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Theorem dchrinvcl 23353
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 23340 . . . . 5  |-  ( X  e.  D  ->  N  e.  NN )
96, 8syl 16 . . . 4  |-  ( ph  ->  N  e.  NN )
10 fveq2 5866 . . . . 5  |-  ( k  =  x  ->  ( X `  k )  =  ( X `  x ) )
1110oveq2d 6301 . . . 4  |-  ( k  =  x  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  x
) ) )
12 fveq2 5866 . . . . 5  |-  ( k  =  y  ->  ( X `  k )  =  ( X `  y ) )
1312oveq2d 6301 . . . 4  |-  ( k  =  y  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  y
) ) )
14 fveq2 5866 . . . . 5  |-  ( k  =  ( x ( .r `  Z ) y )  ->  ( X `  k )  =  ( X `  ( x ( .r
`  Z ) y ) ) )
1514oveq2d 6301 . . . 4  |-  ( k  =  ( x ( .r `  Z ) y )  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  (
x ( .r `  Z ) y ) ) ) )
16 fveq2 5866 . . . . 5  |-  ( k  =  ( 1r `  Z )  ->  ( X `  k )  =  ( X `  ( 1r `  Z ) ) )
1716oveq2d 6301 . . . 4  |-  ( k  =  ( 1r `  Z )  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  ( 1r `  Z ) ) ) )
182, 3, 7, 4, 6dchrf 23342 . . . . . 6  |-  ( ph  ->  X : B --> CC )
194, 5unitss 17122 . . . . . . 7  |-  U  C_  B
2019sseli 3500 . . . . . 6  |-  ( k  e.  U  ->  k  e.  B )
21 ffvelrn 6020 . . . . . 6  |-  ( ( X : B --> CC  /\  k  e.  B )  ->  ( X `  k
)  e.  CC )
2218, 20, 21syl2an 477 . . . . 5  |-  ( (
ph  /\  k  e.  U )  ->  ( X `  k )  e.  CC )
23 simpr 461 . . . . . 6  |-  ( (
ph  /\  k  e.  U )  ->  k  e.  U )
246adantr 465 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  X  e.  D )
2520adantl 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  k  e.  B )
262, 3, 7, 4, 5, 24, 25dchrn0 23350 . . . . . 6  |-  ( (
ph  /\  k  e.  U )  ->  (
( X `  k
)  =/=  0  <->  k  e.  U ) )
2723, 26mpbird 232 . . . . 5  |-  ( (
ph  /\  k  e.  U )  ->  ( X `  k )  =/=  0 )
2822, 27reccld 10314 . . . 4  |-  ( (
ph  /\  k  e.  U )  ->  (
1  /  ( X `
 k ) )  e.  CC )
29 1t1e1 10684 . . . . . . . 8  |-  ( 1  x.  1 )  =  1
3029eqcomi 2480 . . . . . . 7  |-  1  =  ( 1  x.  1 )
3130a1i 11 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
1  =  ( 1  x.  1 ) )
322, 3, 7dchrmhm 23341 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
336adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X  e.  D )
3432, 33sseldi 3502 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
35 simprl 755 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  x  e.  U )
3619, 35sseldi 3502 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  x  e.  B )
37 simprr 756 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
y  e.  U )
3819, 37sseldi 3502 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
y  e.  B )
39 eqid 2467 . . . . . . . . 9  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
4039, 4mgpbas 16961 . . . . . . . 8  |-  B  =  ( Base `  (mulGrp `  Z ) )
41 eqid 2467 . . . . . . . . 9  |-  ( .r
`  Z )  =  ( .r `  Z
)
4239, 41mgpplusg 16959 . . . . . . . 8  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
43 eqid 2467 . . . . . . . . 9  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
44 cnfldmul 18237 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
4543, 44mgpplusg 16959 . . . . . . . 8  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
4640, 42, 45mhmlin 15796 . . . . . . 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 1228 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  (
x ( .r `  Z ) y ) )  =  ( ( X `  x )  x.  ( X `  y ) ) )
4831, 47oveq12d 6303 . . . . 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 9551 . . . . . . 7  |-  1  e.  CC
5049a1i 11 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
1  e.  CC )
5118adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X : B --> CC )
5251, 36ffvelrnd 6023 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  x
)  e.  CC )
5351, 38ffvelrnd 6023 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  y
)  e.  CC )
542, 3, 7, 4, 5, 33, 36dchrn0 23350 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( ( X `  x )  =/=  0  <->  x  e.  U ) )
5535, 54mpbird 232 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  x
)  =/=  0 )
562, 3, 7, 4, 5, 33, 38dchrn0 23350 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( ( X `  y )  =/=  0  <->  y  e.  U ) )
5737, 56mpbird 232 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  y
)  =/=  0 )
5850, 52, 50, 53, 55, 57divmuldivd 10362 . . . . 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 2511 . . . 4  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( 1  /  ( X `  ( x
( .r `  Z
) y ) ) )  =  ( ( 1  /  ( X `
 x ) )  x.  ( 1  / 
( X `  y
) ) ) )
6032, 6sseldi 3502 . . . . . . 7  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
61 eqid 2467 . . . . . . . . 9  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
6239, 61rngidval 16969 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
63 cnfld1 18254 . . . . . . . . 9  |-  1  =  ( 1r ` fld )
6443, 63rngidval 16969 . . . . . . . 8  |-  1  =  ( 0g `  (mulGrp ` fld ) )
6562, 64mhm0 15797 . . . . . . 7  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( X `  ( 1r `  Z ) )  =  1 )
6660, 65syl 16 . . . . . 6  |-  ( ph  ->  ( X `  ( 1r `  Z ) )  =  1 )
6766oveq2d 6301 . . . . 5  |-  ( ph  ->  ( 1  /  ( X `  ( 1r `  Z ) ) )  =  ( 1  / 
1 ) )
68 1div1e1 10238 . . . . 5  |-  ( 1  /  1 )  =  1
6967, 68syl6eq 2524 . . . 4  |-  ( ph  ->  ( 1  /  ( X `  ( 1r `  Z ) ) )  =  1 )
702, 3, 4, 5, 9, 7, 11, 13, 15, 17, 28, 59, 69dchrelbasd 23339 . . 3  |-  ( ph  ->  ( k  e.  B  |->  if ( k  e.  U ,  ( 1  /  ( X `  k ) ) ,  0 ) )  e.  D )
711, 70syl5eqel 2559 . 2  |-  ( ph  ->  K  e.  D )
72 dchrmulid2.t . . . 4  |-  .x.  =  ( +g  `  G )
732, 3, 7, 72, 71, 6dchrmul 23348 . . 3  |-  ( ph  ->  ( K  .x.  X
)  =  ( K  oF  x.  X
) )
74 fvex 5876 . . . . . . 7  |-  ( Base `  Z )  e.  _V
754, 74eqeltri 2551 . . . . . 6  |-  B  e. 
