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Theorem dchrinvcl 22551
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 22538 . . . . 5  |-  ( X  e.  D  ->  N  e.  NN )
96, 8syl 16 . . . 4  |-  ( ph  ->  N  e.  NN )
10 fveq2 5688 . . . . 5  |-  ( k  =  x  ->  ( X `  k )  =  ( X `  x ) )
1110oveq2d 6106 . . . 4  |-  ( k  =  x  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  x
) ) )
12 fveq2 5688 . . . . 5  |-  ( k  =  y  ->  ( X `  k )  =  ( X `  y ) )
1312oveq2d 6106 . . . 4  |-  ( k  =  y  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  y
) ) )
14 fveq2 5688 . . . . 5  |-  ( k  =  ( x ( .r `  Z ) y )  ->  ( X `  k )  =  ( X `  ( x ( .r
`  Z ) y ) ) )
1514oveq2d 6106 . . . 4  |-  ( k  =  ( x ( .r `  Z ) y )  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  (
x ( .r `  Z ) y ) ) ) )
16 fveq2 5688 . . . . 5  |-  ( k  =  ( 1r `  Z )  ->  ( X `  k )  =  ( X `  ( 1r `  Z ) ) )
1716oveq2d 6106 . . . 4  |-  ( k  =  ( 1r `  Z )  ->  (
1  /  ( X `
 k ) )  =  ( 1  / 
( X `  ( 1r `  Z ) ) ) )
182, 3, 7, 4, 6dchrf 22540 . . . . . 6  |-  ( ph  ->  X : B --> CC )
194, 5unitss 16742 . . . . . . 7  |-  U  C_  B
2019sseli 3349 . . . . . 6  |-  ( k  e.  U  ->  k  e.  B )
21 ffvelrn 5838 . . . . . 6  |-  ( ( X : B --> CC  /\  k  e.  B )  ->  ( X `  k
)  e.  CC )
2218, 20, 21syl2an 474 . . . . 5  |-  ( (
ph  /\  k  e.  U )  ->  ( X `  k )  e.  CC )
23 simpr 458 . . . . . 6  |-  ( (
ph  /\  k  e.  U )  ->  k  e.  U )
246adantr 462 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  X  e.  D )
2520adantl 463 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  k  e.  B )
262, 3, 7, 4, 5, 24, 25dchrn0 22548 . . . . . 6  |-  ( (
ph  /\  k  e.  U )  ->  (
( X `  k
)  =/=  0  <->  k  e.  U ) )
2723, 26mpbird 232 . . . . 5  |-  ( (
ph  /\  k  e.  U )  ->  ( X `  k )  =/=  0 )
2822, 27reccld 10096 . . . 4  |-  ( (
ph  /\  k  e.  U )  ->  (
1  /  ( X `
 k ) )  e.  CC )
29 1t1e1 10465 . . . . . . . 8  |-  ( 1  x.  1 )  =  1
3029eqcomi 2445 . . . . . . 7  |-  1  =  ( 1  x.  1 )
3130a1i 11 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
1  =  ( 1  x.  1 ) )
322, 3, 7dchrmhm 22539 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
336adantr 462 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X  e.  D )
3432, 33sseldi 3351 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
35 simprl 750 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  x  e.  U )
3619, 35sseldi 3351 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  x  e.  B )
37 simprr 751 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
y  e.  U )
3819, 37sseldi 3351 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
y  e.  B )
39 eqid 2441 . . . . . . . . 9  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
4039, 4mgpbas 16587 . . . . . . . 8  |-  B  =  ( Base `  (mulGrp `  Z ) )
41 eqid 2441 . . . . . . . . 9  |-  ( .r
`  Z )  =  ( .r `  Z
)
4239, 41mgpplusg 16585 . . . . . . . 8  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
43 eqid 2441 . . . . . . . . 9  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
44 cnfldmul 17783 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
4543, 44mgpplusg 16585 . . . . . . . 8  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
4640, 42, 45mhmlin 15467 . . . . . . 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 1213 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  (
x ( .r `  Z ) y ) )  =  ( ( X `  x )  x.  ( X `  y ) ) )
4831, 47oveq12d 6108 . . . . 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 9336 . . . . . . 7  |-  1  e.  CC
5049a1i 11 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
1  e.  CC )
5118adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  ->  X : B --> CC )
5251, 36ffvelrnd 5841 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  x
)  e.  CC )
5351, 38ffvelrnd 5841 . . . . . 6  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( X `  y
)  e.  CC )
542, 3, 7, 4, 5, 33, 36dchrn0 22548 . . . . . . 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 22548 . . . . . . 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 10144 . . . . 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 2476 . . . 4  |-  ( (
ph  /\  ( x  e.  U  /\  y  e.  U ) )  -> 
( 1  /  ( X `  ( x
( .r `  Z
) y ) ) )  =  ( ( 1  /  ( X `
 x ) )  x.  ( 1  / 
( X `  y
) ) ) )
6032, 6sseldi 3351 . . . . . . 7  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
61 eqid 2441 . . . . . . . . 9  |-  ( 1r
`  Z )  =  ( 1r `  Z
)
6239, 61rngidval 16595 . . . . . . . 8  |-  ( 1r
`  Z )  =  ( 0g `  (mulGrp `  Z ) )
63 cnfld1 17800 . . . . . . . . 9  |-  1  =  ( 1r ` fld )
6443, 63rngidval 16595 . . . . . . . 8  |-  1  =  ( 0g `  (mulGrp ` fld ) )
6562, 64mhm0 15468 . . . . . . 7  |-  ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  ( X `  ( 1r `  Z ) )  =  1 )
6660, 65syl 16 . . . . . 6  |-  ( ph  ->  ( X `  ( 1r `  Z ) )  =  1 )
6766oveq2d 6106 . . . . 5  |-  ( ph  ->  ( 1  /  ( X `  ( 1r `  Z ) ) )  =  ( 1  / 
1 ) )
68 1div1e1 10020 . . . . 5  |-  ( 1  /  1 )  =  1
6967, 68syl6eq 2489 . . . 4  |-  ( ph  ->  ( 1  /  ( X `  ( 1r `  Z ) ) )  =  1 )
702, 3, 4, 5, 9, 7, 11, 13, 15, 17, 28, 59, 69dchrelbasd 22537 . . 3  |-  ( ph  ->  ( k  e.  B  |->  if ( k  e.  U ,  ( 1  /  ( X `  k ) ) ,  0 ) )  e.  D )
711, 70syl5eqel 2525 . 2  |-  ( ph  ->  K  e.  D )
72 dchrmulid2.t . . . 4  |-  .x.  =  ( +g  `  G )
732, 3, 7, 72, 71, 6dchrmul 22546 . . 3  |-  ( ph  ->  ( K  .x.  X
)  =  ( K  oF  x.  X
) )
74 fvex 5698 . . . . . . 7  |-  ( Base `  Z )  e.  _V
754, 74eqeltri 2511 . . . . . 6  |-  B  e. 
