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Theorem dchrzrhmul 20501
Description: A Dirichlet character is completely multiplicative. (Contributed by Mario Carneiro, 4-May-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
)
dchrzrh1.x  |-  ( ph  ->  X  e.  D )
dchrzrh1.a  |-  ( ph  ->  A  e.  ZZ )
dchrzrh1.c  |-  ( ph  ->  C  e.  ZZ )
Assertion
Ref Expression
dchrzrhmul  |-  ( ph  ->  ( X `  ( L `  ( A  x.  C ) ) )  =  ( ( X `
 ( L `  A ) )  x.  ( X `  ( L `  C )
) ) )

Proof of Theorem dchrzrhmul
StepHypRef Expression
1 dchrzrh1.x . . . . . . . . 9  |-  ( ph  ->  X  e.  D )
2 dchrmhm.g . . . . . . . . . 10  |-  G  =  (DChr `  N )
3 dchrmhm.b . . . . . . . . . 10  |-  D  =  ( Base `  G
)
42, 3dchrrcl 20495 . . . . . . . . 9  |-  ( X  e.  D  ->  N  e.  NN )
51, 4syl 15 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
65nnnn0d 10034 . . . . . . 7  |-  ( ph  ->  N  e.  NN0 )
7 dchrmhm.z . . . . . . . 8  |-  Z  =  (ℤ/n `  N )
87zncrng 16514 . . . . . . 7  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
96, 8syl 15 . . . . . 6  |-  ( ph  ->  Z  e.  CRing )
10 crngrng 15367 . . . . . 6  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
119, 10syl 15 . . . . 5  |-  ( ph  ->  Z  e.  Ring )
12 eqid 2296 . . . . . 6  |-  (flds  ZZ )  =  (flds  ZZ )
13 dchrelbas4.l . . . . . 6  |-  L  =  ( ZRHom `  Z
)
1412, 13zrhrhm 16482 . . . . 5  |-  ( Z  e.  Ring  ->  L  e.  ( (flds  ZZ ) RingHom  Z ) )
1511, 14syl 15 . . . 4  |-  ( ph  ->  L  e.  ( (flds  ZZ ) RingHom  Z ) )
16 dchrzrh1.a . . . 4  |-  ( ph  ->  A  e.  ZZ )
17 dchrzrh1.c . . . 4  |-  ( ph  ->  C  e.  ZZ )
18 zsscn 10048 . . . . . 6  |-  ZZ  C_  CC
19 cnfldbas 16399 . . . . . . 7  |-  CC  =  ( Base ` fld )
2012, 19ressbas2 13215 . . . . . 6  |-  ( ZZ  C_  CC  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
2118, 20ax-mp 8 . . . . 5  |-  ZZ  =  ( Base `  (flds  ZZ ) )
22 zex 10049 . . . . . 6  |-  ZZ  e.  _V
23 cnfldmul 16401 . . . . . . 7  |-  x.  =  ( .r ` fld )
2412, 23ressmulr 13277 . . . . . 6  |-  ( ZZ  e.  _V  ->  x.  =  ( .r `  (flds  ZZ ) ) )
2522, 24ax-mp 8 . . . . 5  |-  x.  =  ( .r `  (flds  ZZ ) )
26 eqid 2296 . . . . 5  |-  ( .r
`  Z )  =  ( .r `  Z
)
2721, 25, 26rhmmul 15521 . . . 4  |-  ( ( L  e.  ( (flds  ZZ ) RingHom  Z )  /\  A  e.  ZZ  /\  C  e.  ZZ )  ->  ( L `  ( A  x.  C ) )  =  ( ( L `  A ) ( .r
`  Z ) ( L `  C ) ) )
2815, 16, 17, 27syl3anc 1182 . . 3  |-  ( ph  ->  ( L `  ( A  x.  C )
)  =  ( ( L `  A ) ( .r `  Z
) ( L `  C ) ) )
2928fveq2d 5545 . 2  |-  ( ph  ->  ( X `  ( L `  ( A  x.  C ) ) )  =  ( X `  ( ( L `  A ) ( .r
`  Z ) ( L `  C ) ) ) )
302, 7, 3dchrmhm 20496 . . . 4  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
3130, 1sseldi 3191 . . 3  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
32 eqid 2296 . . . . . 6  |-  ( Base `  Z )  =  (
Base `  Z )
3321, 32rhmf 15520 . . . . 5  |-  ( L  e.  ( (flds  ZZ ) RingHom  Z )  ->  L : ZZ --> ( Base `  Z )
)
3415, 33syl 15 . . . 4  |-  ( ph  ->  L : ZZ --> ( Base `  Z ) )
35 ffvelrn 5679 . . . 4  |-  ( ( L : ZZ --> ( Base `  Z )  /\  A  e.  ZZ )  ->  ( L `  A )  e.  ( Base `  Z
) )
3634, 16, 35syl2anc 642 . . 3  |-  ( ph  ->  ( L `  A
)  e.  ( Base `  Z ) )
37 ffvelrn 5679 . . . 4  |-  ( ( L : ZZ --> ( Base `  Z )  /\  C  e.  ZZ )  ->  ( L `  C )  e.  ( Base `  Z
) )
3834, 17, 37syl2anc 642 . . 3  |-  ( ph  ->  ( L `  C
)  e.  ( Base `  Z ) )
39 eqid 2296 . . . . 5  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
4039, 32mgpbas 15347 . . . 4  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
4139, 26mgpplusg 15345 . . . 4  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
42 eqid 2296 . . . . 5  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
4342, 23mgpplusg 15345 . . . 4  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
4440, 41, 43mhmlin 14438 . . 3  |-  ( ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  ( L `  A )  e.  (
Base `  Z )  /\  ( L `  C
)  e.  ( Base `  Z ) )  -> 
( X `  (
( L `  A
) ( .r `  Z ) ( L `
 C ) ) )  =  ( ( X `  ( L `
 A ) )  x.  ( X `  ( L `  C ) ) ) )
4531, 36, 38, 44syl3anc 1182 . 2  |-  ( ph  ->  ( X `  (
( L `  A
) ( .r `  Z ) ( L `
 C ) ) )  =  ( ( X `  ( L `
 A ) )  x.  ( X `  ( L `  C ) ) ) )
4629, 45eqtrd 2328 1  |-  ( ph  ->  ( X `  ( L `  ( A  x.  C ) ) )  =  ( ( X `
 ( L `  A ) )  x.  ( X `  ( L `  C )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751    x. cmul 8758   NNcn 9762   NN0cn0 9981   ZZcz 10040   Basecbs 13164   ↾s cress 13165   .rcmulr 13225   MndHom cmhm 14429  mulGrpcmgp 15341   Ringcrg 15353   CRingccrg 15354   RingHom crh 15510  ℂfldccnfld 16393   ZRHomczrh 16467  ℤ/nczn 16470  DChrcdchr 20487
This theorem is referenced by:  dchrmusum2  20659  dchrvmasumlem1  20660  dchrvmasum2lem  20661  dchrisum0fmul  20671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-seq 11063  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-imas 13427  df-divs 13428  df-mnd 14383  df-mhm 14431  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-nsg 14635  df-eqg 14636  df-ghm 14697  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-rnghom 15512  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-sra 15941  df-rgmod 15942  df-lidl 15943  df-rsp 15944  df-2idl 16000  df-cnfld 16394  df-zrh 16471  df-zn 16474  df-dchr 20488
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