_V
7675a1i 11 . . . . 5  |-  ( ph  ->  B  e.  _V )
77 ovex 6310 . . . . . . 7  |-  ( 1  /  ( X `  k ) )  e. 
_V
78 c0ex 9591 . . . . . . 7  |-  0  e.  _V
7977, 78ifex 4008 . . . . . 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 6022 . . . . 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 5921 . . . . 5  |-  ( ph  ->  X  =  ( k  e.  B  |->  ( X `
 k ) ) )
8476, 80, 81, 82, 83offval2 6541 . . . 4  |-  ( ph  ->  ( K  oF  x.  X )  =  ( k  e.  B  |->  ( if ( k  e.  U ,  ( 1  /  ( X `
 k ) ) ,  0 )  x.  ( X `  k
) ) ) )
85 oveq1 6292 . . . . . . . 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 6292 . . . . . . . 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 3952 . . . . . . 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 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  ( X `  k )  e.  CC )
896adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  B )  ->  X  e.  D )
90 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  B )  ->  k  e.  B )
912, 3, 7, 4, 5, 89, 90dchrn0 23350 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  B )  ->  (
( X `  k
)  =/=  0  <->  k  e.  U ) )
9291biimpar 485 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  ( X `  k )  =/=  0 )
9388, 92recid2d 10317 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  (
( 1  /  ( X `  k )
)  x.  ( X `
 k ) )  =  1 )
9493ifeq1da 3969 . . . . . . . 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 9778 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  B )  ->  (
0  x.  ( X `
 k ) )  =  0 )
9695ifeq2d 3958 . . . . . . . 8  |-  ( (
ph  /\  k  e.  B )  ->  if ( k  e.  U ,  1 ,  ( 0  x.  ( X `
 k ) ) )  =  if ( k  e.  U , 
1 ,  0 ) )
9794, 96eqtrd 2508 . . . . . . 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 2520 . . . . . 6  |-  ( (
ph  /\  k  e.  B )  ->  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  x.  ( X `
 k ) )  =  if ( k  e.  U ,  1 ,  0 ) )
9998mpteq2dva 4533 . . . . 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 2527 . . . 4  |-  ( ph  ->  .1.  =  ( k  e.  B  |->  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  x.  ( X `
 k ) ) ) )
10284, 101eqtr4d 2511 . . 3  |-  ( ph  ->  ( K  oF  x.  X )  =  .1.  )
10373, 102eqtrd 2508 . 2  |-  ( ph  ->  ( K  .x.  X
)  =  .1.  )
10471, 103jca 532 1  |-  ( ph  ->  ( K  e.  D  /\  ( K  .x.  X
)  =  .1.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   ifcif 3939    |-> cmpt 4505   -->wf 5584   ` cfv 5588  (class class class)co 6285    oFcof 6523   CCcc 9491   0cc0 9493   1c1 9494    x. cmul 9498    / cdiv 10207   NNcn 10537   Basecbs 14493   +g cplusg 14558   .rcmulr 14559   MndHom cmhm 15787  mulGrpcmgp 16955   1rcur 16967  Unitcui 17101  ℂfldccnfld 18231  ℤ/nczn 18347  DChrcdchr 23332
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 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-addf 9572  ax-mulf 9573
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6956  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-ec 7314  df-qs 7318  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-fz 11674  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-starv 14573  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-unif 14581  df-0g 14700  df-imas 14766  df-qus 14767  df-mnd 15735  df-mhm 15789  df-grp 15871  df-minusg 15872  df-sbg 15873  df-subg 16012  df-nsg 16013  df-eqg 16014  df-cmn 16615  df-abl 16616  df-mgp 16956  df-ur 16968  df-rng 17014  df-cring 17015  df-oppr 17085  df-dvdsr 17103  df-unit 17104  df-invr 17134  df-subrg 17239  df-lmod 17326  df-lss 17391  df-lsp 17430  df-sra 17630  df-rgmod 17631  df-lidl 17632  df-rsp 17633  df-2idl 17691  df-cnfld 18232  df-zring 18297  df-zn 18351  df-dchr 23333
This theorem is referenced by:  dchrabl  23354
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