_V
7675a1i 11 . . . . 5  |-  ( ph  ->  B  e.  _V )
77 ovex 6115 . . . . . . 7  |-  ( 1  /  ( X `  k ) )  e. 
_V
78 c0ex 9376 . . . . . . 7  |-  0  e.  _V
7977, 78ifex 3855 . . . . . 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 5840 . . . . 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 5741 . . . . 5  |-  ( ph  ->  X  =  ( k  e.  B  |->  ( X `
 k ) ) )
8476, 80, 81, 82, 83offval2 6335 . . . 4  |-  ( ph  ->  ( K  oF  x.  X )  =  ( k  e.  B  |->  ( if ( k  e.  U ,  ( 1  /  ( X `
 k ) ) ,  0 )  x.  ( X `  k
) ) ) )
85 oveq1 6097 . . . . . . . 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 6097 . . . . . . . 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 3799 . . . . . . 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 462 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  ( X `  k )  e.  CC )
896adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  B )  ->  X  e.  D )
90 simpr 458 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  B )  ->  k  e.  B )
912, 3, 7, 4, 5, 89, 90dchrn0 22548 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  B )  ->  (
( X `  k
)  =/=  0  <->  k  e.  U ) )
9291biimpar 482 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  ( X `  k )  =/=  0 )
9388, 92recid2d 10099 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  B )  /\  k  e.  U )  ->  (
( 1  /  ( X `  k )
)  x.  ( X `
 k ) )  =  1 )
9493ifeq1da 3816 . . . . . . . 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 9563 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  B )  ->  (
0  x.  ( X `
 k ) )  =  0 )
9695ifeq2d 3805 . . . . . . . 8  |-  ( (
ph  /\  k  e.  B )  ->  if ( k  e.  U ,  1 ,  ( 0  x.  ( X `
 k ) ) )  =  if ( k  e.  U , 
1 ,  0 ) )
9794, 96eqtrd 2473 . . . . . . 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 2485 . . . . . 6  |-  ( (
ph  /\  k  e.  B )  ->  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  x.  ( X `
 k ) )  =  if ( k  e.  U ,  1 ,  0 ) )
9998mpteq2dva 4375 . . . . 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 2492 . . . 4  |-  ( ph  ->  .1.  =  ( k  e.  B  |->  ( if ( k  e.  U ,  ( 1  / 
( X `  k
) ) ,  0 )  x.  ( X `
 k ) ) ) )
10284, 101eqtr4d 2476 . . 3  |-  ( ph  ->  ( K  oF  x.  X )  =  .1.  )
10373, 102eqtrd 2473 . 2  |-  ( ph  ->  ( K  .x.  X
)  =  .1.  )
10471, 103jca 529 1  |-  ( ph  ->  ( K  e.  D  /\  ( K  .x.  X
)  =  .1.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970   ifcif 3788    e. cmpt 4347   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   CCcc 9276   0cc0 9278   1c1 9279    x. cmul 9283    / cdiv 9989   NNcn 10318   Basecbs 14170   +g cplusg 14234   .rcmulr 14235   MndHom cmhm 15458  mulGrpcmgp 16581   1rcur 16593  Unitcui 16721  ℂfldccnfld 17777  ℤ/nczn 17893  DChrcdchr 22530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-0g 14376  df-imas 14442  df-divs 14443  df-mnd 15411  df-mhm 15460  df-grp 15538  df-minusg 15539  df-sbg 15540  df-subg 15671  df-nsg 15672  df-eqg 15673  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-subrg 16843  df-lmod 16930  df-lss 16992  df-lsp 17031  df-sra 17231  df-rgmod 17232  df-lidl 17233  df-rsp 17234  df-2idl 17292  df-cnfld 17778  df-zring 17843  df-zn 17897  df-dchr 22531
This theorem is referenced by:  dchrabl  22552
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