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Theorem dchrinvcl 20990
Description: Closure of the group inverse operation on Dirichlet characters. (Contributed by Mario Carneiro, 19-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 )
dchr1cl.o  |-  .1.  =  ( k  e.  B  |->  if ( k  e.  U ,  1 ,  0 ) )
dchrmulid2.t  |-  .x.  =  ( +g  `  G )
dchrmulid2.x  |-  ( ph  ->  X  e.  D )
dchrinvcl.n  |-  K  =  ( k  e.  B  |->  if ( k  e.  U ,  ( 1  /  ( X `  k ) ) ,  0 ) )
Assertion
Ref Expression
dchrinvcl  |-  ( ph  ->  ( K  e.  D  /\  ( K  .x.  X
)  =  .1.  )
)
Distinct variable groups:    B, k    U, k    k, N    ph, k    k, X    k, Z
Allowed substitution hints:    D( k)    .x. ( k)    .1. ( k)    G( k)    K( k)

Proof of Theorem dchrinvcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dchrinvcl.n . . 3  |-  K  =  ( k  e.  B  |->  if ( k  e.  U ,  ( 1  /  ( X `  k ) ) ,  0 ) )
2 dchrmhm.g . . . 4  |-  G  =  (DChr `  N )
3 dchrmhm.z . . . 4  |-  Z  =  (ℤ/n `  N )
4 dchrn0.b . . . 4  |-  B  =  ( Base `  Z
)
5 dchrn0.u . . . 4  |-  U  =  (Unit `  Z )
6 dchrmulid2.x . . . . 5  |-  ( ph  ->  X  e.  D )
7 dchrmhm.b . . . . . 6  |-  D  =  ( Base `  G
)
82, 7dchrrcl 20977 . . . . 5  |-  ( X  e.  D  ->  N  e.  NN )
96, 8syl 16 . . . 4  |-  ( ph  ->  N  e.  NN )
10 fveq2 5687 . . . . 5  |-  ( k  =  x  ->  ( X `  k )  =  ( X `  x ) )
1110oveq2d 6056 . . . 4  |-  ( k  =  x  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  x
) ) )
12 fveq2 5687 . . . . 5  |-  ( k  =  y  ->  ( X `  k )  =  ( X `  y ) )
1312oveq2d 6056 . . . 4  |-  ( k  =  y  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  y
) ) )
14 fveq2 5687 . . . . 5  |-  ( k  =  ( x ( .r `  Z ) y )  ->  ( X `  k )  =  ( X `  ( x ( .r
`  Z ) y ) ) )
1514oveq2d 6056 . . . 4  |-  ( k  =  ( x ( .r `  Z ) y )  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  (
x ( .r `  Z ) y ) ) ) )
16 fveq2 5687 . . . . 5  |-  ( k  =  ( 1r `  Z )  ->  ( X `  k )  =  ( X `  ( 1r `  Z ) ) )
1716oveq2d 6056 . . . 4  |-  ( k  =  ( 1r `  Z )  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  ( 1r `  Z ) ) ) )
182, 3, 7, 4, 6dchrf 20979 . . . . . 6  |-  ( ph  ->  X : B --> CC )
194, 5unitss 15720 . . . . . . 7  |-  U  C_  B
2019sseli 3304 . . . . . 6  |-  ( k  e.  U  ->  k  e.  B )
21 ffvelrn 5827 . . . . . 6  |-  ( ( X : B --> CC  /\  k  e.  B )  ->  ( X `  k
)  e.  CC )
2218, 20, 21syl2an 464 . . . . 5  |-  ( (
ph  /\  k  e.  U )  ->  ( X `  k )  e.  CC )
23 simpr 448 . . . . . 6  |-  ( (
ph  /\  k  e.  U )  ->  k  e.  U )
246adantr 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  X  e.  D )
2520adantl 453 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  k  e.  B )
262, 3, 7, 4, 5, 24, 25dchrn0 20987 . . . . . 6  |-  ( (
ph  /\  k  e.  U )  ->  (
( X `  k
)  =/=  0  <->  k  e.  U ) )
2723, 26mpbird 224 . . . . 5  |-  ( (
ph  /\  k  e.  U )  ->  ( X `  k )  =/=  0 )
2822, 27reccld 9739 . . . 4  |-  ( (
ph  /\  k  e.  U )  ->  (
1  /  ( X `
 k ) )  e.  CC )
29 1t1e1 10082 . . . . . . . 8  |-  ( 1  x.  1 )  =  1
3029eqcomi 2408 . . . . . . 7  |-  1  =  ( 1  x.  1 )
3130a1i 11 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
1  =  ( 1  x.  1 ) )
322, 3, 7dchrmhm 20978 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
336adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X  e.  D )
3432, 33sseldi 3306 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
35 simprl 733 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  x  e.  U )
3619, 35sseldi 3306 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  x  e.  B )
37 simprr 734 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
y  e.  U )
3819, 37sseldi 3306 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
