =  ( X `  A ) )
1413neeq1d 2563 . . . . . 6  |-  ( x  =  A  ->  (
( X `  x
)  =/=  0  <->  ( X `  A )  =/=  0 ) )
15 eleq1 2447 . . . . . 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 2991 . . . 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 8992 . . . . 5  |-  1  =/=  0
2120a1i 11 . . . 4  |-  ( (
ph  /\  A  e.  U )  ->  1  =/=  0 )
229nnnn0d 10206 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
234zncrng 16748 . . . . . . . 8  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
24 crngrng 15601 . . . . . . . 8  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
2522, 23, 243syl 19 . . . . . . 7  |-  ( ph  ->  Z  e.  Ring )
26 eqid 2387 . . . . . . . 8  |-  ( invr `  Z )  =  (
invr `  Z )
27 eqid 2387 . . . . . . . 8  |-  ( .r
`  Z )  =  ( .r `  Z
)
28 eqid 2387 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
296, 26, 27, 28unitrinv 15710 . . . . . . 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 5672 . . . . 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 15708 . . . . . . 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 2387 . . . . . . . 8  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
3837, 5mgpbas 15581 . . . . . . 7  |-  B  =  ( Base `  (mulGrp `  Z ) )
3937, 27mgpplusg 15579 . . . . . . 7  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
40 eqid 2387 . . . . . . . 8  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
41 cnfldmul 16632 . . . . . . . 8  |-  x.  =  ( .r ` fld )
4240, 41mgpplusg 15579 . . . . . . 7  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
4338, 39, 42mhmlin 14672 . . . . . 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 15593 . . . . . . 7  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
46 cnfld1 16649 . . . . . . . 8  |-  1  =  ( 1r ` fld )
4740, 46rngidval 15593 . . . . . . 7  |-  1  =  ( 0g `  (mulGrp ` fld ) )
4845, 47mhm0 14673 . . . . . 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 2427 . . . 4  |-  ( (
ph  /\  A  e.  U )  ->  (
( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =  1 )
51 cnfldbas 16630 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
5240, 51mgpbas 15581 . . . . . . . 8  |-  CC  =  ( Base `  (mulGrp ` fld ) )
5338, 52mhmf 14670 . . . . . . 7  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  X : B --> CC )
5433, 53syl 16 . . . . . 6  |-  ( (
ph  /\  A  e.  U )  ->  X : B --> CC )
5554, 36ffvelrnd 5810 . . . . 5  |-  ( (
ph  /\  A  e.  U )  ->  ( X `  ( ( invr `  Z ) `  A ) )  e.  CC )
5655mul02d 9196 . . . 4  |-  ( (
ph  /\  A  e.  U )  ->  (
0  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =  0 )
5721, 50, 563netr4d 2577 . . 3  |-  ( (
ph  /\  A  e.  U )  ->  (
( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =/=  ( 0  x.  ( X `  ( ( invr `  Z ) `  A ) ) ) )
58 oveq1 6027 . . . 4  |-  ( ( X `  A )  =  0  ->  (
( X `  A
)  x.  ( X `
 ( ( invr `  Z ) `  A
) ) )  =  ( 0  x.  ( X `  ( ( invr `  Z ) `  A ) ) ) )
5958necon3i 2589 . . 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 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   -->wf 5390   ` cfv 5394  (class class class)co 6020   CCcc 8921   0cc0 8923   1c1 8924    x. cmul 8928   NNcn 9932   NN0cn0 10153   Basecbs 13396   .rcmulr 13457   MndHom cmhm 14663  mulGrpcmgp 15575   Ringcrg 15587   CRingccrg 15588   1rcur 15589  Unitcui 15671   invrcinvr 15703  ℂfldccnfld 16626  ℤ/nczn 16704  DChrcdchr 20883
This theorem is referenced by:  dchrinvcl  20904  dchrfi  20906  dchrghm  20907  dchreq  20909  dchrabs  20911  dchrabs2  20913  dchr1re  20914  dchrpt  20918  dchrsum  20920  sum2dchr  20925  rpvmasumlem  21048  dchrisum0flblem1  21069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-addf 9002  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-ec 6843  df-qs 6847  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-0g 13654  df-imas 13661  df-divs 13662  df-mnd 14617  df-mhm 14665  df-grp 14739  df-minusg 14740  df-sbg 14741  df-subg 14868  df-nsg 14869  df-eqg 14870  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-cring 15591  df-ur 15592  df-oppr 15655  df-dvdsr 15673  df-unit 15674  df-invr 15704  df-subrg 15793  df-lmod 15879  df-lss 15936  df-lsp 15975  df-sra 16171  df-rgmod 16172  df-lidl 16173  df-rsp 16174  df-2idl 16230  df-cnfld 16627  df-zn 16708  df-dchr 20884
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