y  e.  B )
39 eqid 2404 . . . . . . . . 9  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
4039, 4mgpbas 15609 . . . . . . . 8  |-  B  =  ( Base `  (mulGrp `  Z ) )
41 eqid 2404 . . . . . . . . 9  |-  ( .r
`  Z )  =  ( .r `  Z
)
4239, 41mgpplusg 15607 . . . . . . . 8  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
43 eqid 2404 . . . . . . . . 9  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
44 cnfldmul 16664 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
4543, 44mgpplusg 15607 . . . . . . . 8  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
4640, 42, 45mhmlin 14700 . . . . . . 7  |-  ( ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  x  e.  B  /\  y  e.  B
)  ->  ( X `  ( x ( .r
`  Z ) y ) )  =  ( ( X `  x
)  x.  ( X `
 y ) ) )
4734, 36, 38, 46syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  (
x ( .r `  Z ) y ) )  =  ( ( X `  x )  x.  ( X `  y ) ) )
4831, 47oveq12d 6058 . . . . 5  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( 1  /  ( X `  ( x
( .r `  Z
) y ) ) )  =  ( ( 1  x.  1 )  /  ( ( X `
 x )  x.  ( X `  y
) ) ) )
49 ax-1cn 9004 . . . . . . 7  |-  1  e.  CC
5049a1i 11 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
1  e.  CC )
5118adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X : B --> CC )
5251, 36ffvelrnd 5830 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  x
)  e.  CC )
5351, 38ffvelrnd 5830 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  y
)  e.  CC )
542, 3, 7, 4, 5, 33, 36dchrn0 20987 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( ( X `  x )  =/=  0  <->  x  e.  U ) )
5535, 54mpbird 224 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  x
)  =/=  0 )
562, 3, 7, 4, 5, 33, 38dchrn0 20987 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( ( X `  y )  =/=  0  <->  y  e.  U ) )
5737, 56mpbird 224 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  y
)  =/=  0 )
5850, 52, 50, 53, 55, 57divmuldivd 9787 . . . . 5  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( ( 1  / 
( X `  x
) )  x.  (
1  /  ( X `
 y ) ) )  =  ( ( 1  x.  1 )  /  ( ( X `
 x )  x.  ( X `  y
) ) ) )
5948, 58eqtr4d 2439 . . . 4  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( 1  /  ( X `  ( x
( .r `  Z
) y ) ) )  =  ( ( 1  /  ( X `
 x ) )  x.  ( 1  / 
( X `  y
) ) ) )
6032, 6sseldi 3306 . . . . . . 7  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
61 eqid 2404 . . . . . . . . 9  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
6239, 61rngidval 15621 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
63 cnfld1 16681 . . . . . . . . 9  |-  1  =  ( 1r ` fld )
6443, 63rngidval 15621 . . . . . . . 8  |-  1  =  ( 0g `  (mulGrp ` fld ) )
6562, 64mhm0 14701 . . . . . . 7  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( X `  ( 1r `  Z ) )  =  1 )
6660, 65syl 16 . . . . . 6  |-  ( ph  ->  ( X `  ( 1r `  Z ) )  =  1 )
6766oveq2d 6056 . . . . 5  |-  ( ph  ->  ( 1  /  ( X `  ( 1r `  Z ) ) )  =  ( 1  / 
1 ) )
6849div1i 9698 . . . . 5  |-  ( 1  /  1 )  =  1
6967, 68syl6eq 2452 . . . 4  |-  ( ph  ->  ( 1  /  ( X `  ( 1r `  Z ) ) )  =  1 )
702, 3, 4, 5, 9, 7, 11, 13, 15, 17, 28, 59, 69dchrelbasd 20976 . . 3  |-  ( ph  ->  ( k  e.  B  |->  if ( k  e.  U ,  ( 1  /  ( X `  k ) ) ,  0 ) )  e.  D )
711, 70syl5eqel 2488 . 2  |-  ( ph  ->  K  e.  D )
72 dchrmulid2.t . . . 4  |-  .x.  =  ( +g  `  G )
732, 3, 7, 72, 71, 6dchrmul 20985 . . 3  |-  ( ph  ->  ( K  .x.  X
)  =  ( K  o F  x.  X
) )
74 fvex 5701 . . . . . . 7  |-  ( Base `  Z )  e.  _V
754, 74eqeltri 2474 . . . . . 6  |-  B  e. 
_V
7675a1i 11 . . . . 5  |-  ( ph  ->  B  e.  _V )
77 ovex 6065 . . . . . . 7  |-  ( 1  /  ( X `  k ) )  e. 
_V
78 c0ex 9041 . . . . . . 7  |-  0  e.  _V
7977, 78ifex 3757 . . . . . 6  |-  if ( k  e.  U , 
( 1  /  ( X `  k )
) ,  0 )  e.  _V
8079a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  B )  ->  if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  e.  _V )
8118ffvelrnda 5829 . . . . 5  |-  ( (
ph  /\  k  e.  B )  ->  ( X `  k )  e.  CC )
821a1i 11 . . . . 5  |-  ( ph  ->  K  =  ( k  e.  B  |->  if ( k  e.  U , 
( 1  /  ( X `  k )
) ,  0 ) ) )
8318feqmptd 5738 . . . . 5  |-  ( ph  ->  X  =  ( k  e.  B  |->  ( X `
 k ) ) )
8476, 80, 81, 82, 83offval2 6281 . . . 4  |-  ( ph  ->  ( K  o F  x.  X )  =  ( k  e.  B  |->  ( if ( k  e.  U ,  ( 1  /  ( X `
 k ) ) ,  0 )  x.  ( X `  k
) ) ) )
85 oveq1 6047 . . . . . . . 8  |-  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  =  ( 1  /  ( X `  k ) )  -> 
( if ( k  e.  U ,  ( 1  /  ( X `
 k ) ) ,  0 )  x.  ( X `  k
) )  =  ( ( 1  /  ( X `  k )
)  x.  ( X `
 k ) ) )
86 oveq1 6047 . . . . . . . 8  |-  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  =  0  -> 
( if ( k  e.  U ,  ( 1  /  ( X `
 k ) ) ,  0 )  x.  ( X `  k
) )  =  ( 0  x.  ( X `
 k ) ) )
8785, 86ifsb 3708 . . . . . . 7  |-  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  x.  ( X `
 k ) )  =  if ( k  e.  U ,  ( ( 1  /  ( X `  k )
)  x.  ( X `
 k ) ) ,  ( 0  x.  ( X `  k
) ) )
8881adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  ( X `  k )  e.  CC )
896adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  B )  ->  X  e.  D )
90 simpr 448 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  B )  ->  k  e.  B )
912, 3, 7, 4, 5, 89, 90dchrn0 20987 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  B )  ->  (
( X `  k
)  =/=  0  <->  k  e.  U ) )
9291biimpar 472 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  ( X `  k )  =/=  0 )
9388, 92recid2d 9742 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  (
( 1  /  ( X `  k )
)  x.  ( X `
 k ) )  =  1 )
9493ifeq1da 3724 . . . . . . . 8  |-  ( (
ph  /\  k  e.  B )  ->  if ( k  e.  U ,  ( ( 1  /  ( X `  k ) )  x.  ( X `  k
) ) ,  ( 0  x.  ( X `
 k ) ) )  =  if ( k  e.  U , 
1 ,  ( 0  x.  ( X `  k ) ) ) )
9581mul02d 9220 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  B )  ->  (
0  x.  ( X `
 k ) )  =  0 )
9695ifeq2d 3714 . . . . . . . 8  |-  ( (
ph  /\  k  e.  B )  ->  if ( k  e.  U ,  1 ,  ( 0  x.  ( X `
 k ) ) )  =  if ( k  e.  U , 
1 ,  0 ) )
9794, 96eqtrd 2436 . . . . . . 7  |-  ( (
ph  /\  k  e.  B )  ->  if ( k  e.  U ,  ( ( 1  /  ( X `  k ) )  x.  ( X `  k
) ) ,  ( 0  x.  ( X `
 k ) ) )  =  if ( k  e.  U , 
1 ,  0 ) )
9887, 97syl5eq 2448 . . . . . 6  |-  ( (
ph  /\  k  e.  B )  ->  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  x.  ( X `
 k ) )  =  if ( k  e.  U ,  1 ,  0 ) )
9998mpteq2dva 4255 . . . . 5  |-  ( ph  ->  ( k  e.  B  |->  ( if ( k  e.  U ,  ( 1  /  ( X `
 k ) ) ,  0 )  x.  ( X `  k
) ) )  =  ( k  e.  B  |->  if ( k  e.  U ,  1 ,  0 ) ) )
100 dchr1cl.o . . . . 5  |-  .1.  =  ( k  e.  B  |->  if ( k  e.  U ,  1 ,  0 ) )
10199, 100syl6reqr 2455 . . . 4  |-  ( ph  ->  .1.  =  ( k  e.  B  |->  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  x.  ( X `
 k ) ) ) )
10284, 101eqtr4d 2439 . . 3  |-  ( ph  ->  ( K  o F  x.  X )  =  .1.  )
10373, 102eqtrd 2436 . 2  |-  ( ph  ->  ( K  .x.  X
)  =  .1.  )
10471, 103jca 519 1  |-  ( ph  ->  ( K  e.  D  /\  ( K  .x.  X
)  =  .1.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916   ifcif 3699    e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   CCcc 8944   0cc0 8946   1c1 8947    x. cmul 8951    / cdiv 9633   NNcn 9956   Basecbs 13424   +g cplusg 13484   .rcmulr 13485   MndHom cmhm 14691  mulGrpcmgp 15603   1rcur 15617  Unitcui 15699  ℂfldccnfld 16658  ℤ/nczn 16736  DChrcdchr 20969
This theorem is referenced by:  dchrabl  20991
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-addf 9025  ax-mulf 9026
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-imas 13689  df-divs 13690  df-mnd 14645  df-mhm 14693  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-nsg 14897  df-eqg 14898  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-sra 16199  df-rgmod 16200  df-lidl 16201  df-rsp 16202  df-2idl 16258  df-cnfld 16659  df-zn 16740  df-dchr 20970
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